FINITENESS RESULTS FOR MODULAR CURVES OF GENUS AT LEAST 2

MATTHEW H. BAKER, ENRIQUE GONZÁLEZ-JIMÉNEZ, JOSEP GONZÁLEZ, AND BJORN POONEN

Abstract. A curve X over ℚ is modular if it is dominated by X_{1}(N) for some N; if in addition the image of its jacobian in J_{1}(N) is contained in the new subvariety of J_{1}(N), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over ℚ of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J_{0}(N)^{new} with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genus-g curves over k dominated by a Fermat curve is finite and computable.

1. Introduction

Let X_{1}(N) be the usual modular curve over ℚ; see Section 3.1 for a definition. (All curves and varieties in this paper are smooth, projective, and geometrically integral, unless otherwise specified. When we write an affine equation for a curve, its smooth projective model is implied.) A curve X over ℚ will be called modular if there exists a nonconstant morphism π: X_{1}(N) → X over ℚ. If X is modular, then X(ℚ) is nonempty, since it contains the image of the cusp ∞ ∈ X_{1}(N)(ℚ). The converse, namely that if X(ℚ) is nonempty then X is modular, holds if the genus g of X satisfies g ≤ 1 [10]. In particular, there are infinitely many modular curves over ℚ of genus 1. On the other hand, we propose the following:

Conjecture 1.1. For each g ≥ 2, the set of modular curves over ℚ of genus g is finite.

Remark 1.2.

(i) When we speak of the finiteness of the set of curves over ℚ satisfying some condition, we mean the finiteness of the set of ℚ-isomorphism classes of such curves.

Date: November 12, 2004. 2000 Mathematics Subject Classification. Primary 11G18, 14G35; Secondary 11F11, 11G10, 14H45. Key words and phrases. Modular curves, modular forms, de Franchis-Severi Theorem, gonality, com-

putability, hyperelliptic curves, automorphisms of curves.

The first author was supported in part by an NSF Postdoctoral Fellowship. The second author was supported in part by DGICYT Grant BHA2000-0180 and a postdoctoral fellowship

from the European Research Training Network “Galois Theory and Explicit Methods in Arithmetic”. The third author was supported in part by DGICYT Grant BFM2003-06768-C02-02. The fourth author was supported by NSF grants DMS-9801104 and DMS-0301280, and a Packard

Fellowship.

(ii) For any fixed N, the de Franchis-Severi Theorem (see Theorem 5.7) implies the finiteness of the set of curves over ℚ dominated by X_{1}(N). Conjecture 1.1 can be thought of as a version that is uniform as one ascends the tower of modular curves X_{1}(N), provided that one fixes the genus of the dominated curve.

(iii) Conjecture 1.1 is true if one restricts the statement to quotients of X_{1}(N) by sub-

groups of its group of modular automorphisms. See Remark 3.16 for details. (iv) If X_{1}(N) dominates a curve X, then the jacobian JacX is a quotient of J_{1}(N) :=

JacX_{1}(N). The converse, namely that if X is a curve such that X(ℚ) is nonempty and JacX is a quotient of J_{1}(N) then X is dominated by X_{1}(N), holds if the genus g of X is ≤ 1, but can fail for g ≥ 2. See Section 8.2 for other “pathologies.”

We remark that throughout this paper, quotients or subvarieties of varieties, and morphisms between varieties, are implicitly assumed to be defined over the same field as the original varieties. If X is a curve over ℚ, and we wish to discuss automorphisms over ℂ, for example, we will write Aut(X_{ℂ}). When we refer to a “quotient of an abelian variety A” we mean a quotient in the category of abelian varieties as opposed to, for instance, the quotient of A by the action of a finite subgroup of Aut(A).

(v) In contrast with Conjecture 1.1, there exist infinitely many genus-two curves over ℚ whose jacobians are quotients of J_{1}(N) for some N. See Proposition 8.2(5).

(vi) In Section 9, we use a result of Aoki [4] to prove an analogue of Conjecture 1.1 in which X_{1}(N) is replaced by the Fermat curve x^{N} + y^{N} = z^{N} in ℙ^{2}. In fact, such an analogue can be proved over arbitrary fields of characteristic zero, not just ℚ.

We prove many results towards Conjecture 1.1 in this paper. Given a variety X over a field k, let Ω = Ω_{X/k}^{1} denote the sheaf of regular 1-forms. Call a modular curve X over ℚ new of level N if there exists a nonconstant morphism π: X_{1}(N) → X (defined over ℚ) such that π^{∗}H^{0}(X, Ω) is contained in the new subspace H^{0}(X_{1}(N), Ω)_{new}, or equivalently if the image of the homomorphism π^{∗}: JacX → J_{1}(N) induced by Picard functoriality is contained in the new subvariety J_{1}(N)_{new} of J_{1}(N). (See Section 3.1 for the definitions of H^{0}(X_{1}(N), Ω)_{new}, J_{1}(N)_{new}, J_{1}(N)^{new}, and so on.) For example, it is known that every elliptic curve E over ℚ is a new modular curve of level N, where N is the conductor of E. Here the conductor cond(A) of an abelian variety A over ℚ is a positive integer ∏_{p} p^{f_{p}}, where each exponent f_{p} is defined in terms of the action of an inertia subgroup of Gal (\overline{ℚ}/ℚ) at the prime p on a Tate module T_{ℓ}A: for the full definition, see [51, §1], for instance. If A is a quotient of J_{1}(N)^{new}, then cond(A) = N^{dim A}, by [12].

Theorem 1.3. For each g ≥ 2, the set of new modular curves over ℚ of genus g is finite and computable.

The first results of this type were proved in [27], which showed that the set of new modular curves of genus 2 over ℚ is finite, and that there are exactly 149 such curves whose jacobian is ℚ-simple. Equations for these 149 curves are given in Tables 1 and 2 of [27].

Remark 1.4.

(i) See Section 5 for the precise definition of “computable.” (ii) An analysis of the proof of Theorem 1.3 would show that as g → ∞, there are at

most exp((6 + o(1))g^{2}) new modular curves of genus g over ℚ. Probably a much smaller bound holds, however. It is conceivable even that there is an upper bound not depending on g.

(iii) If we consider all genera g ≥ 2 together, then there are infinitely many new modular curves. For example, X_{1}(p) is a new modular curve whenever p is prime, and its genus tends to infinity with p.

If we drop the assumption that our modular curves are new, we can still prove results, but (so far) only if we impose restrictions on the level. Given m > 0, let Sparse_{m} denote the set of positive integers N such that if 1 = d_{1} < d_{2} < ⋯ < d_{t} = N are the positive divisors of N, then d_{i+1}/d_{i} > m for i = 1, …, t−1. Define a function B(g) on integers g ≥ 2 by B(2) = 13, B(3) = 17, B(4) = 21, and B(g) = 6g − 5 for g ≥ 5. (For the origin of this function, see the proofs of Propositions 2.1 and 2.8.) A positive integer N is called m-smooth if all primes p dividing N satisfy p ≤ m. Let Smooth_{m} denote the set of m-smooth integers.

Theorem 1.5. Fix g ≥ 2, and let S be a subset of {1, 2, … }. The set of modular curves over ℚ of genus g and of level contained in S is finite if any of the following hold:

(i) S = Sparse_{B(g)}.

(ii) S = Smooth_{m} for some m > 0.

(iii) S is the set of prime powers.

Remark 1.6. Since Sparse_{B(g)} ∪ Smooth_{B(g)} contains all prime powers, parts (i) and (ii) of Theorem 1.5 imply (iii).

Remark 1.7. In contrast with Theorem 1.3, we do not know, even in theory, how to compute the finite sets of curves in Theorem 1.5. The reason for this will be explained in Remark 5.12.

If X is a curve over a field k, and L is a field extension of k, let X_{L} denote X ×_{k} L. The gonality G of a curve X over ℚ is the smallest possible degree of a nonconstant morphism X_{ℂ} → ℙ_{ℂ}^{1}. (There is also the notion of ℚ-gonality, where one only allows morphisms over ℚ. By defining gonality using morphisms over ℂ instead of ℚ, we make the next theorem stronger.) In Section 4.3, we combine Theorem 1.3 with a known lower bound on the gonality of X_{1}(N) to prove the following:

Theorem 1.8. For each G ≥ 2, the set of new modular curves over ℚ of genus at least 2 and gonality at most G is finite and computable.

(We could similarly prove an analogue of Theorem 1.5 for curves of bounded gonality instead of fixed genus.)

Recall that a curve X of genus g over a field k is called hyperelliptic if g ≥ 2 and the canonical map X → ℙ^{g−1} is not a closed immersion: equivalently, g ≥ 2 and there exists a degree-2 morphism X → Y where Y has genus zero. If moreover X(k) ≠ ∅ then Y ≃ ℙ_{k}^{1}, and if also k is not of characteristic 2, then X is birational to a curve of the form y^{2} = f(x) where f is a separable polynomial in k[x] of degree 2g + 1 or 2g + 2. Recall that X(k) ≠ ∅ is automatic if X is modular, because of the cusp ∞.

Taking G = 2 in Theorem 1.8, we find that the set of new modular hyperelliptic curves over ℚ is finite and computable. We can say more:

Theorem 1.9. Let X be a new modular hyperelliptic curve over ℚ of genus g ≥ 3 and level N. Then

(i) g ≤ 16.

(ii) If JacX is a quotient of J_{0}(N), then g ≤ 10. If moreover 3 | N, then X is the genus-3 curve X_{0}(39).

(iii) If JacX is not a quotient of J_{0}(N), then either g is even or g ≤ 9.

Further information is given in Sections 6.3 and 6.5, and in the appendix. See Section 3.1 for the definitions of X_{0}(N) and J_{0}(N).

As we have already remarked, if we consider all genera g ≥ 2 together, there are infinitely many new modular curves. To obtain finiteness results, so far we have needed to restrict either the genus or the gonality. The following theorem, proved in Section 7, gives a different type of restriction that again implies finiteness.

Theorem 1.10. For each prime p, the set of new modular curves over ℚ of genus at least 2 whose jacobian is a quotient of J_{0}(N)^{new} for some N divisible by p is finite and computable.

Question 1.11. Does Theorem 1.10 remain true if J_{0}(N)^{new} is replaced by J_{1}(N)^{new}?

Call a curve X over a field k of characteristic zero k-modular if there exists a nonconstant morphism X_{1}(N)_{k} → X (over k).

Question 1.12. Is it true that for every field k of characteristic zero, and every g ≥ 2, the set of k-modular curves over k of genus g up to k-isomorphism is finite?

Remark 1.13. If X is a k-modular curve over k, and we define k_{0} = k ∩ \overline{ℚ} ⊆ \overline{k}, then X = X_{0} ×_{k_{0}} k for some k_{0}-modular curve X_{0}. This follows from the de Franchis-Severi Theorem.

Remark 1.14. If k and k′ are fields of characteristic zero with [k′ : k] finite, then a positive answer to Question 1.12 for k′ implies a positive answer for k, since Galois cohomology and the finiteness of automorphism group of curves of genus at least 2 show that for each X′ over k′, there are at most finitely many curves X over k with X ×_{k} k′ ≃ X′. But it is not clear, for instance, that a positive answer for \overline{ℚ} implies or is implied by a positive answer for ℚ.

2. Recovering curve information from differentials

2.1. Recovering a curve from partial expansions of its differentials. The goal of this section is to prove the following result, which will be used frequently in the rest of this paper.

Proposition 2.1. Fix an integer g ≥ 2. There exists an integer B > 0 depending on g such that if k is a field of characteristic zero, and w_{1}, …, w_{g} are elements of k[[q]]/(q^{B}), then up to k-isomorphism, there exists at most one curve X over k such that there exist P ∈ X(k) and an analytic uniformizing parameter q in the completed local ring Ô_{X,P} such that w_{1} dq, …, w_{g} dq are the expansions modulo q^{B} of some basis of H^{0}(X, Ω).

Lemma 2.2. Let k be a field, and let X/k be a curve of genus g. Let P ∈ X(k) be a k-rational point, let F ∈ k[t_{1}, …, t_{g}] be a homogeneous polynomial of degree d, and let q be an analytic uniformizing parameter in Ô_{X,P}. Suppose we are given elements ω_{1}, …, ω_{g} ∈ H^{0}(X, Ω), and for each i = 1, …, g, write ω_{i} = w_{i}dq with w_{i} ∈ k[[q]]. Then if F (w_{1}, …, w_{g}) ≡ 0 (mod q^{B}) with B > d(2g − 2), then F (ω_{1}, …, ω_{g}) = 0 in H^{0}(X, Ω^{⊗d}).

Proof. A nonzero element of H^{0}(X, Ω^{⊗d}) has d(2g − 2) zeros on X, so its expansion at any given point can vanish to order at most d(2g − 2). □

The following is a weak form of a theorem of Petri appearing as Theorem 2.3 on page 131 of [5], for example. A curve is trigonal if its gonality is 3.

Theorem 2.3 (Petri, 1923). Let X be a nonhyperelliptic curve of genus g ≥ 4 over a field k of characteristic zero. Then the image of the canonical map X → ℙ^{g−1} is the common zero locus of some set of homogeneous polynomials of degree 2 and 3. Moreover, if X is neither trigonal nor a smooth plane quintic, then degree 2 polynomials suffice.

Corollary 2.4. Let X be a curve of genus g ≥ 2 over a field k of characteristic zero. Then the image X′ of the canonical map X → ℙ^{g−1} is the common zero locus of the set of homogeneous polynomials of degree 4 that vanish on X′.

Proof. We may assume that k is algebraically closed. If X is hyperelliptic of genus g, say birational to y^{2} = f(x) where f has distinct roots, then we may choose { x^{i} dx/y : 0 ≤ i ≤ g − 1 } as basis of H^{0}(X, Ω), and then the image of the canonical map is the rational normal curve cut out by { t_{i}t_{j} − t_{i′}t_{j′} : i + j = i′ + j′ } where t_{0}, …, t_{g−1} are the homogeneous coordinates on ℙ^{g−1}. If X is nonhyperelliptic of genus 3, its canonical model is a plane quartic. In all other cases, we use Petri’s Theorem. (The zero locus of a homogeneous polynomial h of degree d < 4 equals the zero locus of the set of homogeneous polynomials of degree 4 that are multiples of h.) □

Lemma 2.5. Let X be a hyperelliptic curve of genus g over a field k of characteristic zero, and suppose P ∈ X(k). Let {ω_{1}, …, ω_{g}} be a basis of H^{0}(X, Ω) such that ord_{P} (ω_{1}) < ⋯ < ord_{P} (ω_{g}). Then x := ω_{g−1}/ω_{g} and y := dx/ω_{g} generate the function field k(X), and there is a unique polynomial F (x) of degree at most 2g + 2 such that y^{2} = F (x). Moreover, F is squarefree. If P is a Weierstrass point, then deg F = 2g + 1 and ord_{P} (ω_{i}) = 2i − 2 for all i; otherwise deg F = 2g + 2 and ord_{P} (ω_{i}) = i − 1 for all i. Finally, it is possible to replace each ω_{i} by a linear combination of ω_{i}, ω_{i+1}, …, ω_{g} to make ω_{i} = x^{g−i} dx/y for 1 ≤ i ≤ g.

Proof. This follows easily from Lemma 3.6.1, Corollary 3.6.3, and Theorem 3.6.4 of [25]. □

Proof of Proposition 2.1. Suppose that X, P, q, and the w_{i} are as in the statement of the proposition. Let ω_{i} be the corresponding elements of H^{0}(X, Ω). We will show that X is determined by the w_{i} when B = max{8g − 7, 6g + 1}.

Since B > 8g − 8, Lemma 2.2 implies that the w_{i} determine the set of homogeneous polynomial relations of degree 4 satisfied by the ω_{i}, so by Corollary 2.4 the w_{i} determine the image X′ of the canonical map. In particular, the w_{i} determine whether X is hyperelliptic, and they determine X if X is nonhyperelliptic.

Therefore it remains to consider the case where X is hyperelliptic. Applying Gaussian elimination to the w_{i}, we may assume 0 = ord_{q}(w_{1}) < ⋯ < ord_{q}(w_{g}) ≤ 2g − 2 and that the first nonzero coefficient of each w_{i} is 1. We use Lemma 2.5 repeatedly in what follows. The value of ord(w_{2}) determines whether P is a Weierstrass point.

Suppose that P is a Weierstrass point. Then w_{i} = q^{2i−2}(1 + ⋯ + O(q^{B−2i+2})), where each “⋯ ” here and in the rest of this proof represents some known linear combination of positive powers of q up to but not including the power in the big-O term. (“Known” means “determined by the original w_{i}.”)

Define x = w_{g−1}/w_{g} = q^{−2}(1 + ⋯ + O(q^{B−2g+2})). Define y = dx/(w_{g} dq) = −2q^{−(2g+1)}(1 + ⋯ + O(q^{B−2g+2})). Then y^{2} = 4q^{−(4g+2)}(1 + ⋯ + O(q^{B−2g+2})). Since B ≥ 6g + 1, we have −(4g + 2) + (B − 2g + 2) > 0, so there is a unique polynomial F (of degree 2g + 1) such that y^{2} = F (x).

A similar calculation shows that in the case where P is not a Weierstrass point, then B ≥ 3g + 2 is enough. □

Remark 2.6. Let us show that if the hypotheses of Proposition 2.1 are satisfied except that the w_{i} belong to \overline{k}[[q]]/(q^{B}) instead of k[[q]]/(q^{B}), and the ω_{i} are permitted to lie in H^{0}(X\overline{k}, Ω), then the conclusion still holds. Let E be a finite Galois extension of k containing all the coefficients of the w_{i}. The E-span of the w_{i} must be stable under Gal (E/k) if they come from a curve over ℚ, and in this case, we can replace the w_{i} by a k-rational basis of this span. Then Proposition 2.1 applies.

Remark 2.7. We can generalize Proposition 2.1 to the case where q is not a uniformizing parameter on X:

Fix an integer g ≥ 2, and let k be a field of characteristic zero. Let B > 0 be the integer appearing in the statement of Proposition 2.1, and let e be a positive integer. Then if w_{1}, …, w_{g} are elements of k[[q]]/(q^{eB}), then up to k-isomorphism, there exists at most one curve X over k such that there exist P ∈ X(k), an analytic uniformizing parameter q′ ∈ Ô_{X,P} and a relation q′ = c_{e}q^{e} + c_{e+1}q^{e+1} + … with c_{e} ≠ 0, such that w_{1} dq, …, w_{g} dq are the expansions modulo q^{eB} of some basis of H^{0}(X, Ω).

The proof of this statement is similar to the proof of Proposition 2.1, and is left to the reader.

The rest of this section is concerned with quantitative improvements to Proposition 2.1, and is not needed for the general finiteness and computability results of Sections 4 and 5.

Proposition 2.8. Proposition 2.1 holds with B = B(g), where B(2) = 13, B(3) = 17, B(4) = 21, and B(g) = 6g − 5 for g ≥ 5. Moreover, if we are given that the curve X to be recovered is hyperelliptic, then we can use B(g) = 4g + 5 or B(g) = 2g + 4, according as P is a Weierstrass point or not.

Proof. For nonhyperelliptic curves of genus g ≥ 4, we use Theorem 2.3 instead of Corollary 2.4 to see that B > 6g − 6 can be used in place of B > 8g − 8.

Now suppose that X is hyperelliptic. As before, assume ord_{q}(w_{1}) < ⋯ < ord_{q}(w_{g}) and that the first nonzero coefficient of each w_{i} is 1. The value of ord(w_{2}) determines whether P is a Weierstrass point.

Suppose that P is a Weierstrass point. Then w_{i} = q^{2i−2}(1 + ⋯ + O(q^{B−2i+2})). (As in the proof of Proposition 2.1, ⋯ means a linear combination of positive powers of q, whose coefficients are determined by the w_{i}.) Define x̃ = w_{g−1}/w_{g} = q^{−2}(1 + ⋯ + O(q^{B−2g+2})). For 1 ≤ i ≤ g − 2, the expression x̃^{g−i}w_{g} = q^{2i−2}(1 + ⋯ + O(q^{B−2g+2})) is the initial expansion of w_{i} +∑_{j=i+1}^{g} c_{ij}w_{j} for some c_{ij} ∈ k, and all the c_{ij} are determined if 2+(B −2g +2) > 2g −2, that is, if B ≥ 4g − 5. Let w′_{i} = w_{i} + ∑_{j=i+1}^{g} c_{ij}w_{j} = q^{2i−2}(1 + ⋯ + O(q^{B−2i+2})). Define x = w′_{1}/w′_{2} = q^{−2}(1+⋯+O(q^{B−2})). Define y = −2q^{−(2g+1)}(1+⋯+O(q^{B−2})) as the solution to w′_{1} dq = x^{g−1} dx/y. Then y^{2} = 4q^{−(4g+2)}(1 + ⋯ + O(q^{B−2})), and if −(4g + 2) + B − 2 > 0, we can recover the polynomial F of degree 2g + 1 such that y^{2} = F (x). Hence B ≥ 4g + 5 suffices. A similar proof shows that B ≥ 2g+4 suffices in the case that P is not a Weierstrass point.

Hence max{6g − 5, 4g + 5, 2g + 4} suffices for all types of curves, except that the 6g − 5 should be 8g − 7 when g = 3. This is the function B(g). □

Remark 2.9. We show here that for each g ≥ 2, the bound B = 4g + 5 for the precision needed to recover a hyperelliptic curve is sharp. Let F (x) ∈ ℂ[x] be a monic polynomial of degree 2g + 1 such that X: y^{2} = F (x) and X′: y^{2} = F (x) + 1 are curves of genus g that are not birationally equivalent. Let q be the uniformizing parameter at the point at infinity on X such that x = q^{−2} and y = q^{−(2g+1)} + O(q^{−2g}). Define q′ similarly for X′. A calculation shows that the q-expansions of the differentials x^{i} dx/y for 0 ≤ i ≤ g − 1 are even power series in q times dq, and modulo q^{4g+4} dq they agree with the corresponding q′-expansions for X′ except for the coefficient of q^{4g+2} dq in x^{g−1} dx/y. By a change of analytic parameter q = Q + αQ^{4g+3} for some α ∈ ℂ, on X only, we can make even that coefficient agree.

A similar proof shows that in the case that P is not a Weierstrass point, the bound 2g + 4 cannot be improved.

Remark 2.10. When studying new modular curves of genus g, we can also use the multiplicativity of Fourier coefficients of modular forms (see (3.7)) to determine some coefficients from earlier ones. Hence we can sometimes get away with less than B(g) coefficients of each modular form.

2.2. Descending morphisms. The next result will be used a number of times throughout this paper. In particular, it will be an important ingredient in the proof of Theorem 1.9.

Proposition 2.11. Let X, Y, Z be curves over a field k of characteristic zero, and assume that the genus of Y is > 1. Then:

(i) Given nonconstant morphisms π: X → Z and ϕ: X → Y such that ϕ^{∗}H^{0}(Y, Ω) ⊆ π^{∗}H^{0}(Z, Ω), there exists a nonconstant morphism u′: Z → Y making the diagram

\xymatrix{X \ar[d]_{π} \ar[dr]^{ϕ} \\ Z \ar@{.>}[r]_{u′} & Y}

commute. (ii) If π: X → Y is a nonconstant morphism and u is an automorphism of X such that

u^{∗} maps π^{∗}H^{0}(Y, Ω) into itself, then there exists a unique automorphism u′ of Y making the diagram

\xymatrix{X \ar[d]_{π} \ar[r]^{u} & X \ar[d]^{π} \\ Y \ar@{.>}[r]^{u′} & Y}

commute.

Proof. The conclusion of (i) is equivalent to the inclusion ϕ^{∗}k(Y) ⊆ π^{∗}k(Z). It suffices to prove that every function in ϕ^{∗}k(Y) is expressible as a ratio of pullbacks of meromorphic differentials on Z. If Y is nonhyperelliptic, then the field k(Y) is generated by ratios of pairs of differentials in H^{0}(Y, Ω), so the inclusion follows from the hypothesis ϕ^{∗}H^{0}(Y, Ω) ⊆ π^{∗}H^{0}(Z, Ω). When Y is hyperelliptic, we must modify this argument slightly. We have k(Y) = k(x, y), where y^{2} = F (x) for some polynomial F (U) in k[U] without double roots. The field generated by ratios of differentials in H^{0}(Y, Ω) is k(x), so ϕ^{∗}k(x) ⊆ π^{∗}k(Z). To show that ϕ^{∗}y ∈ π^{∗}k(Z) too, write y = x dx/(x dx/y) and observe that x dx/y ∈ H^{0}(Y, Ω).

Now we prove (ii). The hypothesis on u^{∗} lets us apply (i) with ϕ = π ∘ u to construct u′: Y → Y. Since Y has genus > 1 and k has characteristic zero, the Hurwitz formula implies that u′ is an automorphism. Considering function fields proves uniqueness. □

Remark 2.12. Both parts of Proposition 2.11 can fail if the genus of Y is 1. On the other hand, (ii) remains true under the additional assumption that X → Y is optimal in the sense that it does not factor nontrivally through any other genus 1 curve.

Remark 2.13. Proposition 2.11 remains true if k has finite characteristic, provided that one assumes that the morphisms are separable.

3. Some facts about modular curves

3.1. Basic facts about X_{1}(N). We record facts about X_{1}(N) that we will need for the proof of finiteness in Theorem 1.3. See [67] and [20] for a detailed introduction.

Let H = \{z ∈ ℂ : Im z > 0\}. Let q = e^{2πiz}. The group SL_{2}(ℝ) acts on H by (\begin{matrix}a & b \\ c & d \end{matrix})z =

\frac{az + b}{cz + d}. The quotient of H by the subgroup

Γ_{1}(N) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) | (\begin{matrix}a \\ c \end{matrix}) ≡ (\begin{matrix}1 \\ 0 \end{matrix}) (mod N)\}

is isomorphic as Riemann surface to the space of complex points on a smooth affine curve Y_{1}(N) over ℚ. The ℚ-structure on Y_{1}(N) can be specified by the condition that its function field be the subset of the function field of Y_{1}(N) whose Fourier expansions in q have coefficients in ℚ. Alternatively, the ℚ-structure on Y_{1}(N) can be specified by interpreting Y_{1}(N) as a moduli space parametrizing elliptic curves E equipped with an immersion μ_{N} → E: one must use the group scheme μ_{N} of N^{th} roots of unity instead of ℤ/Nℤ in order to get the same ℚ-structure as in the previous sentence. (See [43, Chapter II], where the moduli space (over ℤ) is denoted M(Γ_{00}(N)), or see Variant 9.3.6 and Section 12.3 of [20], where the moduli space is denoted Y_{μ}(N).)

Next, there is a uniquely determined smooth projective curve X_{1}(N) over ℚ having Y_{1}(N) as a dense open subset. The difference X_{1}(N)(ℂ) \ Y_{1}(N)(ℂ) is the (finite) set of cusps on X_{1}(N). One such cusp is the point ∞, which can be defined as the limit in X_{1}(N)(ℂ) as t → +∞ of the image of it ∈ H in X_{1}(N)(ℂ). In fact, ∞ ∈ X_{1}(N)(ℚ) [20, Variant 9.3.6]. Let J_{1}(N) denote the jacobian of X_{1}(N). We have the Albanese morphism X_{1}(N) → J_{1}(N) sending P to the class of the divisor (P) − (∞).

Pulling back 1-forms under H → Y_{1}(N)(ℂ) ↪ X_{1}(N)(ℂ) → J_{1}(N)(ℂ) identifies H^{0}(J_{1}(N)_{ℂ}, Ω) and H^{0}(X_{1}(N)_{ℂ}, Ω) with S_{2}(N)\frac{dq}{q} for some g-dimensional ℂ-subspace S_{2}(N) of qℂ[[q]]. We will not define modular forms in general here, but S_{2}(N) is known as the space of weight 2 cusp forms on Γ_{1}(N).

If M|N and d|\frac{N}{M}, then z ↦ d·z on H induces a morphism X_{1}(N) → X_{1}(M), which in turn induces S_{2}(M) → S_{2}(N) and J_{1}(M) → J_{1}(N). The old subspace S_{2}(N)_{old} of S_{2}(N) is defined as the sum of the images of all such maps S_{2}(M) → S_{2}(N) for all d and M such that M|N, M ≠ N, and d|\frac{N}{M}. Similarly define the old subvariety J_{1}(N)_{old} of J_{1}(N). The space S_{2}(N) has a hermitian inner product called the Petersson inner product. Let S_{2}(N)_{new} denote the orthogonal complement to S_{2}(N)_{old} in S_{2}(N). The identifications above also give us new and old subspaces of H^{0}(X_{1}(N)_{ℂ}, Ω) and H^{0}(J_{1}(N)_{ℂ}, Ω). Let J_{1}(N)^{new} = J_{1}(N)/J_{1}(N)_{old}. There is also an abelian subvariety J_{1}(N)_{new} of J_{1}(N) that can be characterized in two ways: either as the abelian subvariety such that

ker(H^{0}(J_{1}(N)_{ℂ}, Ω) → H^{0}((J_{1}(N)_{new})_{ℂ}, Ω)) = H^{0}(J_{1}(N)_{ℂ}, Ω)_{old},

or as the abelian subvariety such that J_{1}(N) = J_{1}(N)_{old} +J_{1}(N)_{new} with J_{1}(N)_{old} ∩J_{1}(N)_{new} finite. (The latter description uniquely characterizes J_{1}(N)_{new} because of a theorem that no ℚ-simple quotient of J_{1}(N)_{old} is isogenous to a ℚ-simple quotient of J_{1}(N)^{new}; this theorem can be proved by comparing conductors, using [12].) The abelian varieties J_{1}(N)^{new} and J_{1}(N)_{new} are ℚ-isogenous. We define X_{0}(N), J_{0}(N), and J_{0}(N)^{new} similarly, starting with

Γ_{0}(N) := \{ (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) | c ≡ 0 (mod N) \}

instead of Γ_{1}(N).

For n ≥ 1 with gcd(n, N) = 1, there exist well-known correspondences T_{n} on X_{1}(N), and they induce endomorphisms of S_{2}(N) and of J_{1}(N) known as Hecke operators, also denoted T_{n}. By [49, p. 294], there is a unique basis New_{N} of S_{2}(N)_{new} consisting of f = a_{1}q+a_{2}q^{2}+a_{3}q^{3}+… such that a_{1} = 1 and T_{n}f = a_{n}f whenever gcd(n, N) = 1. The elements of New_{N} are called the newforms of level N. (For us, newforms are always normalized: this means that a_{1} = 1.) Each a_{n} is an algebraic integer, bounded by σ_{0}(n)\sqrt{n} in each archimedean absolute value, where σ_{0}(n) is the number of positive integer divisors of n: [19, Théorème 8.2] proves this when n is a prime not dividing N, and then facts (3.7) through (3.11) below imply it for all n ≥ 1. For each field k, let G_{k} = Gal (\overline{k}/k). The Galois group G_{ℚ} acts on New_{N}. For any quotient A of J_{1}(N), let S_{2}(A) denote the image of H^{0}(A_{ℂ}, Ω) → H^{0}(J_{1}(N)_{ℂ}, Ω) ≃ S_{2}(N) (the last isomorphism drops the dq/q); similarly for any nonconstant morphism π: X_{1}(N) → X of curves, define S_{2}(X) := π^{∗}H^{0}(X_{ℂ}, Ω)\frac{q}{dq} ⊆ S_{2}(N). G. Shimura [69, Theorem 1] associated to f ∈ New_{N} an abelian variety quotient A_{f} of J_{1}(N) such that S_{2}(A_{f}) is spanned by the Galois conjugates of f; this association induces a bijection

(3.1)  G_{ℚ}\^{New^{N}} →^{∼} \frac{\{ℚ-simple quotients of J_{1}(N)^{new}\}}{ℚ-isogeny}
f ↦ A_{f}.

(Earlier, in Theorem 7.14 of [67], Shimura had attached to f an abelian subvariety of J_{1}(N).) Shimura proved that J_{1}(N) is isogenous to a product of these A_{f}, and K. Ribet [65] proved that A_{f} is ℚ-simple. This explains the surjectivity of (3.1). The injectivity is well-known to experts, but we could not find a suitable reference, so we will prove it, as part of Proposition 3.2.

The subfield E_{f} = ℚ(a_{2}, a_{3}, …) of ℂ is a number field, and dim A_{f} = [E_{f} : ℚ]. Moreover, End(A_{f})⊗ℚ can be canonically identified with E_{f}, and under this identification the element λ ∈ End A_{f} acts on f as multiplication by λ (considered as element of E_{f}), and on each Galois conjugate ^{σ}f by multiplication by ^{σ}λ. (Shimura [69, Theorem 1] constructed an injection End(A_{f}) ⊗ ℚ ↪ E_{f}, and Ribet [65, Corollary 4.2] proved that it was an isomorphism.)

If A and B are abelian varieties over ℚ, let A ∼^{ℚ} B denote the statement that A and B are isogenous over ℚ.

Proposition 3.2. Suppose f ∈ New_{N} and f′ ∈ New_{N′}. Then A_{f} ∼^{ℚ} A_{f′} if and only if N = N′ and f =^{τ}f′ for some τ ∈ G_{ℚ}.

Proof (K. Ribet). The “if” part is immediate from Shimura’s construction. Therefore it suffices to show that one can recover f, up to Galois conjugacy, from the isogeny class of A_{f}. Let ℓ be a prime. Let V be the ℚ_{ℓ}-Tate module V_{ℓ}(A_{f}) attached to A_{f}. Let \overline{V} = V ⊗_{ℚ_{ℓ}} \overline{ℚ}_{ℓ}.

The proof of Proposition 4.1 of [65] shows that \overline{V} = ⨁_{σ} V_{σ}, where V_{σ} is an irreducible \overline{ℚ}_{ℓ}[G_{ℚ}]-module of dimension 2 over \overline{ℚ}_{ℓ}, indexed by embeddings σ: E_{f} ↪ \overline{ℚ}_{ℓ}. Moreover, for p ∤ ℓN, the trace of the p-Frobenius automorphism acting on V_{σ} equals σ(a_{p}), where a_{p} ∈ E_{f} is the coefficient of q^{p} in the Fourier expansion of f.

If f′ ∈ New′_{N} is another weight 2 newform, and A_{f} ∼^{ℚ} A_{f′}, then (using ′ in the obvious way to denote objects associated with f′), we have isomorphisms of G_{ℚ}-modules V ≃ V′ and \overline{V} ≃ \overline{V}′. Fix σ: E_{f} ↪ \overline{ℚ}_{ℓ}. Then V_{σ} is isomorphic to some irreducible component V′_{σ′} of \overline{V}′, where σ′ is some embedding E_{f′} ↪ \overline{ℚ}_{ℓ}. Taking traces of Frobenius elements, we find that σ(a_{p}) = σ′(a′_{p}) for almost all p. Then Theorem 5 of [49] implies that f and f′ have the same level and are Galois conjugate. □

We have parallel decompositions

S_{2}(N)_{new} = ⨁_{f∈G_{ℚ}\New_{N}}  ⨁_{τ: E_{f} ↪ℂ}ℂ ^{τ}f

J_{1}(N)^{new} ∼^{ℚ} ⨁_{f∈G_{ℚ}\New_{N}} A_{f}

and parallel decompositions

(3.3)  S_{2}(N) = ⨁_{M|N} ⨁_{f∈G_{ℚ}\New_{M}}  ⨁_{d| \frac{N}{M}}  ⨁_{τ: E_{f} ↪ℂ}ℂ ^{τ}f(q^{d})

(3.4)  J_{1}(N) ∼^{ℚ} ⨁_{M|N} ⨁_{f∈G_{ℚ}\New_{M}} A_{f}^{n_{f}}

where n_{f} := σ_{0}(N/M). (When we write f ∈ G_{ℚ}\New_{N}, we mean that f runs through a set of representatives for the orbits of G_{ℚ} acting on New_{M}. See [41, Example 5.5] and [42, Remark 17] for more details concerning the relationship between the decompositions (3.3) and (3.4).) Because of Proposition 3.2, the given decompositions of J_{1}(N)^{new} and J_{1}(N) are exactly the decompositions up to isogeny into nonisogenous ℚ-simple abelian varieties occurring with multiplicity.

Lemma 3.5. Let A be an abelian variety quotient of J_{1}(N). Then S_{2}(A) has a G_{ℚ}-stable basis consisting of cusp forms each of the form

h(q) = ∑_{d| \frac{N}{M}} c_{d}f(q^{d})

for some M | N and f ∈ New_{M} and elements c_{d} ∈ E_{f} (all depending on h).

Proof. By multiplying the quotient map J_{1}(N) → A by a positive integer, we may assume that it factors through the isogeny

J_{1}(N) → ⨁_{M|N} ⨁_{f∈G_{ℚ}\New_{M}} A_{f}^{n_{f}}

of (3.4). We may also assume that A is ℚ-simple, and even that A is isomorphic to A_{f} for some f, so that the quotient map J_{1}(N) → A is the composition of J_{1}(N) → A_{f}^{n_{f}} with a homomorphism A_{f}^{n_{f}} → A. The latter is given by an n_{f}-tuple c = (c_{d}) of elements of End(A_{f}), indexed by the divisors d of N/M. Under

X_{1}(N) ↪ J_{1}(N) → A_{f}^{n_{f}} →^{c} A,

the 1-form on A_{ℂ} ≃ (A_{f})_{ℂ} corresponding to f pulls back to ∑_{d| \frac{N}{M}} c_{d}f(q^{d}) dq/q. Finally, H^{0}(A_{ℂ}, Ω) has a basis consisting of this 1-form and its conjugates, and the pullbacks of these conjugates are sums of the same form. □

Corollary 3.6. Let π: X_{1}(N) → X be a nonconstant morphism of curves over ℚ. Then S_{2}(X) has a G_{ℚ}-stable basis T consisting of cusp forms each of the form

h(q) = ∑_{d| \frac{N}{M}} c_{d}f(q^{d})

for some M | N and f ∈ New_{M}, where c_{d} ∈ E_{f} depends on f and d.

Proof. Apply Lemma 3.5 to the Albanese homomorphism J_{1}(N) → JacX. □

3.2. Automorphisms of X_{1}(N).

3.2.1. Diamonds. The action on H of a matrix (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) induces an automorphism

of X_{1}(N) over ℚ depending only on (d mod N). This automorphism is called the diamond operator ⟨d⟩. It induces an automorphism of S_{2}(N). Let ε be a Dirichlet character modulo N, that is, a homomorphism (ℤ/Nℤ)^{∗} → ℂ^{∗}. Let S_{2}(N, ε) be the ℂ-vector space { h ∈ S_{2}(N) : h|⟨d⟩ = ε(d)h }. A form h ∈ S_{2}(N, ε) is called a form of Nebentypus ε. Every newform f ∈ New_{N} is a form of some Nebentypus, and is therefore an eigenvector for all the diamond operators. Character theory gives a decomposition

S_{2}(N) = ⨁_{ε} S_{2}(N, ε) ,

where ε runs over all Dirichlet characters modulo N. Define S_{2}(N, ε)_{new} = S_{2}(N, ε) ∩ S_{2}(N)_{new}. When we write ε = 1, we mean that ε is the trivial Dirichlet character modulo N, that is,

ε(n) = \begin{cases}1 & if (n, N) = 1 \\ 0 & otherwise. \end{cases}

A form of Nebentypus ε = 1 is a form on Γ_{0}(N).

We recall some properties of a newform f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{2}(N, ε). Let cond ε denote the smallest integer M | N such that ε is a composition (ℤ/Nℤ)^{∗} → (ℤ/Mℤ)^{∗} → ℂ^{∗}. Throughout this paragraph, p denotes a prime, and denotes complex conjugation. Let v_{p}(n) denote the p-adic valuation on ℤ. If v_{p}(cond ε) < v_{p}(N), then ε is a composition

(ℤ/Nℤ)^{∗} → (ℤ/(N/p)ℤ)^{∗} →^{ε′} ℂ^{∗} for some ε′. Then

(3.7)  ∑_{n=1}^{∞} a_{n}n^{−s} = ∏_{p}(1 − a_{p}p^{−s} + p ε(p) p^{−2s})^{−1},
(3.8)  p | N ⟺ ε(p) = 0
(3.9)  v_{p}(cond ε) < v_{p}(N) ≥ 2 ⟹ a_{p} = 0
(3.10)  v_{p}(cond ε) < v_{p}(N) = 1 ⟹ a_{p}^{2} = ε′(p)
(3.11)  1 ≤ v_{p}(cond ε) = v_{p}(N) ⟹ |a_{p}| = \sqrt{p}
(3.12)  v_{p}(N) = 1, ε = 1 ⟹ f|W_{p} = −a_{p}f
(3.13)  p ∤ N ⟹ \overline{a}_{p} = \overline{ε(p)} a_{p}.

(The equivalence (3.8) is trivial. Facts (3.7), (3.9), (3.10), (3.11) follow from parts (i), (iii), (iii), (ii) of [49, Theorem 3], respectively. Fact (3.12) follows from (3.10), [6, Theorem 2.1], and lines 6–7 of [6, p. 224]. Fact (3.13) follows from the first case of [6, equation (1.1)] and the lines preceding it.)

3.2.2. Involutions. For every integer M | N such that (M, N/M) = 1, there is an automorphism W_{M} of X_{1}(N)_{ℂ} inducing an isomorphism between S_{2}(N, ε)_{new} and S_{2}(N, \overline{ε}_{M}ε_{N/M})_{new}, where ε_{M} denotes the character mod M induced by ε. (For more on this action, see [6]). Given a newform f ∈ S_{2}(N, ε), there is a newform \overline{ε}_{M} ⊗ f in S_{2}(N, \overline{ε}_{M}ε_{N/M}) whose p^{th} Fourier coefficient b_{p} is given by

b_{p} = \begin{cases}\overline{ε}_{M}(p)a_{p} & if p ∤ M, \\ \overline{ε}_{N/M}(p)\overline{a_{p}} & if p | M. \end{cases}

Then

(3.14)  f|W_{M} = λ_{M}(f)(\overline{ε}_{M} ⊗ f)

for some λ_{M}(f) ∈ ℂ with |λ_{M}(f)| = 1. Moreover, we have

f|W_{M}^{2} = \overline{ε}_{N/M}(−M) f , f|(W_{M′} W_{M}) = \overline{ε}_{M′}(M)f|W_{M′ M}

whenever (M, M′) = 1. In the particular case M = N, the automorphism W_{N} is an involution defined over the cyclotomic field ℚ(ζ_{N}), called the Weil involution. For all τ ∈ Gal (ℚ(ζ_{N})/ℚ), the automorphisms ^{τ}W_{N} are involutions and are also called Weil involutions by some authors. The involution W_{N} maps each S_{2}(A_{f}) into itself and satisfies

(3.15)  ⟨d⟩W_{N} = W_{N}⟨d⟩^{−1} , ^{τ_{d}}W_{N} = W_{N} ⟨d⟩ for all d ∈ (ℤ/Nℤ)^{∗} ,

where τ_{d} is the element of Gal (ℚ(ζ_{N})/ℚ) mapping ζ_{N} to ζ_{N}^{d}.

Let ε be a Dirichlet character modulo N. Define the congruence subgroup

Γ(N, ε) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) | ε(d) = 1\} .

Let S_{2}(Γ(N, ε)) denote the space of weight 2 cusp forms on Γ(N, ε). Denote by X(N, ε) the modular curve over ℚ such that X(N, ε)(ℂ) contains Γ(N, ε)\H as a dense open subset. We can identify H^{0}(X(N, ε)_{ℂ}, Ω) with S_{2}(Γ(N, ε))\frac{dq}{q}. The Dirichlet characters modulo N whose kernel contains ker(ε) are exactly the powers of ε, so

S_{2}(Γ(N, ε)) = ⨁_{k=1}^{n} S_{2}(N, ε^{k}) ,

where n is the order of ε. The diamond operators and the Weil involution induce automorphisms of X(N, ε)_{ℂ}. If moreover ε = 1, the curve X(N, 1) is X_{0}(N) and the automorphisms W_{M} on X_{1}(N)_{ℂ} induce involutions on X_{0}(N) over ℚ that are usually called the Atkin-Lehner involutions.

Remark 3.16. Define the modular automorphism group of X_{1}(N) to be the subgroup of Aut(X_{1}(N)\overline{ℚ}) generated by the W_{M}’s and the diamond automorphisms. The quotient of X_{1}(N) by this subgroup equals the quotient of X_{0}(N) by its group of Atkin-Lehner involutions, and is denoted X^{∗}(N). The genus of X_{0}(N) is N^{1+o(1)} as N → ∞, so by Proposition 4.4, the gonality of X_{0}(N) is at least N^{1+o(1)}. On the other hand, the degree of X_{0}(N) → X^{∗}(N) is only 2^{#{prime factors of N}} = N^{o(1)}. Hence the gonality of X^{∗}(N) tends to infinity. Thus the genus of X^{∗}(N) tends to infinity. In particular, Conjecture 1.1 is true if one restricts the statement to modular curves X that are quotients of X_{1}(N) by some subgroup of the modular automorphism group.

3.2.3. Automorphisms of new modular curves. By Proposition 2.11, if X is a new modular curve of level N and genus g ≥ 2, then the diamond operators and the Weil involution W_{N} induce automorphisms of X\overline{ℚ}, which we continue to denote by ⟨d⟩ and W_{N} respectively. Throughout the paper, D will denote the abelian subgroup of Aut(X\overline{ℚ}) consisting of diamond automorphisms, and D_{N} := ⟨D, W_{N}⟩ will denote the subgroup of Aut(X\overline{ℚ}) generated by D and W_{N}. If moreover X is hyperelliptic, and w is its hyperelliptic involution, then the group generated by D and W_{N}.w will be denoted by D′_{N}.

Note that D is a subgroup of Aut(X), and the groups D_{N}, D′_{N} are G_{ℚ}-stable by (3.15). If JacX ∼^{ℚ} A_{f} for some f with nontrivial Nebentypus, then D_{N} is isomorphic to the dihedral group with 2n elements, D_{2·n}, where n is the order of the Nebentypus of f.

For every nonconstant morphism π: X_{1}(N) → X of curves over ℚ such that S_{2}(X) ⊆ ⊕_{i=1}^{n}S_{2}(N, ε^{i}) for some Nebentypus ε of order n, there exists a nonconstant morphism π(ε): X(N, ε) → X over ℚ. This is clear when the genus of X is ≤ 1, and follows from Proposition 2.11(i) if the genus of X is > 1. In particular, for a new modular curve X of genus > 1, there exists a surjective morphism X_{0}(N) → X if and only if D is the trivial group. More generally, we have the following result.

Lemma 3.17. Let X be a new modular curve of level N, and let G be a G_{ℚ}-stable subgroup of Aut(X\overline{ℚ}). Let X′ = X/G. If the genus of X′ is at least 2, then the group D′ of diamonds of X′ is isomorphic to D/(G∩D). In particular, if G = D or D_{N}, then there is a nonconstant morphism π′: X_{0}(N) → X′ defined over ℚ.

Proof. By Proposition 2.11, each diamond automorphism of X induces an automorphism of X′. Hence we obtain a surjective homomorphism ρ: D → D′. Let K = ker(ρ). Since G∩D ⊆ K, it suffices to show that K ⊆ G. Now H^{0}(X′, Ω) pulls back to the space H^{0}(X, Ω)^{G} of G-invariant regular differentials of X, so K acts trivially on the latter. Therefore the result follows from Lemma 3.18 below. □

Lemma 3.18. Let X be a curve over a field of characteristic zero, and let G be a finite subgroup of Aut(X). Assume that the genus g′ of X′ := X/G is at least 2. Let

\overline{G} := \{ ϕ ∈ Aut(X) : ϕ^{∗}ω = ω for all ω ∈ H^{0}(X, Ω)^{G} \}.

Then \overline{G} = G.

Proof. It is clear that G ⊆ \overline{G}. Now suppose ϕ ∈ \overline{G}, and let H := ⟨G, ϕ⟩. Also, set X′′ := X/H, and let g′′ be the genus of X′′. Then there is a natural dominant morphism π: X′ → X′′. Moreover,

g′′ = dim H^{0}(X, Ω)^{H} = dim H^{0}(X, Ω)^{⟨G,ϕ⟩} = dim H^{0}(X, Ω)^{G} = g′.

Since g′ ≥ 2 by hypothesis, the Hurwitz formula implies deg(π) = 1. Therefore ϕ ∈ G, so that \overline{G} ⊆ G as required. □

3.3. Supersingular points. We will use a lemma about curves with good reduction.

Lemma 3.19. Let R be a discrete valuation ring with fraction field K. Suppose f: X → Y is a finite morphism of smooth, projective, geometrically integral curves over K, and X extends to a smooth projective model X over R (in this case we say that X has good reduction). If Y has genus ≥ 1, then Y extends to a smooth projective model Y over R, and f extends to a finite morphism X → Y over R.

Proof. This result is Corollary 4.10 in [50]. See the discussion there also for references to earlier weaker versions. □

The next two lemmas are well-known, but we could not find explicit references, so we supply proofs.

Lemma 3.20. Let p be a prime. Let Γ ⊆ SL_{2}(ℤ) denote a congruence subgroup of level N not divisible by p. Let X_{Γ} be the corresponding integral smooth projective curve over \overline{ℚ}, and let ψ be the degree of the natural map X_{Γ} → X(1). Then X_{Γ} has good reduction at any place above p, and the number of \overline{𝔽}_{p}-points on the reduction mapping to supersingular points on X(1) is at least (p − 1)ψ/12.

Proof. By [39], the curve X(N) admits a smooth model over ℤ[1/N], and has a rational point (the cusp ∞). Since p ∤ N and X(N) dominates X_{Γ}, Lemma 3.19 implies that X_{Γ} has good reduction at p, at least if X_{Γ} has genus ≥ 1. If X_{Γ} has genus 0, then the rational point on X(N) gives a rational point on X_{Γ}, so X_{Γ} ≃ ℙ^{1}, so X_{Γ} has good reduction at p in any case. Replacing Γ by the group generated by Γ and − id does not change X_{Γ}, so without loss of generality, we may assume that − id ∈ Γ. Then ψ = (SL_{2}(ℤ) : Γ).

If E is an elliptic curve, then Γ naturally acts on the finite set of ordered symplectic bases of E[N]. The curve Y_{Γ} := X_{Γ} − {cusps} classifies isomorphism classes of pairs (E, L), where E is an elliptic curve and L is a Γ-orbit of symplectic bases of E[N].

Fix E. Since SL_{2}(ℤ) acts transitively on the symplectic bases of E[N], the number of Γ-orbits of symplectic bases is (SL_{2}(ℤ) : Γ) = ψ. Two such orbits L and L′ correspond to the same point of X_{Γ} if and only if L′ = αL for some α ∈ Aut(E). Then ψ is the sum of the sizes of the orbits of Aut(E) acting on the Γ-orbits, so

ψ = ∑_{(E,L)∈X_{Γ}} \frac{#Aut(E)}{#Aut(E, L)},

where the sum is over representatives (E, L) of the points on X_{Γ} above E, and Aut(E, L) is the subgroup of Aut(E) stabilizing L.

Dividing by #Aut(E) and summing over all supersingular E over \overline{𝔽}_{p}, we obtain

∑_{supersingular points (E, L) ∈ X_{Γ}(\overline{𝔽}_{p})} \frac{1}{#Aut(E, L)} = ψ ∑_{supersingular E/\overline{𝔽}_{p}} \frac{1}{#Aut(E)} = \frac{(p − 1)ψ}{24} ,

where the last step is the mass formula of Deuring and Eichler (see Chapter 13, §4 of [38] for a proof). But [−1] ∈ Aut(E, L), so #Aut(E, L) ≥ 2 at each (E, L) ∈ X_{Γ}(\overline{𝔽}_{p}). Therefore the number of supersingular points on X_{Γ} must be at least 2(p − 1)ψ/24 = (p − 1)ψ/12. □

Lemma 3.21. Let p be a prime. Given a supersingular elliptic curve E over \overline{𝔽}_{p}, there exists an elliptic curve E′ over 𝔽_{p^{2}} such that E ≃ E′_{\overline{𝔽}_{p}} and the p^{2}-power Frobenius endomorphism of E′ equals −p.

Proof. Honda-Tate theory supplies an elliptic curve E over 𝔽_{p} such that the characteristic polynomial of the p-power Frobenius endomorphism Frob_{p} satisfies Frob_{p}^{2} = −p. All supersingular elliptic curves over \overline{𝔽}_{p} are isogenous, so there exists an isogeny ϕ: E_{\overline{𝔽}_{p}} → E. The inseparable part of this isogeny is a power of Frob_{p}, so without loss of generality, we may assume that ϕ is separable. The kernel K of ϕ is preserved by −p = Frob_{p}^{2}, so K is defined over 𝔽_{p^{2}}. Take E′ = E_{𝔽_{p^{2}}}/K. □

The following is a generalization of inequalities used in [60].

Lemma 3.22. Let X be a new modular hyperelliptic curve of level N and genus g over ℚ. If p is a prime not dividing N, then (p − 1)(g − 1) < 2(p^{2} + 1).

Proof. We may assume g ≥ 2. Since p ∤ N, Lemma 3.19 implies that X_{1}(N) and X have good reduction at p, and the morphism π: X_{1}(N) → X induces a corresponding morphism of curves over 𝔽_{p}. By Proposition 2.11(ii), the diamond automorphism ⟨−p⟩ of X_{1}(N) induces an automorphism of X, which we also call ⟨−p⟩. These automorphisms induces automorphisms of the corresponding curves over 𝔽_{p}. For the rest of this proof, X_{1}(N), X, π, ⟨−p⟩ represent these objects over 𝔽_{p}. Also denote by ⟨−p⟩ the induced morphism ℙ^{1} → ℙ^{1} on the canonical image of the hyperelliptic curve X. On each curve over 𝔽_{p} or 𝔽_{p^{2}}, let F denote the p^{2}-power Frobenius morphism.

Let s be the number of x ∈ X_{1}(N)(\overline{𝔽}_{p}) satisfying F x = ⟨−p⟩x. We will bound s in two different ways. On the one hand, Lemma 3.21 implies that all supersingular points on X_{1}(N)(\overline{𝔽}_{p}) have this property. By Lemma 3.20, there are at least (p − 1)ψ/12 such points, where ψ is the degree of X_{1}(N) → X(1). Thus s ≥ (p − 1)ψ/12.

On the other hand, any solution x to F x = ⟨−p⟩x in X_{1}(N)(\overline{𝔽}_{p}) maps to a solution to the same equation on X or on ℙ^{1}. The number of solutions to F x = ⟨−p⟩x in ℙ^{1}(\overline{𝔽}_{p}) is at most p^{2} + 1, because they can be counted by intersecting the graphs of F and ⟨−p⟩, which are curves of type (1, p^{2}) and (1, 1), respectively, on ℙ^{1} × ℙ^{1}. Therefore s ≤ (2 deg π)(p^{2} + 1).

Combining the inequalities from the previous two paragraphs gives (p−1)ψ/12 ≤ (2 deg π) (p^{2} + 1). By Proposition 1.40 of [67], the genus g_{1} of X_{1}(N) satisfies g_{1} − 1 < ψ/12. By Hurwitz, deg π ≤ (g_{1}−1)/(g−1). Combining the last three inequalities yields (p−1)(g−1) < 2(p^{2} + 1). □

Remark 3.23. One can extend the above arguments to prove the following more general result:

Let X be a new modular curve of genus g and level N, with p ∤ N. Suppose X admits a degree d map (defined over ℚ) to a curve X′/ℚ of genus g′. Finally, suppose that the diamond automorphisms on X_{1}(N) are compatible with automorphisms of X′. (This is automatic if g′ ≥ 2, or if g ≥ 2, g′ = 0, and d = 2.) Then (g − 1)(p − 1) ≤ d(p^{2} + 1 + 2pg′).

We omit the proof, since we will not use this result.

Corollary 3.24. Let X be a new modular hyperelliptic curve of level N and genus g over ℚ. If g > 10, then 6|N. If g > 13, then 30|N.

Proof. Take p = 2, 3, and 5 in Lemma 3.22. □

In the case where X is a curve dominated by X_{0}(N), ⟨−p⟩ is the identity, so in the proof of Lemma 3.22 we may replace the bound (2 deg π)(p^{2} + 1) by (deg π)#X(𝔽_{p^{2}}) to obtain the following:

Lemma 3.25. Suppose that X is a curve of genus g ≥ 2 over ℚ, and π: X_{0}(N) → X is a dominant morphism over ℚ. Suppose that p is a prime not dividing N. Then X has good reduction at p, and (p − 1)(g − 1) < #X(𝔽_{p^{2}}). In particular, if G_{ℚ} denotes the ℚ-gonality of X, then g < G_{ℚ} \frac{p^{2}+1}{p−1} + 1.

3.4. Previous work on curves dominated by modular curves. Except for the determination of all new modular genus-2 curves with ℚ-simple jacobian in [27], all work we know of on modular curves of genus ≥ 2 has focused on X_{0}(N), X_{1}(N), or quotients of these by a subgroup of the group of Atkin-Lehner involutions or diamonds, respectively. Here we summarize some of this work.

The 19 values of N for which X_{0}(N) is hyperelliptic (of genus ≥ 2) were determined in [60], and all of their equations were given in [26]. The corresponding determination of the three values of N for which X_{1}(N) is hyperelliptic was carried out in [55], and equations for these (and a few other X_{1}(N)) were given in [63]. In [40] it is proved that a curve strictly between X_{0}(N) and X_{1}(N) (that is, a quotient of X_{1}(N) by a nontrivial proper subgroup of the diamond group) is never hyperelliptic. A series of works [46], [31], [30] led up to the determination of all 64 values of N for which the quotient of X_{0}(N) by its Atkin-Lehner group, X^{∗}(N), is hyperelliptic, and this was generalized in [23] to quotients of X_{0}(N) by an arbitrary subgroup of the Atkin-Lehner group. Similar results determining all trigonal curves of the form X_{0}(N), X_{0}(N)/W_{d} for a single Atkin-Lehner involution W_{d}, and X^{∗}(N), can be found in [32], [33], and [34], respectively. Some of these curves are not new, and hence do not appear in our tables.

The method of constructing equations for modular curves one at a time in terms of a basis of cusp forms has been used by many authors: in addition to the papers already mentioned, we have [58], [70], [24], for instance. In particular, [70] gives methods for both the hyperelliptic and nonhyperelliptic cases, using Petri’s Theorem in the latter, as we do. F. Klein [45] gave an explicit model for X(p) for every prime p in 1879. See [75] for a modern construction of X(N) for every N. An analogous result for X_{1}(p) can be found in [7].

4. Finiteness theorems

Our goal in this section is to prove the finiteness statements in Theorems 1.3, 1.5, and 1.8. Algorithmic and practical issues will be dealt with only in later sections, because we assume that some readers will be interested only in the finiteness. In particular, the computability of the sets in question will be proved in Section 5, and practical algorithms in the hyperelliptic case will be given in Section 6.

4.1. New modular curves of fixed genus.

Proof of finiteness in Theorem 1.3. Fix g ≥ 2. Let X be a new modular curve of level N and genus g, given by the nonconstant morphism π: X_{1}(N) → X. Since the ℚ-simple factors A_{f} of J_{1}(N)^{new} are pairwise nonisogenous, any abelian subquotient of J_{1}(N)^{new} is isogenous to a product of some subset of these A_{f}. Applying this to J = JacX and then comparing 1-forms shows that there exists a G_{ℚ}-stable subset T ⊆ New_{N} that is a basis for S_{2}(X).

¿From now on, let f = q+a_{2}q^{2}+a_{3}q^{3}+⋯ denote an element of T. The map π is unramified at ∞, since there exists ω ∈ H^{0}(X_{ℂ}, Ω) such that π^{∗}ω = f dq/q, which is nonvanishing at ∞. In other words, the analytic uniformizer q at ∞ on X_{1}(N) serves also as analytic uniformizer at π(∞) on X.

The field E_{f} generated by the coefficients of f satisfies [E_{f} : ℚ] = dim A_{f} ≤ dim J = g. Let B be the integer of Proposition 2.1. Each a_{n} for 1 ≤ n ≤ B is an algebraic integer of degree at most g, bounded in every archimedean absolute value by σ_{0}(n)\sqrt{n}, so there are only finitely many possibilities for a_{n}. Thus there are only finitely many possibilities for { f mod q^{B} | f ∈ T }, given g. By Proposition 2.1 and Remark 2.6, each such possibility arises for at most one curve over ℚ. □

Remark 4.1. For this proof, any bound on the absolute values of |a_{n}| in terms of n would have sufficed. But when we do computations, it will be useful to have the strong bound σ_{0}(n)\sqrt{n}.

4.2. Non-new modular curves of fixed genus. Our goal in this section is to prove the finiteness assertion of Theorem 1.5.

Proof of finiteness in Theorem 1.5(i). Let X/ℚ be a modular curve of genus g ≥ 2 and level N, so that there exists a map π: X_{1}(N) → X over ℚ. Let B = B(g) be the integer of Proposition 2.1.

We will follow the basic strategy of the proof of Theorem 1.3, but two complications arise: the map π need not be unramified at ∞, and the cusp forms whose Fourier coefficients we need to bound need not be newforms. The sparseness assumption will allow us to get around both of these difficulties.

Let T be as in Corollary 3.6. Choose j ∈ T with ord_{q}(j) minimal. Then j dq/q corresponds to a regular differential on X_{ℂ} not vanishing at π(∞). Since q is an analytic parameter at ∞ on X_{1}(N), the ramification index e of π at ∞ equals ord_{q}(j). In particular, e|N. By Remark 2.7, the curve X is determined by the set { h mod q^{eB} | h ∈ T }.

Fix h = ∑_{n=1}^{∞} a_{n}q^{n} ∈ T. Since ord_{q}(h) ≥ e, Corollary 3.6 implies that there exist an integer M|N, a newform f ∈ New_{M}, and c_{d} ∈ ℂ such that

h = ∑_{d| \frac{N}{M} ,d≥e} c_{d}f(q^{d}).

Since N ∈ Sparse_{B}, we have d > eB for all d | N such that d > e. If c_{e} = 0, then h mod q^{eB} = 0; otherwise we may scale to assume that c_{e} = 1. In this case, the sparseness of N implies that a_{n}(h(q)) = a_{n}(f(q^{e})) for 1 ≤ n < eB. In other words, for 1 ≤ n < eB we have

a_{n}(h) = \begin{cases}a_{n/e}(f) & if e | n, \\ 0 & if e ∤ n. \end{cases}

Since f is a newform, each a_{n}(h) with n ≤ eB and e|n is an algebraic integer satisfying |a_{n}(h)| ≤ σ_{0}(n/e)\sqrt{n/e} for all archimedean absolute values. As before, it then follows that there are only finitely many possibilities for { h mod q^{eB} | h ∈ T }, and therefore for X. □

Proof of finiteness in Theorem 1.5(ii). Let X/ℚ be a modular curve of genus g ≥ 2 and level N ∈ Smooth_{m}, with m > 0. Then JacX is isogenous to a subvariety of J_{1}(N), so in particular JacX has good reduction outside the finite set Σ, where Σ is the set of primes p such that p ≤ m. By combining the Shafarevich conjecture (proved by Faltings [22, Theorem 6]) with a well-known finiteness result for polarizations (see [56, Theorem 18.1]), it follows that for any number field K, any positive integer d, and any finite set S of places of K, there are only finitely many K-isomorphism classes of principally polarized abelian varieties of dimension d over K with good reduction outside S. In particular, there are only finitely many possible ℚ-isomorphism classes for JacX as a principally polarized abelian variety. By the Torelli theorem [57, Corollary 12.2], it follows that there are only finitely many possible ℚ-isomorphism classes for X. □

As mentioned in the introduction, part (iii) of Theorem 1.5 (concerning prime power levels) follows from (i) and (ii).

To close this section, we remark that for modular curves of prime power level (as opposed to general non-new modular curves), one has good control over the ramification at the cusp infinity. More precisely:

Proposition 4.2. Suppose that N = p^{r} is a prime power. Suppose that X is modular of level N but not modular of level M for any M < N. Then the map π: X_{1}(N) → X is unramified at ∞.

Proof. Corollary 3.6 says that S_{2}(X) has a basis in which each element h has the form ∑_{d| \frac{N}{M}} c_{d}f(q^{d}) for some M|N and f ∈ New_{M}. If π is ramified at ∞, then c_{1} = 0 in each such h. Then ∑_{d| \frac{N}{M} , d>1} c_{d}f(q^{d/p}) ∈ S_{2}(p^{r−1}), according to the decomposition (3.3). Thus π^{∗}H^{0}(X_{ℂ}, Ω) ⊆ B_{p}^{∗}H^{0}(X_{1}(p^{r−1})_{ℂ}, Ω), where B_{p}: X_{1}(p^{r}) → X_{1}(p^{r−1}) denotes the degeneracy map induced by q ↦ q^{p}. By Proposition 2.11(i), π factors through X_{1}(p^{r−1}), contradicting the minimality of N. □

4.3. New modular curves of fixed gonality. In [2, p. 1006], one finds the following lower bound on the gonality of X_{1}(N).

Theorem 4.3. Let g′ and G′ be the genus and gonality, respectively, of X_{1}(N). Then G′ ≥ \frac{21}{200}(g′ − 1).

The linearity of the bound in the genus of X_{1}(N) is what enables us to deduce the following.

Proposition 4.4. If X is a ℂ-modular curve of genus g ≥ 2 and gonality G, then g ≤ \frac{200}{21}G + 1.

Proof. Let d be the degree of the given morphism π: X_{1}(N)_{ℂ} → X. Let g′ and G′ be the genus and gonality of X_{1}(N), so that G′ ≥ \frac{21}{200}(g′ − 1) by Theorem 4.3. Any morphism X → ℙ_{ℂ}^{1} can be composed with π to obtain a morphism X_{1}(N)_{ℂ} → ℙ_{ℂ}^{1}, so G′ ≤ dG. The Hurwitz formula implies g′ − 1 ≥ d(g − 1). Now combine these three inequalities. □

Proposition 4.4 and Theorem 1.3 together imply the finiteness in Theorem 1.8.

Remark 4.5. Abramovich’s result used the lower bound 0.21 for the positive eigenvalues of the Laplacian on Γ\H for congruence subgroups Γ. In Appendix 2 to [44], Kim and Sarnak recently improved this to 975/4096 > 0.238. This means that 200/21 can be improved to 2/0.238 < 17/2. In particular, taking G = 2, we find that a ℂ-modular hyperelliptic curve has genus at most 17.

5. Computability

5.1. The meaning of computable. “Computable” will mean computable by a Turing machine. (See [37] for a definition of Turing machine.) To give a precise sense to each statement in our introduction, we must specify what the input and output of the Turing machine are to be. In particular, we will need to choose how to represent various objects, such as curves over number fields. In many cases, there exist algorithms for converting between various possible representations, so then the particular representation chosen is not important. A number field k can be represented by f ∈ ℤ[x] such that k ≃ ℚ[x]/(f(x)). The representation is not unique, but we do not care, since a Turing machine can decide, given f_{1}, f_{2} ∈ ℤ[x], whether f_{1} and f_{2} define isomorphic number fields (and if so, find an isomorphism) [48, §2.9]. An element α ∈ k can then be represented by g ∈ ℚ[x] (of degree at most deg f − 1) whose image in ℚ[x]/(f(x)) corresponds to α. Turing machines can also handle arithmetic over \overline{ℚ}, thought of as a subfield of ℂ, by representing each α ∈ \overline{ℚ} by its minimal polynomial over ℚ, together with decimal approximations to its real and imaginary parts to distinguish α from its conjugates. If k_{0} = ℚ or k_{0} = \overline{ℚ}, then a field finitely generated over k_{0} can be represented as the fraction field of a domain k_{0}[t_{1}, …, t_{n}]/(f_{1}, …, f_{m}), or alternatively as a finite extension of the rational function field k_{0}(t_{1}, …, t_{n}) (in the same way that we handled finite extensions of ℚ).

A curve X over a field k finitely generated over ℚ or \overline{ℚ} can be represented by f ∈ k[x, y] such that the (possibly singular) affine curve f(x, y) = 0 is k-birational to X. Meromorphic differentials on X represented by f ∈ k[x, y] can be expressed as g(x, y) dx or g(x, y) dy, where g = g_{1}/g_{2} with g_{1}, g_{2} ∈ k[x, y] and g_{2} not divisible by f. A finite set of k-isomorphism classes of curves over k can be represented by a finite list of curves over k in which each class is represented exactly once.

A closed subvariety of ℙ^{n} over a field k finitely generated over ℚ or \overline{ℚ} can be represented by a finite set of generators of its homogeneous ideal. A constructible subset of ℙ^{n} (in the Zariski topology) can be represented as a Boolean combination of closed subvarieties. Morphisms or rational maps between quasiprojective varieties can be defined locally by rational functions on a finite number of affine open subsets. Elimination of quantifiers over algebraically closed fields is effective, and it follows that the image of a constructible subset of a quasiprojective variety under a morphism can be computed. (The elimination of quantifiers is usually attributed to Tarski; for a proof, see Section 3.2 of [53], especially Theorem 3.2.2 and Corollary 3.2.8(ii).) Irreducible components of a variety can also be computed: this follows from the primary decomposition algorithm in [35].

When in one of our theorems we claim that some set is computable, what we really mean is that there exists a Turing machine that takes as input the various parameters on which the set depends (such as a ground field, a curve, and/or an integer g), and, after a finite but unspecified amount of computation, terminates and outputs the set in question.

5.2. Computability lemmas for curves.

Lemma 5.1. A Turing machine can solve the following problems: Given a field k finitely generated over ℚ or \overline{ℚ}, and given curves X and Y over k,

(1) Compute the genus g of (the smooth projective model of) X.

(2) If g ≥ 2, decide whether or not X is hyperelliptic.

(3) If either of the following holds:

(i) g(X) ≥ 2 (ii) g(X) = 0 and k is a number field or \overline{ℚ}, decide whether X and Y are k-birational.

Proof.

(1) See [1].

(2) See [74].

(3) By (1) we may assume that X and Y have the same genus g, and that g is known. If g ≥ 2, use [36]. From now on, suppose that g = 0. If k = \overline{ℚ}, then X and Y are automatically birational. So from now on, suppose that k is a number field. By [73], we can find plane conics birational to X and Y, respectively. Diagonalize the corresponding quadratic forms to put each curve in the form x^{2} − ay^{2} − bz^{2} = 0 for some a, b ∈ k^{∗}. By the Hasse principle, the curves are isomorphic over k if and only if the order-2 Hilbert symbols (a, b)_{v} match at every place v. In fact, one need only check the places above 2 and ∞ and the places occuring in the factorizations of the a and b for each curve. Each Hilbert symbol is computable: see [18] or II.7.1.5 and II.7.5 in [29]. (Alternatively, the question of whether the conic x^{2} −ay^{2} −bz^{2} = 0 has a k_{v}-point can be answered by taking its restriction of scalars down to ℚ and using a general algorithm for deciding whether a variety over ℚ has a ℚ_{p}-point: see [72] or [59] for the archimedean and nonarchimedean cases, respectively.) □

Remark 5.2. Most of these algorithms have been fully implemented. For instance, Magma [8] can compute the genus of a plane curve, and PARI-GP at http://www.parigp-home.de/ has a function nfhilbert for computing Hilbert symbols.

Remark 5.3. Lemma 5.1(3) holds in the genus 1 case if and only if there exists an algorithm to decide, given a genus 1 curve Z over k, whether Z(k) = ∅. Such an algorithm exists trivially if k = \overline{ℚ}. If k is a number field, the existence of such an algorithm is implied by the conjecture that the Shafarevich-Tate group � (E) is finite for all elliptic curves E over k.

Lemma 5.4. Let X be a curve over a number field k, and let ℓ ≥ 1 be an integer. Let A = JacX, and let M = A[ℓ] be the Galois module of ℓ-torsion points. Then we can compute a description of M of the following type: a finite set equipped with an addition table and an action of Gal (L/k) for some explicit finite Galois extension L of k over which all points of M are defined.

Proof. Compute the genus g of X. We represent points of A(\overline{ℚ}) (nonuniquely) by divisors on X\overline{ℚ} whose support avoids any singularities of the given model. A solution of the “Riemann-Roch problem” decides whether a given divisor is principal: see the references cited in [61].

Hence we can decide when two given divisors represent the same element of A(\overline{ℚ}). Now enumerate the countably many divisors on X\overline{ℚ} avoiding the singularities, and continue until finding a set S of ℓ^{2g} such divisors D such that ℓD is principal, and such that they represent distinct elements of A(\overline{ℚ}). For each pair in S, we can determine which divisor in S is linearly equivalent to their sum. Finally, we can compute a finite Galois extension L of k containing the coordinates of all points appearing in the divisors in S, and for each σ ∈ Gal (L/k) and D ∈ S, we can determine which divisor in S is linearly equivalent to ^{σ}D. □

Lemma 5.5. Given a Galois extension of number fields L/k, the set of ramified primes in k and the sequence of ramification groups for each ramified prime are computable.

Proof. By Sections 2.2.4 and 2.3.5 respectively of [14], we can compute the relative discriminant ideal and factor it to obtain the ramified primes. For each ramified prime 𝔭 of k, we use [14, Algorithm 2.5.4] to extend it to an ideal of L, factor it to obtain a prime 𝔓 of L above 𝔭, and use [14, §2.2.3] find an element α ∈ 𝔓 of 𝔓-adic valuation 1. For each σ ∈ Gal (L/k), we compute the valuation of σα − α: this determines the ramification groups. □

Proposition 5.6. The conductor of the jacobian of a curve X over a number field k is computable.

Proof. Let ℓ ∈ ℤ be a prime, and let A = JacX. Compute A[ℓ] and L as in Lemma 5.4. Use Lemma 5.5 to compute the ramification groups of Gal (L/k) at each ramified prime of k. Their action on A[ℓ] gives us the exponent of 𝔭 in cond(A) for each 𝔭 not dividing ℓ. Repeating the above with a different ℓ lets us compute the remaining exponents. □

5.3. The de Franchis-Severi Theorem. The following result, which we have already mentioned, is known as the de Franchis-Severi Theorem; we show in addition that the finite set it promises is computable. We thank Matthias Aschenbrenner, Brian Conrad, Tom Graber, Tom Scanlon, and Jason Starr for discussions related to the proof.

Theorem 5.7 (de Franchis-Severi, computable version). Let k be a number field or \overline{ℚ}. Let X be a curve over k. Then the set of pairs (Y, π) where Y is a curve over k of genus at least 2 and π: X → Y is a morphism, up to k-isomorphism, is finite and computable.

Remark 5.8. We consider (Y, π) and (Y′, π′) to be isomorphic if and only if there is an isomorphism Y → Y′ whose composition with π gives π′.

Proof. For the finiteness, see pp. 223–224 of [47]. We assume first that k = \overline{ℚ}. The genus g of each Y is bounded by the genus g_{X} of X, so it suffices to show that for each fixed g ≥ 2, we can compute the set C_{g} of isomorphism classes of pairs (Y, π) where Y has genus g.

View X as a subvariety of ℙ^{n} using the tricanonical embedding. Compute equations defining X in ℙ^{n}. If (Y, π) ∈ C_{g} and Y ⊆ ℙ^{m} is the tricanonical embedding of Y, then 3K_{X} − π^{∗}(3K_{Y}) is linearly equivalent to an effective divisor, so the theory of linear systems implies that there is a linear subspace L ⊆ ℙ^{n} of dimension n − m − 1 such that π and the linear projection π_{L}: ℙ^{n} −−→ ℙ^{m} coincide on X − L. Riemann-Roch gives n = 5g_{X} − 6 and m = 5g −6. Let G be the Grassmannian variety whose points correspond to linear subspaces L ⊆ ℙ^{n} of dimension n − m − 1. Then (Y, π) ↦ L defines an injection ι: C_{g} → G(\overline{ℚ}).

Conversely, if we start with a linear subspace L, corresponding to a point s ∈ G, let Y_{s} be the Zariski closure of π_{L}(X −L) in ℙ^{m}, and let π_{s}: X → Y_{s} denote the morphism induced by π_{L}. The map s ↦ (Y_{s}, π_{s}) restricted to ι(C_{g}) is an inverse of ι, but in general (Y_{s}, π_{s}) need not be in C_{g}. Moreover, the Y_{s} need not form the fibers of a smooth (or even flat) family.

Claim 1: Each closed subvariety H ⊆ G can be (computably) partitioned into a finite number of irreducible locally closed subsets H_{i} such that for each i, either

(1) For all s ∈ H_{i}, the curve Y_{s} is not smooth over the residue field of s, or

(2) There is a smooth family Y → H_{i} of curves, and an H_{i}-morphism X ×_{k} H_{i} → Y whose fiber above s ∈ H_{i} is π_{s}: X → Y_{s}.

Proof: We use induction on dim H. Because irreducible components can be computed, we may reduce to the case where H is irreducible.

We next use the principle that “whatever happens at the generic point also happens over some computable dense Zariski open subset.” Let η be the generic point of H, and let L be the corresponding linear subspace defined over the function field κ of H. Choose a dense open affine subset SpecA of H, and write elements of κ as ratios of elements of A. Working over κ, we compute the intersection of L with X, the image of X −L under the projection ℙ^{n} −−→ ℙ^{m}, its closure Y_{η}, and the morphism π_{η}: X → Y_{η}. Using partial derivatives, we compute also whether or not Y_{η} is smooth over κ. Compute the localization A′ of A obtained by adjoining to A the inverses of the numerators and denominators of the finitely many elements of κ that appear during these computations. Then the formulas computed over κ make sense over SpecA′, so that all the constructions can be performed over SpecA′. Moreover, for each s ∈ SpecA′, the curve Y_{s} is smooth over the residue field of s if and only if Y_{η} is smooth over κ.

Let H_{1} = SpecA′ ⊆ H. The complement H − H_{1} is a closed subvariety of lower dimension than H, so using the inductive hypothesis, we can partition H − H_{1} into H_{2}, …, H_{n} with the desired properties. This completes the proof of Claim 1.

Apply Claim 1 to G, and discard the H_{i} in which the Y_{s} are not smooth. For each remaining H_{i}, the genus of Y_{s} is constant for s ∈ H_{i}, so we compute the genus of the generic fiber Y_{η} and discard H_{i} if this genus is not g.

Claim 2: For each remaining H_{i}, let J_{i} be the set of s ∈ H_{i} for which the linear subspace L ⊆ ℙ^{n} corresponding to s equals the linear subspace L′ ⊆ ℙ^{n} defined as the common zeros of the sections in the image of H^{0}(Y_{s}, ω^{⊗3}) ↪ H^{0}(X, ω^{⊗3}). Then J_{i} is constructible and computable.

Proof: The strategy of proof is the same as that of Claim 1. The equality of L and L′ can be tested at the generic point η of H_{i}, and the outcome will be the same on some computable dense Zariski open subset of H_{i}. We finish the proof of Claim 2 by an induction on the dimension.

Let J be the (computable) union of the J_{i}. By definition, J = ι(C_{g}). Since C_{g} is finite, J is finite. Therefore C_{g} can be computed by computing (Y_{s}, π_{s}) for s ∈ J. This completes the proof of Theorem 5.7 in the case where k = \overline{ℚ}.

Finally, suppose that k is a number field. Compute the finite subset J for \overline{k} = \overline{ℚ} as above. Since ι is G_{k}-equivariant, it suffices to compute the G_{k}-invariant elements of J. For each L ∈ J ⊆ G, we compute the Plücker coordinates for L relative to a k-basis of H^{0}(X, ω^{⊗3}), and discard those for which the Plücker embedding does not map L to a k-point of projective space. The (Y_{s}, π_{s}) corresponding to the remaining elements of J are the ones over k, each appearing once. □

Remark 5.9. Suppose that k is a number field. Then the finiteness of the set of Y (without the morphism) in Theorem 5.7 holds even if we include curves Y of genus ≤ 1! It is also possible to compute a finite set of curves over k representing all the k-isomorphism classes of such Y, but with some classes represented more than once. Eliminating the redundancy in genus 1 would require an algorithm as in Remark 5.3.

5.4. Computability of modular curves. Here we use the results of the previous section to prove that the finite sets in Theorems 1.3, 1.8, and 1.10 are computable.

Proposition 5.10. Fix N ≥ 1 and a number field k. The set of k-modular curves of level dividing N up to k-isomorphism is finite and computable.

Proof. By Theorem 5.7, it suffices to compute X_{1}(N). First compute the genus g of X_{1}(N) (a formula can be found in Example 9.1.6 on page 77 of [20], for example). Using modular symbols we can compute a basis of S_{2}(N), with each q-expansion computed up to an error of O(q^{B(g)+1}). (This follows from [52], and an explicit method was described first in [54, §4.3]: see also the earlier paper [16] and the later thesis [71] for some related work.) Multiplying each basis element by dq/q results in expansions for a basis of differentials, and then Section 2.1 explains how to recover either an equation y^{2} = f(x) if X_{1}(N) is hyperelliptic, or equations defining the image of the canonical embedding if X_{1}(N) is not hyperelliptic. In the latter case, we can try various linear projections and use elimination theory until we find one that yields a plane curve birational to X_{1}(N): we can detect whether a linear projection mapped the canonical model birationally by checking the genus of the image. Since we can enumerate all linear projections, we will eventually find one that will work. (In practice, almost any projection will work.) □

Question 5.11. The de Franchis-Severi Theorem lets us prove the finiteness of modular curves of fixed level whether or not they are new. Can the proof of the de Franchis-Severi Theorem somehow be combined with our proof of Theorem 1.3 to prove Conjecture 1.1 in general?

Proof of computability in Theorems 1.3, 1.8, and 1.10. The proof of finiteness in Theorem 1.3 produces a finite list of candidate curves X. For each X, we compute N := cond(JacX)^{1/g}, which will be the level of X if X is a new modular curve of genus g and level N. (If N is not an integer, discard X immediately.) By Proposition 5.10, we can list all modular curves of level N. Lemma 5.1(3) determines if X is birational to a curve in this list; discard X if not. If X is in this list, then it is modular of genus g and level N, and it must be new, since otherwise cond(JacX) would be less than N^{g}. Thus we can obtain a list of all new modular curves of genus g. Finally, we can use Lemma 5.1(3) to eliminate redundancy.

Computability in Theorem 1.8 now follows from computability in Theorem 1.3 and Proposition 4.4. Computability in Theorem 1.10 follows from computability in Theorem 1.8 and Proposition 7.4. □

Remark 5.12. We now explain the difficulty in proving computability of the sets in Theorem 1.5. In the proof of finiteness in part (i) (with S = Sparse_{B(g)}), we obtained a finite list of candidates X. The problem is that we do not know how to bound the level of a given non-new modular curve; in fact, we do not even know how to test if a curve is modular. The levels of the modular forms corresponding to the regular differentials on X can be bounded by the conductor of the jacobian of X, but this is not enough, since the level of X can be higher than the levels of these forms: see Section 8.2.

Our proof of Theorem 1.5(ii) is ineffective, because there is no known effective proof of the Shafarevich conjecture.

6. New modular hyperelliptic curves

By Theorem 1.8, the set of the new modular hyperelliptic curves over ℚ is finite. By Remark 4.5, the genus of such a curve is ≤ 17. The goal of this section is to improve this by proving Theorem 1.9 and other restrictions on these curves. The main results are summarized in Table 1 and proved in Section 6.3. The computational results are given in tables in the appendix, and summarized in Section 6.5.

6.1. Criterion to determine new modular hyperelliptic curves. In this subsection we will characterize effectively the class of new modular hyperelliptic curves. From now on, for a hyperelliptic curve X we denote by WP(X) the set of Weierstrass points of X.

Lemma 6.1. Assume that there exists a nonconstant morphism π: X_{1}(N)_{ℂ} → X of curves over ℂ such that X is hyperelliptic of genus g and π^{∗}H^{0}(X, Ω) = H^{0}(A_{ℂ}, Ω), for some abelian variety A over ℚ which is a quotient of J_{1}(N)^{new}. We denote by f^{(j)} = ∑_{n≥1} a_{n}^{(j)} q^{n}, 1 ≤ j ≤ g, a basis for S_{2}(A) consisting of elements of New_{N}, and we set P = π(∞). Then:

(1) There exists a unique basis {h_{1}, …, h_{g}} of S_{2}(A) such that for all 1 ≤ j ≤ g:

\begin{cases}h_{j} ≡ q^{j}  (mod q^{g+1}) & if P ∉ WP(X) , \\ h_{j} ≡ q^{2j−1} + ∑_{i=j}^{g−1} C_{j,2i} q^{2i} (mod q^{2g}) & if P ∈ WP(X). \end{cases}

(2) Moreover, (i) If P ∉ WP(X), then det(a_{i}^{(j)})_{1≤i,j≤g} ≠ 0 . (ii) If P ∈ WP(X), then det(a_{2i−1}^{(j)})_{1≤i,j≤g} ≠ 0 .

Proof. Applying Lemma 2.5 and the fact that π is unramified at the cusp ∞, we obtain the existence of a basis {h1^{′}, …, hg^{′}} of S_{2}(A) which satisfies:

\begin{cases}hj^{′} ≡ q^{2j−1} (mod q^{2j}) & if P ∈ WP(X) \\ hj^{′} ≡ q^{j}  (mod q^{j+1}) & if P ∉ WP(X) \end{cases}

for all 1 ≤ j ≤ g. Therefore, (1) is obtained by Gaussian elimination. To obtain (2), it suffices to observe that the matrices in (i) and (ii) are the change of basis matrices from {f^{(j)}} to the basis {h_{j}}. □

Remark 6.2. Later, in Proposition 6.7, we prove that if P ∈ WP(X) then a_{2n}^{(j)} = 0 for all n ≥ 1 and j ≤ g. In particular, 4 | N and the basis h_{j} satisfies

h_{j} ≡ q^{2j−1} (mod q^{2g}) .

Remark 6.3. If moreover X comes from a curve over ℚ having jacobian isogenous to A_{f} with f = q + ∑_{n≥2} a_{n}q^{n} ∈ New_{N}, then part (2) of the lemma implies that {1, a_{2}, …, a_{g}} (resp. {1, a_{3}, …, a_{2g−1}}) is a ℚ-basis of E_{f} if P ∉ WP(X) (resp. P ∈ WP(X)).

Remark 6.4. If A is a quotient of J_{1}(N) (over ℚ), and the ℂ-vector space S_{2}(A) has a basis {h_{1}, …, h_{g}} as in part (1) of the previous lemma, then each h_{i} has Fourier coefficients in ℚ, since S_{2}(A) has a basis contained in ℚ[[q]].

The following proposition provides us with an effective criterion to determine when a ℚfactor of J_{1}(N)^{new} is ℚ-isogenous to the jacobian of a modular hyperelliptic curve of level N. Let ⟨v_{1}, …, v_{n}⟩ denote the span of elements v_{1}, …, v_{n} of a vector space.

Proposition 6.5. Let A be a quotient of J_{1}(N)^{new} of dimension g > 1. The following conditions are equivalent:

(1) There exists a modular hyperelliptic curve X of level N over ℚ such that JacX ∼^{ℚ} A.

(2) There exists a hyperelliptic curve X′ over ℂ and a nonconstant morphism π′: X_{1}(N)_{ℂ} → X′ such that π′^{∗}H^{0}(X′, Ω) = H^{0}(A_{ℂ}, Ω).

(3) There exists a basis {h_{1}, … h_{g}} of S_{2}(A) as in part (1) of Lemma 6.1 such that for every pair g_{1}, g_{2} ∈ S_{2}(A) satisfying ⟨g_{1}, g_{2}⟩ = ⟨h_{g−1}, h_{g}⟩ and g_{2} ∈ ⟨h_{g}⟩, there exists F (U) ∈ ℂ[U] of degree 2g + 1 or 2g + 2 without double roots such that the functions on X_{1}(N) given by

x = \frac{g_{1}}{g_{2}} , y = \frac{q dx/dq}{g_{2}}

satisfy the equation y^{2} = F (x).

Proof. It is clear that (1) implies (2). Also, (3) implies (1), because when we apply (3) with g_{1} = h_{g−1} and g_{2} = h_{g}, the modular functions x and y have rational q-expansion, so the corresponding polynomial F has coefficients in ℚ.

We now assume (2) and prove (3). By Lemma 6.1, there exists such a basis {h_{1}, …, h_{g}}. As before, we put P = π′(∞). Let u and v be nonconstant functions on X′ such that:

• div u = (Q) + (w(Q)) − (P) − (w(P)) for some Q ∈ X′ and where w denotes the hyperelliptic involution of X′.

• v^{2} = G(u), where G(U) is a polynomial in ℂ[U] of degree 2g + 1 or 2g + 2, without

double roots. By looking at ord_{P} du/v and ord_{P} udu/v, and using the fact that π′ is unramified at ∞, we have:

⟨π′^{∗}(du/v), π′^{∗}(udu/v)⟩ = ⟨h_{g−1}dq/q, h_{g}dq/q⟩ , ⟨π′^{∗}(du/v)⟩ = ⟨h_{g}dq/q⟩ .

Thus, for every pair g_{1}, g_{2} as in part (3), there exists a matrix (\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ GL_{2}(ℂ) such that

\{ \begin{matrix}g_{1}dq/q = a π′^{∗}(udu/v) + b π′^{∗}(du/v) \\ g_{2}dq/q = d π′^{∗}(du/v) . \end{matrix}

Now, one can check easily that the modular functions

x := \frac{g_{1}}{g_{2}} , y := \frac{q dx/dq}{g_{2}}

satisfy the equation y^{2} = F (x), where

F (U) = \frac{a^{2}}{d^{4}} G (\frac{dU − b}{a}) .

□

In practice, the previous proposition will be used together with the next result.

Lemma 6.6. Let Y be a curve of genus g_{Y} > 0 over ℂ. Let q be an analytic uniformizing parameter in Ô_{Y,P} for a point P ∈ Y (ℂ) and let F ∈ ℂ[t] be a polynomial of degree d > 0. Suppose we are given ω_{1}, ω_{2} ∈ H^{0}(Y, Ω) such that the functions x = ω_{1}/ω_{2} and y = dx/ω_{2} satisfy

y^{2} − F (x) ≡ 0 (mod q^{c}) for some c ≥ (2g_{Y} − 2) Max{6, d} + 1 .

Then we have y^{2} = F (x).

Proof. Put d′ = Max{6, d}. Now, it suffices to observe that y ω_{2}^{3} ∈ H^{0}(Y, Ω^{⊗3}) and, thus, (y^{2} − F (x))ω_{2}^{d′} ∈ H^{0}(Y, Ω^{⊗d′}) and has at P a zero of order d′(2g_{Y} − 2) + 1 at least. □

The following proposition improves part (1) of Lemma 6.1 and is useful for computations.

Proposition 6.7. Let X be a new modular hyperelliptic curve of genus g and level N over ℚ and let π: X_{1}(N) → X be the corresponding morphism. If π(∞) ∈ WP(X) and f = q + ∑_{n≥2} a_{n}q^{n} ∈ New_{N} ∩ S_{2}(X), then a_{2n} = 0 for all n ≥ 1 and 4 | N.

Proof. Write S_{2}(X) = ⊕_{i=1}^{m}S_{2}(A_{f^{(j)}}), where f^{(j)} = q +∑_{n≥2} a_{n}^{(j)}q^{n} is a newform in S_{2}(N, ε_{j}). The condition that a_{2n}^{(j)} = 0 for all n and j is equivalent to the condition that a_{2}^{(j)} = 0 for all j and 2 | N. We consider three cases.

\underline{Case 1}: g = 2 and JacX is ℚ-simple.

Put f := f^{(1)} = ∑_{n≥0} a_{n}q^{n}. The coefficients of f generate some quadratic field ℚ(\sqrt{d}). Let σ denote the nontrivial element of Gal (ℚ(\sqrt{d})/ℚ), so that f^{(2)} = ^{σ}f. By parts (1) and (2)(ii) of Lemma 6.1, respectively, we have a_{2} ∈ ℚ and a_{3} ∉ ℚ. Write a_{n} := A_{n} + B_{n}\sqrt{d}, where A_{n}, B_{n} ∈ ℚ. Define ε := ε_{1}. By (3.7) the q-expansion of f is:

q + A_{2}q^{2} + (A_{3} + B_{3}\sqrt{d})q^{3} + (A_{2}^{2} − 2 ε(2))q^{4} + (A_{5} + B_{5}\sqrt{d})q^{5} + A_{2}(A_{3} + B_{3}\sqrt{d})q^{6} + O(q^{7}),

where B_{3} ≠ 0. Put ε(2) = C +D\sqrt{d}. Following Lemma 6.1 and Proposition 6.5, we compute

h_{1} = \frac{f +^{σ}f}{2} , h_{2} = \frac{f −^{σ}f}{2B_{3}\sqrt{d}} , x = \frac{h_{1}}{h_{2}} = \frac{1}{q^{2}} + …, y = −\frac{q dx/dq}{2h_{2}} = \frac{1}{q^{5}} + ….

For F ∈ ℂ((q)), let Coeff[q^{n}, F] denote the coefficient of q^{n} in F. Since y^{2} − x^{5} is at most a quartic polynomial in x, and x is O(q^{−2}), we have Coeff[q^{−9}, y^{2} − x^{5}] = 0. On the other hand, we compute Coeff[q^{−9}, y^{2} − x^{5}] = −4(A_{2}B_{3} + D)/B_{3}, so D = −A_{2}B_{3}. Thus D = 0 if and only if A_{2} = 0. We claim that A_{2} = D = 0. If 2 | N then ε(2) = 0, so D = 0. If 2 ∤ N, then \overline{a}_{2} = \overline{ε(2)}a_{2} by (3.13), so A_{2} = (C ± D\sqrt{d})A_{2} and equating coefficients of \sqrt{d} yields DA_{2} = 0. Thus A_{2} = D = 0 in both cases.

Now y^{2} − x^{5} − Coeff[q^{−8}, y^{2} − x^{5}]x^{4} is at most a cubic polynomial in x, so

0 = Coeff [q^{−7}, y^{2} − x^{5} − Coeff[q^{−8}, y^{2} − x^{5}]x^{4}] = 12C.

Thus C = 0, so ε(2) = 0. By (3.7), a_{2} = ε(2) = 0 implies a_{2n} = 0 for all n. By (3.10) and (3.11), we have 4 | N.

\underline{Case 2}: g = 2 and JacX is not ℚ-simple.

Then JacX ∼^{ℚ} A_{f^{(1)}} × A_{f^{(2)}} with dim A_{f^{(1)}} = dim A_{f^{(2)}} = 1. Since dim A_{f^{(1)}} is odd, the Nebentypus ε of f^{(1)} is the trivial character modulo N (cf. the proof of Lemma 6.17). The same holds for f^{(2)}. By part (1) of Lemma 6.1, f^{(1)} and f^{(2)} share the same coefficient of q^{2}, say a. By (3.7),

f^{(1)} = q + aq^{2} + a_{3}q^{3} + (a^{2} − 2 ε(2))q^{4} + a_{5}q^{5} + a a_{3}q^{6} + O(q^{7}),
f^{(2)} = q + aq^{2} + b_{3}q^{3} + (a^{2} − 2 ε(2))q^{4} + b_{5}q^{5} + a b_{3}q^{6} + O(q^{7}),

with a, a_{n}, b_{n} ∈ ℤ. By part (2) of Lemma 6.1, a_{3} ≠ b_{3}. As in Case 1, we compute

h_{1} = \frac{f^{(1)} + f^{(2)}}{2} , h_{2} = \frac{f^{(1)} − f^{(2)}}{a_{3} − b_{3}} , x = \frac{h_{1}}{h_{2}} = \frac{1}{q^{2}} + …, y = −\frac{q dx/dq}{2h_{2}} = \frac{1}{q^{5}} + ….

From 0 = Coeff[q^{−9}, y^{2} − x^{5}] = −4a we obtain a = 0. From

0 = Coeff [q^{−7}, y^{2} − x^{5} − Coeff[q^{−8}, y^{2} − x^{5}]x^{4}] = 12ε(2)

we obtain ε(2) = 0. The result follows from a = ε(2) = 0, as in Case 1.

\underline{Case 3}: g > 2.

Put

f^{(j)} = ∑_{n≥0} a_{n}^{(j)}q^{n} , E_{j} = ℚ({a_{n}^{(j)}}) .

Let 𝔼 denote the ℚ-algebra E_{1} ×⋯×E_{m}. Let 𝔼^{∨} denote the dual vector space Hom_{ℚ}(𝔼, ℚ), and let ϕ.â denote the evaluation of a functional ϕ ∈ 𝔼^{∨} at an element â = (a^{(j)}) of 𝔼. For n ≥ 1, set â_{n} = (a_{n}^{(j)}) ∈ 𝔼. Let ε̂(2) = (ε_{i}(2)) ∈ 𝔼. By (3.7), we have

(6.8)  â_{nm} = â_{n}â_{m} when (n, m) = 1,
(6.9)  â_{2^{n}} = â_{2^{n−1}}â_{2} − 2 ε̂(2) â_{2^{n−2}}  when n ≥ 2.

By part (2)(ii) of Lemma 6.1, {â_{1}, â_{3}, â_{5}, …, â_{2g−1}} is a ℚ-basis of 𝔼. Let {ϕ_{1}, ϕ_{2}, …, ϕ_{g}} denote the dual basis of 𝔼^{∨}. Let h_{i} = ∑_{n=1}^{∞}(ϕ_{i}.â_{n})q^{n} for 1 ≤ i ≤ g. Then h_{i} is a ℚcombination of conjugates of the f^{(j)}, and hence {h_{1}, …, h_{g}} is the basis of S_{2}(X) promised by part (1) of Lemma 6.1. Thus the q-expansion of h_{i} has the form

h_{i} = q^{2i−1} + ∑_{j=i}^{g−1} C_{i,2j} q^{2j} + ∑_{j=2g}^{∞} C_{i,j}q^{j}.

In particular, ϕ_{i}.â_{2} = 0 for i > 1, so â_{2} ∈ ℚâ_{1} = ℚ. That is, a_{2}^{(j)} has a value a_{2} ∈ ℚ independent of j. Let γ equal g or g − 1, depending on whether g is odd or even. Then by (6.8),

C_{γ,2γ} = ϕ_{γ}.â_{2γ} = a_{2}ϕ_{γ}.â_{γ} = 0,

since 1 < γ < 2γ − 1. The same computation shows C_{1,2} = a_{2}. Define x = h_{g−1}/h_{g} and y = \frac{q dx/dq}{−2h_{g}} so that X has the equation y^{2} = F (x) for some monic F of degree 2g + 1. Then

x = q^{−2} + bq^{−1} + O(q^{0}) and y = q^{−(2g+1)} + cq^{−2g} + O(q^{−(2g−1)}),

for some b, c ∈ ℚ. Also, y^{2} = F (x) implies 2c = (2g+1)b. Moreover h_{i} dq/q = P_{g−i}(x) dx/(−2y) for some monic P_{g−i}(x) ∈ ℚ[x] of degree g −i, and equating coefficients of q^{2i−1} dq (just after the monic leading term) yields

C_{i,2i} = (g − i)b + (b/2) − c = −ib.

Setting i = γ and i = 1 yields 0 = −γb and a_{2} = −b, so a_{2} = b = 0.

Setting i = 2 yields ϕ_{2}.â_{4} = C_{2,4} = 0. But ϕ_{i}â_{4} = 0 also for i > 2, so â_{4} ∈ ℚ. By (6.9), ε̂(2) ∈ ℚ. Induction using (6.9) shows that â_{2^{n}} ∈ ℚ for all n ≥ 0. By (6.8), â_{n} ∈ ℚâ_{odd(n)}, where odd(n) is the largest odd divisor of n. Using this, g ≥ 3, and the definition of ϕ_{i}, we now prove that C_{g,2g} = C_{g,2g+2} = C_{g−1,2g−2} = C_{g−1,2g} = 0. Indeed,

(6.10)

C_{g,2g} = ϕ_{g}.â_{2g}  = 0, since odd(2g) < 2g − 1;
C_{g,2g+2} = ϕ_{g}.â_{2g+2} = 0, since odd(2g + 2) < 2g − 1;
C_{g−1,2g−2} = ϕ_{g−1}.â_{2g−2} = 0, since odd(2g − 2) < 2g − 3;
C_{g−1,2g} = ϕ_{g−1}.â_{2g} = 0, since odd(2g) < 2g − 3;

except that in the last equation, when g = 3, odd(2g) < 2g − 3 fails so we use â_{2g} = a_{2}â_{3} = 0 instead to deduce the same result. Therefore

h_{g−1} = q^{2g−3}(1 + O(q^{4})),
h_{g} = q^{2g−1}(1 + C_{g,2g+1}q^{2} + O(q^{4})),
x = h_{g−1}/h_{g} = q^{−2}(1 − C_{g,2g+1}q^{2} + O(q^{4})),

and the new basis defined by h′_{i} := x^{g−i}h_{g} for 1 ≤ i ≤ g satisfies

h′_{i} = q^{2i−1}(1 + c_{i}q^{2} + O(q^{4}))

for some c_{i} ∈ ℚ. The coefficient of q^{4} in h′_{i} is zero for all i, so â_{4} = 0. By (6.9), ε̂(2) = 0. The result follows from â_{2} = ε̂(2) = 0, as in Case 1. □

6.2. Automorphisms of hyperelliptic curves. Throughout this subsection we make the following assumptions:

• k is a field of characteristic zero,

• X is the smooth projective model of a curve y^{2} = F (x) for some squarefree F ∈ k[x] of degree 2g + 1 or 2g + 2 for some g ≥ 2 (so in particular X is a hyperelliptic curve of genus g), and

• w is the hyperelliptic involution of X.

Proposition 6.11. Let X′ be a curve of the same type as X, that is, a genus-g curve y′^{2} = F′(x′), where F′ ∈ k[U] is squarefree. Every isomorphism u: X\overline{k} → X′ ; \overline{k} is given by an expression of the following form:

(x, y) ↦ (\frac{ax + b}{cx + d}, \frac{e y}{(cx + d)^{g+1}} ) ,

for some M = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(\overline{k}) and e ∈ \overline{k}^{∗}. The pair (M, e) is unique up to replacement by

(λ M, eλ^{g+1}) for λ ∈ \overline{k}^{∗}. If u is defined over k, then one can take M ∈ GL_{2}(k) and e ∈ k^{∗}. Moreover, if u′: X′ ; \overline{k} → X′′ ; \overline{k} is another isomorphism, given by (M′, e′), then the composition u′u is given by (M′M, e′e).

Proof. An isomorphism X → X′ induces an isomorphism on the canonical images ℙ^{1} → ℙ^{1}. An automorphism of ℙ^{1} over \overline{k} is given by some M ∈ GL_{2}(\overline{k}). The functions f ∈ \overline{k}(X) such that w^{∗}f = −f are those in \overline{k}(x)y, so y′ corresponds to ey/(cx + d)^{g+1} for some e ∈ \overline{k}(x). The image of (x, y) ∈ X must be on X′, so

(\frac{ey}{(cx + d)^{g+1}} )^{2} = F′ (\frac{ax + b}{cx + d}) ,

or equivalently,

e^{2}F (x) = (cx + d)^{2g+2}F′ (\frac{ax + b}{cx + d}) .

The right hand side is a squarefree polynomial, since it can be obtained from F′ by homogenizing to a polynomial z^{2g+2}F′(x/z) of degree 2g + 2, performing a linear change of variable on x and z, and dehomogenizing. But F (x) on the left is a squarefree polynomial too. Thus e ∈ \overline{k}^{∗}. The rest of the statements follow easily. □

Recall that Aut(X) denotes the group of automorphisms of X defined over k.

Lemma 6.12. Let u ∈ Aut(X). We use the (M, e) notation as in Proposition 6.11. Then:

(1) If g is even, u is represented by a unique pair (M, e) ∈ GL_{2}(k) × k^{∗} satisfying e = (det M)^{g/2}. Then u ↦ M defines an injective homomorphism Aut(X) → GL_{2}(k) mapping w to − id.

(2) If g is odd, u is represented by exactly two pairs (±M, e) ∈ GL_{2}(\overline{k}) × \overline{k}^{∗} satisfying e^{2} = (det M)^{g}. Then u ↦ e defines a homomorphism Aut(X) → μ(k) mapping w to −1, where μ(k) denote the group of roots of unity in k.

Proof. Suppose g is even. For a fixed pair (M_{0}, e_{0}) ∈ GL_{2}(k) × k^{∗}, the condition e_{0}λ^{g+1} = (det(λM_{0}))^{g/2} on λ ∈ \overline{k}^{∗} is equivalent to λ = (det M_{0})^{g/2}/e_{0}. Thus λ is unique and, in particular, λ ∈ k^{∗}. Uniqueness also implies that u ↦ M is a homomorphism. Since e = (det M)^{g/2}, M determines (M, e), so the homomorphism is injective.

Now suppose g is odd. If g is odd, the condition (e_{0}λ^{g+1})^{2} = (det(λM_{0}))^{g} determines λ up to sign. It follows that λ^{2} ∈ k^{∗} and that e := e_{0}λ^{g+1} is in k. Then u ↦ e defines a homomorphism Aut(X) → k^{∗}. Its image is contained in μ(k), since Aut(X) is finite. □

Lemma 6.13. Suppose k = ℚ. If g is even, then Aut(X) is isomorphic to a subgroup of D_{2·4} or D_{2·6}. If g is odd, then Aut(X) is the direct product of ⟨w⟩ and a subgroup isomorphic to a subgroup of D_{2·4} or D_{2·6}. In either case, every element of Aut(X) has order 1, 2, 3, 4, or 6.

Proof. Suppose g is even. By Lemma 6.12, Aut(X) is isomorphic to a finite subgroup G of GL_{2}(ℚ). By averaging an inner product on ℚ^{2}, we obtain a G-invariant inner product, so G can be embedded in the orthogonal group O_{2}(ℝ). All finite subgroups of O_{2}(ℝ) are cyclic or dihedral, so it remains to show that elements of G have order dividing 4 or 6. This follows since the eigenvalues of G are roots of unity in a number field of degree at most 2.

Now suppose g is odd. Let \overline{Aut(X)} denote the image of the homomorphism Aut(X) → Aut(ℙ^{1}) ≃ PGL_{2}(ℚ) mapping u ∈ Aut(X) to its action on the canonical image of X. The first map in the exact sequence

0 → ⟨w⟩ → Aut(X) → \overline{Aut(X)} → 0

has a section Aut(X) → μ(ℚ) = {±1} ≃ ⟨w⟩ mapping u to e as in Lemma 6.12. Thus Aut(X) ≃ ⟨w⟩ × \overline{Aut(X)}. Every finite subgroup of PGL_{2}(ℚ) is isomorphic to a subgroup of D_{2·4} or D_{2·6}, by Proposition A in [28]. □

Proposition 6.14. Suppose k = \overline{k}. Then (1) Aut(X) does not contain (ℤ/2ℤ)^{4}.

(2) If Aut(X) contains (ℤ/2ℤ)^{3}, then g is odd.

(3) If Aut(X) contains ℤ/2ℤ × D_{2·4}, then 4|(g + 1).

(4) If Aut(X) contains ℤ/2ℤ × D_{2·6}, then 6|(g + 1).

(5) If Aut(X) contains D_{2·6}, then g ≢ 1 (mod 3).

Proof. This follows from Proposition 2.1 and Satz 5.1 of [9]. See [11] for more, including a complete classification of the groups that can be the automorphism group of a hyperelliptic curve of given genus. □

Lemma 6.15. Let π: X → X′ be a degree-d morphism between curves of genus g and g′, respectively, over a field of characteristic zero. Assume that g ≥ 2 and X is hyperelliptic. Then

(1) If g′ ≥ 2, then X′ is hyperelliptic.

(2) If g′ ≥ 2, then π(WP(X)) ⊆ WP(X′).

(3) Suppose that X′ = X/⟨α⟩ for some α ∈ Aut(X) of order d. Assume that the hyperelliptic involution w of X is not in ⟨α⟩. Let g′′ be the genus of X′′ = X/⟨αw⟩.

(a) If d is odd, then g′ = ⌊g/d⌋.

(b) If d is even and g ≢ −1 (mod d), then g′ = ⌊g/d⌋.

(c) If d is even and g ≡ −1 (mod d), then g′ and g′′ equal ⌊g/d⌋ and ⌈g/d⌉ in some order.

Proof. (1) The image Y of the canonical map X → ℙ^{g−1} has genus zero and dominates the corresponding image for X′, which implies that its genus is zero too. Therefore, X′ is hyperelliptic.

(2) Let w and w′ denote the hyperelliptic involutions on X and X′, respectively. Since the function field k(X) is of degree 2 over k(Y), it must equal the compositum of k(X′) and k(Y) over k(Y′), and the unique nontrivial element w^{∗} of Gal (k(X)/k(Y)) must restrict to the unique nontrivial element (w′)^{∗} of Gal (k(X′)/k(Y′)). Thus πw = w′π. Hence fixed points of w map to fixed points of w′. (Alternatively, (1) and (2) could have been deduced from Proposition 2.11(ii).)

(3) We may assume k = \overline{k}. Since w ∉ ⟨α⟩, the automorphisms α and w induce an automorphism of Y of degree d and an involution on X′ respectively. Let us denote Y′ = X′/⟨w⟩, where w stands for the involution induced on X′. Up to conjugacy, Aut(Y) = PGL_{2}(k) contains only one cyclic subgroup of order d. Thus we may choose coordinates so that the morphism ℙ^{1} ≃ Y → Y′ ≃ ℙ^{1} induced by X → X′ is x ↦ x′ = x^{d}. In particular, Y → Y′ is ramified above two points 0, ∞ ∈ Y′, each with ramification index d.

Let r′ ∈ {0, 1, 2} be the number of points of Y′ among 0 and ∞ that ramify in X′ → Y′. Concerning the behavior of the points in X above 0 and ∞ in X → X′: if d is odd, we have 2(2 − r′) + r′ points with ramification index d; if d is even, we have 2(2 − r′) points with ramification index d, and 2r′ points with ramification index d/2. The Hurwitz formula gives

2g − 2 = d(2g′ − 2) + \begin{cases}(4 − r′)(d − 1) & if d is odd \\ (4 − 2r′)(d − 1) + (2r′)(d/2 − 1) & if d is even. \end{cases}

This implies g = dg′ + (d − 1) − r′⌊d/2⌋. If d is odd, then we deduce g′ = ⌊g/d⌋. If d is even and g ≢ −1 (mod d), then r′ = 1, and g′ = ⌊g/d⌋.

Finally suppose that d is even and g ≡ −1 (mod d). Then r′ ∈ {0, 2}. Since X is birational to y^{2} = f(x^{d}) for some polynomial f, and d is even, the points 0 and ∞ have a total of 4 preimages in X. Each of these four preimages is fixed by α or αw, but not both. There are 2(2 − r′) fixed points of α, so there are 2r′ fixed points of αw. Applying the same arguments to the analogous number r′′ defined for X′′, we find 2(2 − r′′) fixed points of αw, so r′ + r′′ = 2. Thus either (r′, r′′) = (0, 2), in which case (g′, g′′) = (⌊g/d⌋, ⌈g/d⌉), or (r′, r′′) = (2, 0), in which case (g′, g′′) = (⌈g/d⌉, ⌊g/d⌋). □

Remark 6.16. It is not necessarily true that π(WP(X)) = WP(X′).

6.3. Restrictions on new modular hyperelliptic curves.

Proof of part (ii) of Theorem 1.9. If 3 ∤ N, then Lemma 3.22 implies g < 11. Therefore assume 3 | N from now on. We are given that Jac X is a quotient of J_{0}(N). By Proposition 2.11(i), the morphism π: X_{1}(N) → X factors through a morphism π_{0} : X_{0}(N) → X. Let {f^{(1)}, …, f^{(g)}} be the basis of newforms of S_{2}(X). Write f^{(j)} = ∑_{n≥1} a_{n}^{(j)} q^{n}. Then a_{1}^{(j)} = 1 for all j, and g ≥ 3, so Lemma 6.1(2) implies that a_{3}^{(j)} cannot also be independent of j. In particular there exists j such that a_{3}^{(j)} ≠ 0. By (3.9), it follows that 9 ∤ N. Hence we have the Atkin-Lehner involution W_{3} on X_{0}(N), which is defined over ℚ. By (3.10), a_{3}^{(j)} ∈ {1, −1}. By (3.12), f^{(j)}|W_{3} = −a_{3}^{(j)}f^{(j)} for all j. Set X′ := X/⟨W_{3}⟩, where W_{3} denotes the automorphism of X induced from W_{3} on X_{0}(N) by Proposition 2.11(ii). Then X′ also is new of level N and S_{2}(X′) is spanned by the f^{(j)} with a_{3}^{(j)} = −1. There is at least one such j, since otherwise a_{3}^{(j)} = 1 for all j, contradicting Lemma 6.1(2). Applying Lemma 6.1(2) to X′ shows that the genus g′ of X′ is ≤ 2. Similarly, if X′′ = X/⟨wW_{3}⟩ then S_{2}(X′′) is spanned by the f^{(j)} with a_{3}^{(j)} = 1, and the genus g′′ of X′′ is ≤ 2. But g′ + g′′ = g ≥ 3, so either g′ = 2 or g′′ = 2 (maybe both). Hence X is a new modular hyperelliptic curve of genus 3 or 4 having the same level as some new modular curve of genus 2. We know the latter levels, so we can compute all such curves using the methods to be discussed in Section 6.4, one level at a time. We find only the curve C_{39}^{A,B} in Table 8. This must be X_{0}(39), because X_{0}(39) is a new modular genus-3 curve of level divisible by 3. □

Lemma 6.17. Let X be a new modular curve of genus g and let D′ be a subgroup of D. If the quotient X′ := X/D′ has genus g′, then g − g′ is even. In particular, if X is hyperelliptic and the hyperelliptic involution w belongs to D, then g is even.

Proof. Assume D′ ≠ {1}. If a newform f ∈ S_{2}(X) lies outside the image of S_{2}(X′) then f has nontrivial Nebentypus, so the number field E_{f} is a CM field and then dim A_{f} is even. Indeed, it was proved in [64] (see Proposition 3.3 and the subsequent remark) that given a newform f of weight k and with nontrivial Nebentypus ε, the number field E_{f} (which must be either a totally real field or a CM field) is totally real if and only if f has complex multiplication by an imaginary quadratic field K and ε is the quadratic character χ attached to K. By Theorem 3.4 of [64], if f has complex multiplication by K then the character η = ε · χ satisfies η(−1) = (−1)^{k−1}. In particular, if k is even then the character ε is different from χ and E_{f} must be a CM field.

The quotient of JacX by the image of JacX′ is isogenous to a product of such A_{f}, so it has even dimension. □

Proposition 6.18. Let X be a new modular hyperelliptic curve over ℚ of genus g ≥ 2. Then D is cyclic of order 1, 2, 3, 4 or 6. If #D = 4 or 6, then the hyperelliptic involution w is in D. If #D = 6, then g is 2, 12 or 14.

Proof. We may assume D ≠ {1}. If D is cyclic, let u denote a generator.

Since ⟨D, w⟩ is an abelian subgroup of Aut(X) of even order, Lemma 6.13 implies that it is isomorphic to one of the following:

(6.19) ℤ/2ℤ, ℤ/4ℤ, ℤ/6ℤ, (ℤ/2ℤ)^{2}, ℤ/2ℤ × ℤ/4ℤ, ℤ/2ℤ × ℤ/6ℤ, (ℤ/2ℤ)^{3}.

We label these Cases 1 through 7, respectively.

By (3.15), W_{N} ∉ Aut(X), and the group D′′_{N} = ⟨D, w, W_{N}⟩ is a semidirect product ⟨D, w⟩ ⋊ ⟨W_{N}⟩ with W_{N} acting on the normal subgroup ⟨D, w⟩ as −1. Hence in Cases 4, 5, 6, 7, the group D′′_{N} is isomorphic to

(ℤ/2ℤ)^{3}, ℤ/2ℤ × D_{2·4}, ℤ/2ℤ × D_{2·6}, (ℤ/2ℤ)^{4},

respectively. Case 7 is impossible by Proposition 6.14(1).

In Cases 4, 5, 6, Proposition 6.14 implies 2|(g + 1), 4|(g + 1), 6|(g + 1), respectively. In particular, g is odd, so Lemma 6.17 implies that w ∉ D, so D is isomorphic to ℤ/2ℤ, ℤ/4ℤ, ℤ/6ℤ, respectively. This proves the first statement of Proposition 6.18. To prove the second statement, it suffices to rule out Cases 5 and 6.

\underline{Case 5}: D ≃ ℤ/4ℤ, w ∉ D.

Combining 4|(g+1) with the inequality g ≤ 17 of Remark 4.5 shows that g is 3, 7, 11, or 15. Lemma 6.17 implies that the genus g′ of X′ := X/D is odd. Hence Lemma 6.15(3) implies that (g, g′) is (3, 1), (7, 1), (11, 3), or (15, 3). In the last two cases (with g′ = 3), parts (1) and (3) of Lemma 6.15 imply that either X′′ = X′/W_{N} = X/D_{N} or X′′′ = X′/⟨W_{n}w⟩ = X/D′_{N} has genus 2 and is dominated by X_{0}(N). In the first two cases, Lemma 6.15(3) implies that the curve X/⟨u^{2}w⟩ (resp. X/⟨uw⟩) has genus 2 and has a nontrivial diamond of order 4 (resp. 2) by Lemma 3.17 (the nontrivial diamond prevents the genus from being 1). Using the methods of Section 6.4 to check all levels where there are such new genus-2 curves, we find that no X in Case 5 exists.

\underline{Case 6}: D ≃ ℤ/6ℤ, w ∉ D

Combining 6|(g + 1) with g ≤ 17 shows that g is 5, 11, or 17. In the first (resp. second or third) case, Lemma 6.15(3) and Lemma 6.17 together imply that the curve X/⟨u^{3}w⟩ (resp. X/⟨uw⟩) has genus 2, and by Lemma 3.17 it has a nontrivial diamond of order 6 (resp. 2). We check as before that such curves do not exist.

To finish the proof of Proposition 6.18, we must show that if D ≃ ℤ/6ℤ and w ∈ D, then g is 2, 12 or 14. By Lemmas 3.17 and 6.17, X′′ := X/⟨u^{2}⟩ has diamond group ⟨w⟩ and its genus g′′ is even. On the other hand, g is even, g ≤ 17, and g′′ = ⌊g/3⌋ by Lemma 6.15(3), so (g, g′′) is one of (2, 0), (6, 2), (8, 2), (12, 4), or (14, 4). We rule out (6, 2) and (8, 2) by checking all levels of genus 2 curves with diamond group of order 2. □

Remark 6.20. Assume g ≥ 3 and D = ⟨u⟩. If #D = 2 and w ∉ D (resp. if #D = 6), then X/⟨u.w⟩ (resp. X/⟨u^{2}⟩) is a new modular hyperelliptic curve of genus ≥ 2 of the same level, and by Lemma 3.17 its group of diamonds has 2 elements and contains w. Therefore, in order to find all levels of new modular hyperelliptic curves with #D > 1, it suffices to determine the cases such that #D = 2, 3 or 4 with the additional requirement that w ∈ D when #D is even.

Proposition 6.21. Let X be a new modular hyperelliptic curve over ℚ of genus g ≥ 2 such that 2 | #D and w ∉ D. Then #D = 2, and g is 3, 7, 8, or 9. If g = 3, then X is one of the seven curves in Table 12. If g > 3, then 3 ∤ N.

Proof. Proposition 6.18 implies #D = 2. Let u be the generator of D. Let X′ = X/D and X′′ = X/⟨uw⟩, and let g′ and g′′ be their genera, respectively. By Lemma 6.15(3), g′ and g′′ equal ⌊g/2⌋ or ⌈g/2⌉ in some order. Moreover g′′ = g − g′ is even (by Lemma 6.17) and positive (since X ≄ X/D). In particular g ≢ 2 (mod 4), so g ∉ {2, 6, 10, 14}.

Suppose g′′ = 2. For each new modular curve of genus 2 with diamond group of order 2, we compute all new modular curves at that level; we find only the seven curves in Table 12, all of genus 3. Therefore we may assume g′′ ≠ 2 from now on, so g′′ ≥ 4. Since |g′ − g′′| ≤ 1, we have g′ ≥ 3, and g ≥ 7.

\underline{Case 1}: 3 | N.

Since X/D is of genus g′ ≥ 3 with trivial diamond group, Theorem 1.9(ii) implies X/D ≃ X_{0}(39), so N = 39. Computations show that X_{0}(39) is the only new modular hyperelliptic curve of level 39 and genus ≥ 3.

\underline{Case 2}: 3 ∤ N.

Then g ≤ 10 by Corollary 3.24. The possibility g = 10 was already ruled out. □

Proposition 6.22. Let X be a new modular hyperelliptic curve over ℚ of genus g ≥ 3 such that #D = 3. Then g = 3 or g = 5. If g = 3, then X is one of the five curves in Table 11. If g = 5 there are at least two such curves, which are given in Table 13.

Proof. Since Aut(X\overline{ℚ}) contains D′′_{N} := ⟨D, w, W_{N}⟩ ≃ D_{2·6}, Proposition 6.14 implies g ≢ 1 (mod 3). Also g ≤ 17, so g must be 17, 15, 14, 12, 11, 9, 8, 6, 5 or 3.

Suppose g > 5. Then we claim that there is a quotient X′ of X, such that X′ is a new modular curve of genus 2, of the same level N, and dominated by X_{0}(N). Indeed, by repeated application of Lemma 6.15(3), at least one of the curves X/D_{N}, X/D′_{N} or X/D has genus 2 in each possible case. We now consult the list of new modular curves of genus-2 to check that X does not exist.

Finally we consider the case g = 3. By Lemma 6.17, we have JacX ∼^{ℚ} A_{f} × A_{h}, where dim A_{f} = 2 and dim A_{h} = 1. If ^{σ}f denotes the nontrivial Galois conjugate of f, then {f,^{σ}f, h} is a basis of eigenvectors of a generator u of D acting on S_{2}(X). Since the Dinvariant subspace of S_{2}(X) is 1-dimensional, corresponding to A_{h}, and since u is defined over ℚ, the eigenvalues must be {ζ, ζ^{2}, 1}, respectively, where ζ is a primitive cube root of 1. In particular, the basis of eigenvectors is unique up to scalar multiples. Moreover, the field E_{f} contains the eigenvalue of u acting on f, so E_{f} = ℚ(ζ).

On the other hand, all elements of order 3 in PGL_{2}(ℂ) are conjugate, so we can choose a coordinate function x on ℙ_{ℂ}^{1} = X_{ℂ}/⟨w⟩ so that u induces x ↦ ζx. Then X_{ℂ} is the smooth projective model of y^{2} = F (x) for some squarefree polynomial F, and {dx/y, x dx/y, x^{2} dx/y} is a basis of eigenvectors of u acting on H^{0}(X, Ω) = S_{2}(X)\frac{dq}{q}. In particular, there exists a unique eigenvector whose square equals the product of the other two, at least up to a constant factor c.

The same must be true of {f,^{σ}f, h} and the corresponding constant factor c is in ℚ(ζ). For such a relation to be consistent with the action of σ, it must be that h^{2} = cf^{σ}f. Comparing leading coefficients of q-expansions shows that c = 1. Thus h^{2} = f^{σ}f.

Using the methods of Section 6.4 we run through all possibilities with M = 18, f = ∑_{i=1}^{M} a_{n}q^{n} + O(q^{M+1}) with a_{n} ∈ ℤ[ζ], and h = ∑_{i=1}^{M} b_{n}q^{n} + O(q^{M+1}) with b_{n} ∈ ℤ satisfying h^{2} = f^{σ}f. We obtained only the five curves of Table 11. □

Proof of parts (i) and (iii) of Theorem 1.9. Part (i) follows from parts (ii) and (iii) together with Remark 4.5. Part (ii) was already proved, so we need only prove (iii). Suppose #D > 1, and g is odd. Lemma 6.17 implies w ∉ D, and Proposition 6.18 then implies that #D is 2 or 3. Propositions 6.21 and 6.22 complete the proof in these two cases, respectively. □

Remark 6.23. In order to make the process of discarding levels easier in the previous propositions, we used some additional information, such as Corollary 3.24. In cases where we know that there are quotient curves of some smaller genus ≥ 2, and we have already determined all the hyperelliptic curves of that level and genus, we have used this information.

We summarize most of the results of this section in Table 1. For each possibility for D and for each possible answer to the question “Is w ∈ D?”, we give all integers ≥ 2 which might be the genus of a new modular hyperelliptic curve over ℚ having that D and such a w. A bold number indicates that our computations have found a curve of that genus. The other numbers might not actually occur; in fact, most of them probably do not.

D w ∈ D? Potential values of g
{1} no 2, 3, 4, 5, 6, 7, 8, 9, 10
ℤ/2ℤ yes  2, 4, 6, 8, 10, 12, 14, 16
no  3, 7, 8, 9
ℤ/3ℤ no  3, 5
ℤ/4ℤ yes  2, 4, 6, 8, 10, 12, 14, 16
ℤ/6ℤ yes  2, 12, 14

Table 1. Possibilities for the diamond group and genus of a new modular hyperelliptic curve. See Theorem 1.9 and Propositions 6.18, 6.21, and 6.22.

6.4. Computational methods. Recall that [27] computed all new modular genus-two curves with ℚ-simple jacobian. Using similar reasoning, and using some of the sieves described there, we compute all new modular genus-two curves with jacobian not ℚ-simple: there are exactly 64 of such curves (see Table 4). In principle, we can also compute all the equations, levels and newforms for new modular hyperelliptic curves of genus g for each g > 2. But the enormous number of possibilities for the coefficients a_{p} prevents us from completing these computations in practice.

We demonstrate the method in the case of new modular hyperelliptic genus-3 curves with ℚ-simple jacobian. Let f be a corresponding newform. Then dim A_{f} is odd, so f has trivial Nebentypus ε. The only subfields of E_{f} are ℚ and E_{f} itself, so part (2) of Lemma 6.1 implies:

E_{f} = \begin{cases}ℚ(a_{2}) = ℚ(a_{3}) , & if π(∞) ∉ WP(X), \\ ℚ(a_{3}) = ℚ(a_{5}) (a_{2n} = 0) , & otherwise. \end{cases}

For simplicity, we outline the computation in the case where a_{2} ≠ 0 (or equivalently, π(∞) ∉ WP(X)). In this case, E_{f} = ℚ(a_{2}), so a_{2} ∉ {−1, 0, 1}, and (3.9) and (3.10) imply that 2 ∤ N.

(1) We determine all possible polynomials H_{2}(x) = ∏_{i=1}^{3}(x −^{σ_{i}}a_{2}), that is, all the monic irreducible cubic polynomials in ℤ[x] such that all zeros are real and of absolute value ≤ 2\sqrt{2}. In total, there are 80 such polynomials.

(2) For each H_{2}(x), we fix a zero a_{2}. Let M = 2g + 5 = 11, which is the bound in Proposition 2.8 plus 1 (since we multiply newforms by dq/q instead of dq). For every prime p satisfying 3 ≤ p ≤ M, the possible values of ε(p) and a_{p} are those such that ε(p) ∈ \{0, 1\} and a_{p} is an algebraic integer in ℚ(a_{2}) with |a_{p}| ≤ 2\sqrt{p} with respect to every archimedean absolute value. We may restrict the possibilities by imposing ℚ(a_{3}) = ℚ(a_{2}) and ε(2) = ε(3) = 1.

(3) For each possible f = q + ∑_{n=2}^{M} a_{n} q^{n} + O(q^{M+1}), we write f = g_{0} + a_{2}g_{1} + a_{2}^{2}g_{2} with g_{i} ∈ ℚ[[q]]. The g_{i} have the same span as the conjugates of f. Applying linear algebra to the g_{i}, we compute the basis {h_{1}, h_{2}, h_{3}} of part (1) of Lemma 6.1. Next we compute x̃ = h_{2}/h_{3} and h′_{1} = x̃^{2}h_{3}.

(4) We impose the condition h′_{1} ∈ ⟨h_{1}, h_{2}, h_{3}⟩ to sieve out some possibilities. Also we use this condition to extend the precision of h′_{1}, to determine the first M coefficients of h′_{1}. Next compute x = h_{2}/h′_{1} and y = (q dx/dq)/h′_{1}.

(5) We impose the condition that y^{2} = F (x) for a polynomial F of degree 8 without double roots. In the cases that survive, we compute F (x).

(6) Now we have a list of candidate curves. In principle, we should compute the conductor N of each jacobian and keep only those such that N := N^{1/3} ∈ ℤ and there exists a newform in S_{2}(X_{0}(N)) giving rise to y^{2} = F (x). In practice, since computing the conductor can be difficult, it is easier to try to recognize the candidates in a list of new modular curves calculated from newforms of small level (as discussed later in this subsection) and hope that all candidates show up.

Remark 6.24. If X is a new modular curve of level N, and p is a prime not dividing N, then Eichler-Shimura theory shows that the product P (x) of x^{2} − a_{p}x + pε(p) over all f ∈ S_{2}(X) ∩ New_{N} must equal the characteristic polynomial of Frobenius acting on the Tate module of the jacobian of the mod p reduction of X.

In the calculation above, we know 2 ∤ N, so we can compute the characteristic polynomials for all genus-3 hyperelliptic curves over 𝔽_{2}, in order to restrict the possibilities for H_{2}(x) in step (1). Moreover, we can restrict attention to the curves such that #X(𝔽_{2}) ≥ 1 (because of π(∞)) and such that #X(𝔽_{4}) > 2 (by Lemma 3.25).

Similar ideas restrict the possibilities for a_{3}, even when π(∞) ∈ WP(X), i.e., when N is even. For larger g, if it becomes too time-consuming to list all genus-g hyperelliptic curves over 𝔽_{3}, one can at least rule out certain a_{3} by translating the following into conditions on a_{3}:

• #X(𝔽_{3}) ≥ 1, because of the image of the cusp ∞,

• #X(𝔽_{3}) ≤ 8 and #X(𝔽_{9}) ≤ 20, since X is hyperelliptic,

• #X(𝔽_{3}) ≤ #X(𝔽_{9}),

• #X(𝔽_{9}) > 2g − 2, by Lemma 3.25, if Jac X is a quotient of J_{0}(N)^{new}.

The computation in steps (1) through (6) (of new modular hyperelliptic genus-3 curves with a_{2} ≠ 0 and with ℚ-simple jacobian) shows that only two such curves exist. These are given in Table 5 as C_{41}^{A} and C_{95}^{A}.

In the a_{2} = 0 case, the number of possibilities to analyze is considerably higher, because there are more possible values for a_{3}, hence more possibilities for E_{f}. Furthermore, in this case, Proposition 2.8 requires knowledge of a_{p} for primes p ≤ 4g + 6 (which is 18 for g = 3) in order to determine the relation y^{2} = F (x).

Performing all these computations would be extremely time-consuming. Therefore instead we conducted a search of all new modular hyperelliptic curves X of some bounded level. We used W. A. Stein’s Modular Symbols package to implement a program in Magma [8] that detects whether a set of newforms corresponds to a new modular hyperelliptic curve X and that computes an equation for this curve if so. This program, based on Proposition 6.5, was used to determine the ≥ entries in Table 2.

Remark 6.25. If there is a nonconstant morphism π(ε): X(N, ε) → X, then we compute the bound c as in Lemma 6.6 replacing g_{Y} by the genus of X(N, ε). If moreover ε = 1 and Jac X is ℚ-simple, then for each Atkin-Lehner involution W_{M} we have ^{σ}f|W_{M} = λ(M)^{σ}f for all σ, where λ(M) ∈ {−1, 1}. Let B′(N) = {W_{M} ∈ B(N) | λ(M) = 1}, where B(N) is the group of Atkin-Lehner involutions. Then X is dominated by X_{0}(N)/B′(N) and the bound c is computed taking g_{Y} as the genus of X_{0}(N)/B′(N). Note that B′(N) = B(N) or B(N)/B′(N) ≃ ℤ/2ℤ, so B′(N) = {id} if and only if N is a power of a prime with λ(N) = −1.

6.5. Computational results. Table 2 summarizes the results of the computations. Each entry indicates the number of new modular hyperelliptic curves over ℚ with the genus prescribed by the column, and with jacobian satisfying the conditions prescribed by the row heading. In the first (resp. second, third, fourth) row, a number following ≥ is the number of curves of that type of level ≤ 3000 (resp. 569, 2000, 569). We do not know if there are others of higher level. The zeros in the second row for odd g are from Lemma 6.17.

g = 2 g = 3 g = 4 g = 5 g = 6 g ≥ 7
ℚ-simple, #D = 1  120 ≥ 14 ≥ 13 ≥ 3 ≥ 0 ≥ 0
ℚ-simple, #D > 1 29 0 ≥ 1 0 ≥ 0 ≥ 0

not ℚ-simple, #D = 1 64 ≥ 32 ≥ 7 ≥ 0 ≥ 1 ≥ 0 not ℚ-simple, #D > 1 0 ≥ 12 ≥ 3 ≥ 2 ≥ 0 ≥ 0

Total  213 ≥ 58 ≥ 23 ≥ 5 ≥ 1 ≥ 0

Table 2. The number of new modular hyperelliptic curves of genus g.

Equations for the 120+29 genus-two curves with ℚ-simple jacobian have appeared in [27]. Equations for all the rest of the curves are in the tables of the appendix of this paper.

Of the 120 genus-two curves counted in row 1, only six have level > 3000: their levels are 3159, 4160, 7280, 7424 (twice), and 7664. Of the 64 genus-two curves counted in row 3, only two have level > 2000: both are of level 2208. Similarly, of the 29 genus-two curves counted in rows 2 and 4, only four have level > 569: their levels are 768 (twice), and 928 (twice). Moreover, for g ≥ 3, the largest level we found in the curves of row 1 (resp. row 3, resp. rows 2 and 4) was 1664 (resp. 944, resp. 512 even though we computed up to level 3000 (resp. 2000, resp. 569). This leads us to believe that the lower bounds in Table 2 are close to the exact numbers.

Finally, we note that all levels obtained when g > 2 have at most two different odd prime divisors and only the levels N = 36, 72, 144, 784 are divisible by the square of an odd prime.

7. New modular curves with trivial character 7.1. A finiteness result for curves with p in the level. The goal of this section is to prove the finiteness statement in Theorem 1.10. We first give some equivalent characterizations of what it means for a curve to have “trivial character”.

Lemma 7.1. Let X be a new modular curve of genus g ≥ 2, and let N be a positive integer. The following are equivalent:

(i) There exists a morphism π_{0} : X_{0}(N) → X with S_{2}(X) ⊆ S_{2}(X_{0}(N))^{new}.

(ii) There exists a morphism π_{1} : X_{1}(N) → X with S_{2}(X) ⊆ S_{2}(X_{0}(N))^{new}.

(iii) The jacobian J of X is a quotient of J_{0}(N)^{new}. We say that a new modular curve X has trivial character (and level N) if any of the above conditions are satisfied.

Proof. It is clear that (i) implies (iii). Furthermore, it follows from Lemma 2.11(i) that (ii) implies (i). It remains to show that (iii) implies (ii). Since X is a new modular curve, there exists an integer M and a morphism π: X_{1}(M) → X with S_{2}(X) ⊆ S_{2}(X_{1}(M)^{new}). Then J is ℚ-isogenous to a product of abelian varieties of the form A_{f} with each f a newform of level M. By assumption J is a quotient of J_{0}(N)^{new}, so J is ℚ-isogenous also to a product of abelian varieties of the form A_{f′} with each f′ a newform of level N and trivial Nebentypus character. Applying Proposition 3.2, we deduce that N = M and S_{2}(X) ⊆ S_{2}(X_{0}(N))^{new}. □

The following lemma will be used in the proof of Proposition 7.4 below, and will be used repeatedly in Section 7.2.

Lemma 7.2. Suppose X is a curve of genus g > 0 over a field k and that q is an analytic local uniformizing parameter at the point P ∈ X(k). Let ω_{1}, …, ω_{g} be a basis for H^{0}(X, Ω) with ω_{i} = (∑_{j=1}^{∞} a_{j}^{(i)}q^{j})\frac{dq}{q} and a_{1}^{(i)} = 1 for all i. Let m ≥ 2 be an integer, and suppose that

a_{m}^{(i)} = a_{m}^{(i′)} for all 1 ≤ i, i′ ≤ g. Then there exists a rational function f of degree m, defined over k, with poles only at P.

Proof. As usual, if D is a divisor on X, let l(D) denote the dimension of the vector space L(D) consisting of 0 and the rational functions on X whose divisor is ≥ −D. Let K denote a canonical divisor on X.

By hypothesis, if ω = (∑_{j=1}^{∞} a_{j}q^{j})\frac{dq}{q} is any linear combination of ω_{1}, …, ω_{g} with a_{1} = 0, then a_{m} = 0. Therefore no regular differential on X vanishes to order exactly m − 1 at P. In other words, we have l(K − (m − 1)P) = l(K − mP), which by Riemann-Roch is equivalent to l(mP) − l((m − 1)P) = 1. Any f ∈ L(mP) − L((m − 1)P) works. □

Corollary 7.3. With the same notation as in Lemma 7.2, we have:

(i) If a_{2}^{(i)} = a_{2}^{(i′)} for all 1 ≤ i, i′ ≤ g, then either g = 1, or X is hyperelliptic and P is a Weierstrass point.

(ii) If a_{2}^{(i)} = a_{2}^{(i′)} and a_{3}^{(i)} = a_{3}^{(i′)} for all 1 ≤ i, i′ ≤ g, then g = 1.

(iii) If g ≥ 2 and every differential in H^{0}(X, Ω) vanishing at P vanishes to order at least r at P, then r ≤ 2, and r = 2 if and only if X is hyperelliptic and P is a Weierstrass point.

Proof. Part (i) follows from Lemma 7.2, and (iii) follows from (i) and (ii). Therefore we need only prove (ii).

The argument of Lemma 7.2 yields rational functions in L(2P) − L(P) and in L(3P) − L(2P). Taking products, we find rational functions in L(mP) − L((m − 1)P) for all m ≥ 2.

By induction, we prove l(mP) ≥ m for m ≥ 1. If we take m large, Riemann-Roch implies g ≤ 1. □

Proposition 7.4. Let X be a new modular curve of level N and trivial character. If p is a prime divisor of N, then the ℚ-gonality of X is at most p^{2}.

Proof. The value of a_{p^{2}} is the same for each newform f = ∑_{n≥1} a_{n}q^{n} in New_{N}: it is 0 or 1, depending on whether p^{2} | N or not. (This follows from the more general statements (3.7), (3.8), (3.9), and (3.10) by taking ε to be the trivial Dirichlet character modulo N.) Also, recall from the proof of Theorem 1.3 that q serves as an analytic uniformizing parameter at the image P of the cusp ∞ under π: X_{1}(N) → X. The result therefore follows from Lemma 7.2. □

Proof of Theorem 1.10. Combine Proposition 7.4 with Theorem 1.8. □

7.2. Curves with level divisible by small primes. Theorem 1.10 shows that there are only finitely many new modular curves X with trivial character and level divisible by a given prime number p. In this section, we prove some further results about the levels of new modular curves with trivial character.

In addition to relying heavily on Lemma 7.2 and Corollary 7.3, we make use of a classical lemma of Castelnuovo and Severi. Before stating it, we make the following definition. If f_{1} : C → C_{1} and f_{2} : C → C_{2} are nonconstant morphisms (of degree d_{1}, d_{2}, respectively) between curves, we say that f_{1} and f_{2} are independent if the product morphism (f_{1}, f_{2}): C → C_{1} × C_{2} maps C birationally to its image C′. A consideration of degrees shows that if gcd(d_{1}, d_{2}) = 1, then f_{1} and f_{2} are automatically independent.

Lemma 7.5. Let C_{1}, C_{2}, C be curves of genera g_{1}, g_{2}, and g, respectively, over the field k of characteristic zero. Let f_{i}: C → C_{i} be morphisms of degree d_{i}, i = 1, 2. Assume that f_{1} and f_{2} are independent. Then

g ≤ (d_{1} − 1)(d_{2} − 1) + d_{1}g_{1} + d_{2}g_{2}.

Proof. See Theorem 3.5 of [3]. □

Remark 7.6. Lemma 7.5 is true over any field k, but the proof in [3] assumes that k has characteristic zero. See Exercise VIII.C-1 in [5] for an approach that can be generalized to any characteristic.

We now investigate some restrictions on the powers of 2 and 3 dividing the level of new modular curves with trivial character.

Proposition 7.7. Let X be a new modular curve over ℚ of genus g ≥ 2, level N, and trivial character. Then:

(i) If 2 | N, then X admits a map of degree 2, defined over ℚ, to a curve of genus at most 1, and g ≤ 16.

(ii) 4 | N if and only if X is hyperelliptic and π(∞) is a Weierstrass point.

(iii) If 6 | N, then g ≤ 5.

(iv) If 12 | N, then g = 2.

(v) If 18 | N, then g ≤ 4.

(vi) 36 ∤ N.

Proof. Let f_{1}, …, f_{g} be a basis of newforms for Γ_{0}(N) spanning S_{2}(X).

To prove (ii), first note that, as in the proof of Theorem 1.3, the map π is unramified at ∞, so that q serves as an analytic uniformizing parameter at P := π(∞). If 4 | N, then a_{2}(f_{i}) = 0 for all i by (3.9), so it follows from Corollary 7.3 that X is hyperelliptic and P is a Weierstrass point. The other direction of (ii) is a special case of Proposition 6.7.

Before proving (i), note that if 2 | N, then Proposition 7.4 implies that the ℚ-gonality G_{ℚ} of X satisfies G_{ℚ} ≤ 4. Remark 4.5 then implies that g ≤ 34. We seek to sharpen both of these statements.

If X is hyperelliptic, then Theorem 1.9(ii) implies g ≤ 10. Therefore we may suppose that X is not hyperelliptic. We may then assume by (ii) that 4 ∤ N, so that it makes sense to consider the Atkin-Lehner involution W_{2} on X.

Let X′ = X/W_{2}, let P′ be the image in X′ of P under the natural map, and let f′_{1}, …, f′_{g′} be a basis of newforms spanning S_{2}(X′). Then since f_{i}|W_{2} = −a_{2}(f_{i})f_{i} by (3.12), we have a_{2}(f′_{i}) = −1 for all i. Therefore every regular differential on X′ vanishing at P′ vanishes to order at least 2 at P′. We claim that X′ has genus 1. Indeed, if the genus of X′ were at least 2, then Corollary 7.3 would then imply that X′ is hyperelliptic and that P′ is a Weierstrass point, but then we find by applying (ii) to X′ that 4 | N, contrary to what we just assumed. Also, X′ cannot have genus 0 or else X would be hyperelliptic, contrary to assumption. Thus X′ has genus 1 as claimed, and X is a degree 2 cover (over ℚ) of the elliptic curve X′.

If 3 ∤ N, then we have #X(𝔽_{9}) ≤ 2#X′(𝔽_{9}) ≤ 2(9+1+2\sqrt{9}) = 32. But also (3−1)(g−1) < #X(𝔽_{9}) by Lemma 3.25. Thus g ≤ 16.

Finally, if 3 | N, then the conclusion g ≤ 16 follows from the stronger conclusion of (iii) proved below.

To prove (v), we suppose that 18 | N. It follows by Theorem 1.9(ii) that if X is hyperelliptic then g < 3. Thus by (i), we may assume that X is a double cover of an elliptic curve X′.

Since 9 | N, we have a_{3}(f_{i}) = 0 for all i, so Lemma 7.2 implies that X is trigonal (i.e., X admits a degree 3 map to ℙ^{1}). It therefore follows from Lemma 7.5 that g ≤ 4.

To prove (iii), we may assume by (v) that 3 | N but 9 ∤ N. As before, we may also assume that 2 | N but 4 ∤ N, that X is not hyperelliptic, and that X is a double cover of the elliptic curve X′ = X/W_{2}.

Let w_{j}(f_{i}) denote the eigenvalue of W_{j} on the newform f_{i} for j = 2, 3, 6 and i = 1, …, g. We have w_{2}(f_{i}) = −a_{2}(f_{i}), w_{3}(f_{i}) = −a_{3}(f_{i}), and w_{6}(f_{i}) = a_{6}(f_{i}) for all i. Let g_{j} be the genus of X/W_{j} (j = 2, 3, 6), so g_{j} = #{ i | w_{j}(f_{i}) = +1 }. Since X/W_{2} has genus 1, all the w_{2}(f_{i})’s are equal to −1 except for one, which is equal to +1. Suppose that f is the newform on which W_{2} acts with eigenvalue +1. The identity w_{2}w_{3} = w_{6} and the fact that the w_{2}’s are +1, −1, −1, −1, …, −1 implies that g_{3} + g_{6} is equal to either g − 1 or g + 1, depending on whether f|W_{3} equals −f or +f. We consider these two cases separately.

\underline{Case 1}: f|W_{3} = −f.

If g_{3} ≥ 2, then X/W_{3} is a curve of genus at least 2 with all w_{2}’s equal to −1, so as before we would have 4 | N, contrary to assumption. Thus g_{3} ≤ 1. Similarly, g_{6} ≤ 1. But g_{3} + g_{6} = g − 1, so g ≤ 3.

\underline{Case 2}: f|W_{3} = f.

In this case, we have g_{3} ≥ 1. Also f|W_{6} = f, so g_{6} ≥ 1. Assume first that g_{3} = 1. Then w_{2}(f_{j}) = w_{3}(f_{j}) for all j, so a simple argument using Riemann-Roch (as in the proof of Lemma 7.2) shows that there is a rational function on X in L(3P) − L(2P). Since (as in the proof of Proposition 7.4) all a_{4}’s are equal to +1, there is also a function in L(4P) − L(3P). Taking products, we also find functions in L(mP) − L((m − 1)P) for all m ≥ 6, so that the Weierstrass gap sequence at P is contained in {1, 2, 5}. By Riemann-Roch, the gap sequence at any point contains exactly g integers, so g ≤ 3.

Assume now that g_{6} = 1. Then w_{2}(f_{j}) = w_{6}(f_{j}) for all j, so w_{3}(f_{j}) = 1 for all j. Corollary 7.3 gives a function in L(3P) − L(2P), and as before we deduce that g ≤ 3.

In the general case, we may apply the previous reasoning to X/W_{3}, since (X/W_{3})/W_{6} is of genus 1 (its only newform is f). The conclusion is that g_{3} ≤ 3. Similarly, since (X/W_{6})/W_{3} has genus 1, we conclude that g_{6} ≤ 3. Since g_{3} + g_{6} = g + 1 in Case II, we have g ≤ 5 as desired.

To prove (vi), suppose that 36 | N. Then a_{2}(f_{i}) = a_{3}(f_{i}) = 0 for all i. By the same argument as in the proof of (ii), it follows that every differential on X vanishing at P vanishes to order at least 3 at P, contradicting Corollary 7.3.

For (iv), assume 12 | N. Then X is hyperelliptic by part (ii). By Theorem 1.9(ii), it follows that g = 2 as desired. □

We can deduce stronger results if we assume furthermore that the jacobian of X is ℚsimple.

Proposition 7.8. Let X be a new modular curve of genus g ≥ 2, level N, and trivial character. Assume furthermore that the jacobian J of X is ℚ-simple. Then:

(i) If 2 | N, then 4 | N.

(ii) 6 ∤ N.

Proof. Let f_{1}, …, f_{g} as before be a basis of newforms for Γ_{0}(N) spanning S_{2}(X). Since J is ℚ-simple, the f_{i}’s are all Galois conjugates of one another. To prove (i), suppose 2 | N but 4 ∤ N. Then a_{2}(f_{i}) = ±1 for all i. Since in particular a_{2}(f_{i}) ∈ ℚ, all the a_{2}(f_{i})’s must be equal to one another. It follows by Corollary 7.3 that X is hyperelliptic and P is a Weierstrass point. But then 4 | N by Proposition 7.7(ii), contradicting our assumption.

To prove (ii), suppose that 6 | N. By (i) we have in fact that 12 | N. Also, we know by Proposition 7.7(vi) that 36 ∤ N. Therefore we have a_{2}(f_{i}) = 0 and a_{3}(f_{i}) = ±1 for all i. As above, it follows that all the a_{3}(f_{i})’s are in fact equal, contradicting Corollary 7.3.

□

8. Examples and Pathologies

8.1. Examples of non-new hyperelliptic curves. We can construct non-new modular hyperelliptic curves from certain pairs of new modular hyperelliptic curves. For some levels, there exist at least two new modular hyperelliptic curves of level N such that it is possible to take the same modular function x in both their equations. In other words, ℚ(x, \sqrt{P_{1}(x)}, \sqrt{P_{2}(x)}) is a subfield of the function field of X_{1}(N), where the two new modular curves are

C_{N,1} : y_{1}^{2} = P_{1}(x) , C_{N,2} : y_{2}^{2} = P_{2}(x).

Then ℚ(x, \sqrt{P_{1}(x)P_{2}(x)}) also is a subfield of the function field of X_{1}(N), so y^{2} = P_{1}(x)P_{2}(x) is a modular hyperelliptic curve C′_{N} dominated by X_{1}(N). (If desired, one may discard square factors of P_{1}(x)P_{2}(x) to make the right hand side squarefree.) In each example we give, C′_{N} is not new of level N, although sometimes it is new of a smaller level.

We found 33 non-new modular curves C′_{N} of this type. Five of these 33 are not primitive (cf. definition 2.2 in [27]): this means that the minimum N′ such that JacC′_{N} is a quotient of J_{1}(N′) is different from the minimum M such that C′_{N} is dominated by X_{1}(M). The following table shows these five curves.

C′_{N} Decomposition of JacC M g′
C_{184}^{3,3} A_{23A} = J_{0}(23)  46 2
C_{248}^{2,3} A_{31A} = J_{0}(31)  62 2
C_{376}^{2,2} A_{47A} = J_{0}(47)  94 4
C_{544}^{2,3} E_{34A} × A_{68A}  136 3
C_{704}^{3,4} E_{44A} × E_{88A} × A_{88B}  176 4

In the first column appears the label for each curves. For each curve the subscript denotes the level N and the superscript denotes the genus of the curves C_{N,1} and C_{N,2}. In the second column appears the decomposition over ℚ of the jacobians of the curves in the first column (throughout this section we use the labelling of modular forms and abelian varieties described in the appendix). In the third column appears the minimum level M such that X_{1}(M) dominates C′_{N}, and in the last column appears the genus of C′_{N}.

In the rest of this subsection, we discuss the surprising case N = 376. We have that J_{0}(376)^{new} splits over ℚ as the product of the ℚ-simple modular abelian varieties A_{376A}, A_{376B}, A_{376C} and A_{376D}. Furthermore, each of these abelian varieties is ℚ-isogenous to the jacobian of a new modular hyperelliptic curve. Namely, we have

A_{376A} ∼^{ℚ} Jac C_{376}^{A} , A_{376B} ∼^{ℚ} Jac C_{376}^{B} , A_{376C} ∼^{ℚ} Jac C_{376}^{C} , A_{376D} ∼^{ℚ} Jac C_{376}^{D} ,

where the four curves have the following affine models:

\begin{matrix}C_{376}^{A}: y_{A}^{2} = P_{A}(x), & C_{376}^{C}: y_{C}^{2} = P_{B}(x)Q(x), \\ C_{376}^{B}: y_{B}^{2} = P_{B}(x), & C_{376}^{D}: y_{D}^{2} = P_{A}(x)Q(x), \end{matrix}

where

\begin{aligned}x & = & q^{−2} + q^{2} + q^{6} + q^{8} + q^{10} + …, \\ P_{A}(x) & = & x^{5} − x^{3} + 2x^{2} − 2x + 1 , \\ P_{B}(x) & = & x^{5} + 4x^{4} + 3x^{3} − 2x^{2} + 2x + 5 , \\ Q(x) & = & x^{4} − 2x^{3} − 3x^{2} + 4x − 4 . \end{aligned}

Since the modular function x is the same for all four curves, X_{0}(376) also dominates the hyperelliptic curves C_{376}^{2,2} : (y_{A}y_{B})^{2} = P_{A}(x)P_{B}(x) and C_{376}^{2,4} : (y_{B}y_{D})^{2} = P_{A}(x)P_{B}(x)Q(x) (and the genus-one curve (y_{B}y_{C}/P_{B}(x))^{2} = Q(x) isomorphic to the curve labelled 94A1 in [17]). In addition, X_{0}(376) dominates X_{0}(47), which also is hyperelliptic. We will identify the smallest levels of C_{376}^{2,2} and C_{376}^{2,4}.

First consider C_{376}^{2,2}. Let f be the newform labeled by 47A. It is easy to check that there is a modular hyperelliptic curve X over ℚ of level 94 attached to the ℂ-vector space spanned by the Galois conjugates of the eigenform h(q) = f(q) − 2f(q^{2}) ∈ S_{2}(94) and that X is the curve C_{376}^{2,2}. Therefore, JacC_{376}^{2,2} and J_{0}(47) are ℚ-isogenous. But one can check that C_{376}^{2,2} and X_{0}(47) are not ℚ-isomorphic. Hence 94 is the minimum level for C_{376}^{2,2} while 47 is the minimum level for its jacobian.

Now consider C_{376}^{2,4}. We compute that JacC_{376}^{2,4} ∼^{ℚ} J_{0}(47)×J_{0}(94)^{new}. More precisely, C_{376}^{2,2} is a non-new modular curve of minimum level 94: it is attached to the ℂ-vector space generated by the Galois conjugates of f(q) + 2f(q^{2}) and g, where f is as above and g is the newform labeled by 94B with dim A_{94B} = 2. The curve C_{376}^{2,4} has genus 6, and its jacobian has one factor in the new part and another in the old part.

Finally, we note that the highest genus yet found for a modular hyperelliptic curve is 7: the curve is given by the equation

C_{1664}^{2,5} : y^{2} = (x^{2} − 2x + 2)(x^{2} + 2x + 2)(x^{3} − x + 2)(x^{3} − x − 2)(x^{6} + 2x^{4} + x^{2} + 4) ,

and has level N = 1664. Its jacobian splits as JacC_{1664}^{2,5} ∼^{ℚ} E_{26A} × A_{104B} × A_{416F}.

8.2. “Pathologies”. There are many statements about elliptic curve quotients of J_{1}(N) that fail for modular curves of higher genus or for higher-dimensional abelian quotients of J_{1}(N). We begin with a statement about abelian quotients.

Proposition 8.1. There exists an abelian variety A over ℚ such that the following statement is false: “If A is a quotient of J_{1}(N) and N is the smallest integer with this property, and M is any other integer such that A is a quotient of J_{1}(M), then N divides M.”

Proof. Take A = J_{0}(22). One computes that there are no weight 2 newforms on Γ_{0}(22), so A ∼^{ℚ} J_{0}(11)^{2}. Now A is not a quotient of J_{1}(M) for any M prime to 11 because A has bad reduction at 11, but A is also not a quotient of J_{1}(11), since dim J_{1}(11) = 1. Hence N = 22 is the smallest integer for which A is a quotient of J_{1}(N). On the other hand, (3.4) shows that A is also a quotient of J_{1}(11m) for any m ≥ 2. □

Next we list a few false statements about modular curves.

Proposition 8.2. For each of the statements below, there is a curve X over ℚ of genus g ≥ 2 for which the statement is false. Let J = JacX.

(1) “If X is modular, then X is a new modular curve of some level.”

(2) “If X is non-new modular of level N, then J is a quotient of J_{1}(N)_{old}.”

(3) “If X is modular and N is the smallest level such that J is a quotient of J_{1}(N) defined over ℚ, then N is the smallest level of X.”

(4) “If X is modular, then the smallest level of X and cond(J) are divisible by the same rational primes.”

(5) “If J is a ℚ-simple quotient of J_{1}(N)^{new} and X(ℚ) is nonempty, then X is a new

modular curve of level N.” In fact, there exist infinitely many X over ℚ of genus 2 for which (5) fails.

Proof.

(1) Let X = X_{1}(p^{2}) where p ≥ 11 is prime. It follows from an explicit formula for the genus of X_{1}(N) (see Example 9.1.6 of [20]) that 0 < 2g(X_{1}(p)) < g(X_{1}(p^{2})). Therefore, dim J_{1}(p^{2})^{old} = 2g(X_{1}(p)) > 0 and dim J_{1}(p^{2})^{new} = g(X_{1}(p^{2})) − dim J_{1}(p^{2})^{old} > 0. Now, by Proposition 3.2 we obtain that if J_{1}(p^{2})^{new} is a quotient of J_{1}(N)^{new} over ℚ then N must be p^{2}. Hence, X is not new for any level because J_{1}(p^{2})^{old} is not a quotient of J_{1}(p^{2})^{new}.

(2) Let X = X_{1}(p^{2}) for prime p ≥ 11 again. Then X is modular of level p^{2} and not new of any level, but JacX is not a quotient of J_{1}(p^{2})^{old}.

(3) Let X be the genus 4 curve C_{376}^{2,2} constructed above. Then JacX is a quotient of J_{0}(47) (they are isogenous), but the smallest N for which X is modular of level N is 94.

(4) Again let X = C_{376}^{2,2}. Then cond(JacX) = 47^{4} but the smallest N for which X is modular of level N is 94. A simpler example is the genus 2 curve X = X_{0}(22): then cond(JacX) = 11^{2}, but the smallest N for which X is modular of level N is 22.

(5) Let a ∈ ℤ − {0}, and let X_{a} be the smooth projective model of the affine curve

y^{2} = x^{5} + ax^{3} − 4x

over ℚ of genus two. Then X_{a}(ℚ) ≠ ∅, since (0, 0) ∈ X_{a}(ℚ). We will show that for infinitely many values of a, the jacobian J_{a} := JacX_{a} is a ℚ-simple quotient of some J_{1}(N). It can be checked that the group Aut(X_{a}) is isomorphic to D_{2·4} and is generated by the hyperelliptic involution and the automorphisms u, v over ℚ(i), represented as in Lemma 6.12(1) by

M_{u} = (\begin{matrix}0 & 1 + i \\ (1 − i)/2 & 0 \end{matrix}), M_{v} = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) (and e = det M). It can be checked that the

Galois action on Aut(X_{a}) corresponds to the case C_{2}^{C} of section 3 of [13]. By Proposition 5.3 of [13], we obtain (End J_{a}) ⊗ ℚ ≃ ℚ(\sqrt{2}). In particular, J_{a} is ℚ-simple. On the other hand, the quotient E_{a} = X_{a}/u is the elliptic curve

y^{2} = x^{3} + 27(3a − 20i)x − 108(28 + 9ai)(1 − i) ,

whose j-invariant j_{a} equals 64(3a − 20i)^{3}(a − 4i)/(a^{2} + 16)^{2}. Thus J_{a} is isogenous to the Weil restriction of E_{a} over ℚ. Solving j_{a} = \overline{j}_{a} for a shows that j_{a} is never real, so E_{a} does not have CM. Thus (End J_{a}) ⊗ ℚ is a real quadratic field, and the sign of the 2-cocycle attached to E_{a} by Ribet in [66] is trivial (see Theorem 5.4 of [62]). In particular, its local component at 3 is trivial. For every nonzero a ∈ ℤ, E_{a} has good ordinary reduction at 3, so Theorem 5.1 of [21] implies that E_{a} is modular (over ℚ(i)), so J_{a} is modular. Since J_{a} is ℚ-simple, it must be a quotient of J_{1}(N)^{new} for some N depending on a. Since j_{a} is nonconstant, the family X_{a} is not isotrivial. (In fact, it can be proved that X_{a} and X_{b} are isomorphic over \overline{ℚ} if and only if a = ±b.) Hence by Theorem 1.3, at most finitely many of the X_{a} can be new modular curves. (In fact, X_{a} is a new modular curve exactly for a = ±1, ±2, ±3, ±4, ±5, ±7, ±8, ±10, ±22.) □

Question 8.3. Is there a curve analogue of Proposition 8.1? More precisely, does there exist a curve X dominated by some X_{1}(N), but such that the set of such N for X are not all multiples of the smallest N?

9. Curves dominated by Fermat curves

Here we prove an analogue of Conjecture 1.1 for Fermat curves. Let k be a field of characteristic zero and let N ≥ 1. The N^{th} Fermat curve X_{N,k} is the smooth plane curve over k given by the homogeneous equation x^{N} + y^{N} = z^{N} in ℙ_{k}^{2}. Our goal in this section is to prove the following.

Theorem 9.1. For each field k of characteristic zero, and integer g ≥ 2, the set of genus-g curves over k dominated by X_{N,k} for some N ≥ 1 is finite. If k is a number field or \overline{ℚ}, then the set is also computable.

Let J_{N} denote the jacobian of X_{N,k}. For each M|N, there is a morphism ψ_{N,M}: X_{N,k} → X_{M,k} mapping (x : y : z) to (x^{N/M} : y^{N/M} : z^{N/M}). Define J_{N,old} = ∑_{M|N} ψ_{N,M}^{∗}J_{M}, and J_{N}^{new} = J_{N}/J_{N,old}. We may consider X_{M,k} as the quotient of X_{N,k} by a G_{k}-stable subgroup Γ_{N,M} ⊆ Aut(X_{N,\overline{k}}). If we decompose the space H^{0}(X_{N,\overline{k}}, Ω) into characters of Γ_{N,1}, and group those for which a particular divisor M is the smallest M for which Γ_{N,M} acts trivially, then we see that J_{N} is k-isogenous to ∏_{M|N} J_{M}^{new}.

Lemma 9.2. If N > 180, then each k-simple quotient of J_{N}^{new} has dimension at least ϕ(N)/8, where ϕ(N) := #(ℤ/Nℤ)^{∗}.

Proof. We may assume k = ℂ. According to Theorem 1.3 of [4], J_{N}^{new} ∼ ∏ A_{S}, where each A_{S} is an abelian variety of dimension ϕ(N)/2, and S ranges over the elements of a certain finite set Σ. Moreover, if N is not in an explicit finite set of natural numbers (whose largest element is 180), then Theorem 0.2 of [4] shows that for each S ∈ Σ, A_{S} = B_{S}^{W_{S}}, where B_{S} is a simple abelian variety and W_{S} ≤ 4. The result follows. □

Lemma 9.3. There exists a function M = M(g) such that the following holds: If k is a field of characteristic zero, and X is a curve of genus g ≥ 2 over k dominated by X_{N,k} for some N ≥ 1, then X is dominated also by X_{M′,k} for some M′ ≤ M.

Proof. Choose m > 180 such that ϕ(n)/8 > g for all n > m. Let M = m!. Suppose X is a curve of genus g ≥ 2 over k dominated by X_{N,k} for some N ≥ 1. Let M′ = gcd(M, N). By Lemma 9.2, the composition JacX → J_{N} → J_{m′}^{new} is trivial for each m′|N greater than m, so the image of JacX → J_{N} is contained in an abelian subvariety of J_{N} isogenous to ∏_{m′|N,m′≤m} J_{m′}^{new}, which in turn is contained in the image of J_{M′} → J_{N}. Considering differentials and applying Proposition 2.11(i), we find that the morphism X_{N,k} → X factors through X_{M′,k}. □

Proof of Theorem 9.1. Use Lemma 9.3, and then apply Theorem 5.7 (our de Franchis-Severi Theorem) to X_{M′,k} for each M′ ≤ M(g) to obtain a finite list of curves dominated by Fermat curves over k, guaranteed to contain all those of genus g. If we now assume that k is a number field or \overline{ℚ}, then this list is also computable. To eliminate curves of genus not g, and to eliminate possible redundancy, apply parts (1) and (3) of Lemma 5.1. □

Remark 9.4.

(i) If k is a number field, we can obtain finiteness and computability also for dominated curves of genus ≤ 1 in Theorem 9.1. This follows from Remark 5.9: the genus-1 isomorphism problem is not a problem here, because each dominated curve has a rational point, thanks to the point (1 : 0 : 1) on each X_{N,k}.

(ii) The natural analogue of Theorem 9.1 for bounded gonality is false. In affine coordinates, (x, y) ↦ (xy, y^{N}) defines a dominant morphism from x^{N} + y^{N} = 1 to the hyperelliptic curve u^{N} = v(1 − v). This gives infinitely many Fermat-dominated curves of gonality 2. These curves are even new, if N is prime. Perhaps the failure of the gonality result is not surprising, since in contrast with Theorem 4.3, the gonality of high-degree smooth plane curves is small compared to the genus.

Question 9.5. Are there finiteness results for curves over ℚ of fixed genus g ≥ 2 that are dominated by curves in some other families, such as the family of Shimura curves associated to orders in a fixed or varying indefinite quaternion algebra over ℚ?

Appendix

Labeling. We use a deterministic procedure to label newforms, and in particular to fix an ordering of (Galois conjugacy classes of) newforms having a given level and Nebentypus. An ordering was introduced by J. Cremona in [17] in the case of trivial Nebentypus and weight 2 and generalized by W. A. Stein in [71] to the case of weight greater than 2. We are going to use this last labeling for the case of weight 2 and arbitrary Nebentypus.

We will define a function mapping each newform f ∈ ⋃_{N=1}^{∞} New_{N} to a label of the form NX_{ε} (for example, 13A_{{2}}), where N is the level of f, where X is a letter or string in {A, B, …, Z, AA, BB, … } and ε is (some encoding) of the Nebentypus of f. If ε = 1, we omit the subscript ε and use a label of the form NX (for example, 11A), and if the Fourier coefficients of f are integers we will use the labeling in [17]. The labeling function will not be injective: our definition will be such that if f has label NX_{ε} and σ ∈ G_{ℚ}, then ^{σ}f will have label NX_{σ(ε)}, which could be the same as NX_{ε}, even if ^{σ}f ≠ f.

Two things must be explained: how is X constructed from f, and how is ε encoded? First we construct X. Fix N and ε. To f = ∑ a_{n}q^{n} ∈ New_{N} ∩ S_{2}(N, ε) associate the infinite sequence of integers t_{f} = (Tr_{E_{f} /ℚ} a_{1}, Tr_{E_{f} /ℚ} a_{2}, …), where E_{f} is the number field ℚ(a_{1}, a_{2}, …). Choose X ∈ {A, B, …, Z, AA, BB, … } according to the position of t_{f} in the set { t_{g} : g ∈ New_{N} ∩ S_{2}(N, ε) } sorted in increasing dictionary order. Notice that t_{f} determines the Galois conjugacy class of f.

The encoding of Dirichlet characters we now describe was suggested by J. Quer. Suppose ε: (ℤ/Nℤ)^{∗} → ℂ^{∗} is a Dirichlet character. Let N = ∏ p_{n}^{α_{n}} be the prime-ordered factorization. Then there exist unique ε_{p_{n}}: (ℤ/p_{n}^{α_{n}} ℤ)^{∗} → ℂ^{∗} such that ε = ∏ ε_{p_{n}}. If p is an odd prime, let g_{p} be the smallest positive integer that generates (ℤ/p^{α} ℤ)^{∗}, and if p = 2 and α ≤ 2, let g_{p} = −1; in these cases ε_{p} is determined by the integer e_{p} ∈ [0, φ(p^{α})) such that ε_{p}(g_{p}) = e^{2πie_{p}/φ(p^{α})}. If p = 2 and α > 2, then ε_{2} is determined by the integers e′_{2}, e′′_{2} ∈ [0, φ(2^{α})) such that ε_{2}(−1) = e^{2πie′_{2}/φ(2^{α})} and ε_{2}(5) = e^{2πie′′_{2}/φ(2^{α})}, and we write e_{2} = {e′_{2}, e′′_{2}}. (Note: here and in the next sentence, we use set notation although we mean sequences.) Assuming that N is implicit, we denote ε by {e_{p} : p|N}.

If f ∈ S_{2}(N, ε) is a newform with label NX_{ε}, then A_{NX_{ε}} will denote the corresponding modular abelian variety A_{f}, except that when dim A_{f} = 1, we instead follow the labeling in [15] and use the letter E instead of A to denote the modular elliptic curve A_{f}.

We give an example illustrating the above notation: the 2-dimensional space S_{2}(13) splits as S_{2}(13, ε) ⊕ S_{2}(13, ε^{−1}), where ε = {2} is a Dirichlet character modulo 13 (of order 6), and there is only one Galois conjugacy class of newforms of level 13. Thus J_{1}(13) ∼^{ℚ} A_{13A_{{2}}}.

Each new modular hyperelliptic curve in our tables will be denoted C_{NL_{1},...,L_{n}_{ε}}^{M_{1},...,M_{m}} where N is the level, the L_{i} and M_{j} are letters (or in one case, the string AA) indicating the simple factors of the jacobian J of the curve, and ε is (the label of) a Dirichlet character (ℤ/Nℤ)^{∗} → ℂ^{∗}. More precisely, the notation indicates that J is isogenous to the product of the ℚ-simple modular abelian varieties with the following labels: A_{NM_{j}} for each superscript M_{j} and A_{NL_{i}_{ε}} for each subscript L_{i}. (It turns out that in all known cases, there is at most one Galois conjugacy class of nontrivial characters ε involved.)

Tables. Table 4 shows the 64 new modular curves of genus two with jacobian not ℚ-simple. (The 149 new modular curves of genus two with ℚ-simple jacobian were already listed in [27].) Tables 5, 6 and 7 show the 30 new modular hyperelliptic curves of genus > 2 with ℚ-simple jacobian, trivial character, and level ≤ 3000, grouped by their genus. Recall that by Proposition 1.3 of [68], a new modular curve with ℚ-simple jacobian and nontrivial character must have even genus; this explains why we may omit the hypothesis #D = 1 in Tables 5 and 7.

Tables 8, 9 and 10 show the 40 new modular hyperelliptic curves that have genus > 2 and level ≤ 2000 such that the jacobian is a quotient of J_{0}(N)^{new} that is not ℚ-simple. Table 11 shows the five curves with genus 3 and #D = 3. Table 12 contains all new modular hyperelliptic curves with 2 | #D, w ∉ D and genus 3 ≤ g ≤ 4. All of them have genus 3 and their jacobians are neither ℚ-simple nor a ℚ-factor of J_{0}(N)^{new}. Table 13 contains the remaining curves with #D > 1 and level ≤ 569. These curves have #D = 2, 3 or 4.

Remark 9.6. By inspection, the first curve in Table 13, namely C_{52A,B_{{0,4}}}^{A}, has a degree-2 unramified cover that dominates y^{2} = x^{6} + 4x^{5} + 6x^{4} + 2x^{3} + x^{2} + 2x + 1, which is a equation for X_{1}(13). The fourth curve, C_{208A,D_{{{0,0},4}}}^{C}, similarly has a degree-2 unramified cover that dominates X_{1}(13). The second and third curves in Table 13, namely C_{160A,E_{{{1,0},1}}} and C_{160B,F_{{{1,0},1}}}, have degree-2 unramified covers dominating y^{2} = p(x) and y^{2} = −p(−x), respectively, where p(x) = (x − 1)(x^{2} − 2x + 2)(x^{2} − x − 1); the latter two curves are the new modular curves C_{160,C} and C_{160,D} of genus 2 from [27]. We do not have a good explanation for this phenomenon.

Table 4: Not ℚ-simple, g = 2

C  : y^{2} = F (x)
C_{26}^{A,B} : y^{2} = x^{6} + 4x^{5} − 12x^{4} − 114x^{3} − 308x^{2} − 384x − 191
C_{37}^{A,B} : y^{2} = x^{6} − 4x^{5} − 40x^{4} + 348x^{3} − 1072x^{2} + 1532x − 860
C_{50}^{A,B} : y^{2} = x^{6} + 2x^{5} − 5x^{4} − 30x^{3} − 55x^{2} − 48x − 16
C_{54}^{A,B} : y^{2} = x^{6} − 34x^{3} + 1
C_{56}^{A,B} : y^{2} = x^{5} + 6x^{4} − 45x^{3} − 490x^{2} − 1503x − 1564
C_{58}^{A,B} : y^{2} = x^{6} − 2x^{5} + 11x^{4} − 22x^{3} + 21x^{2} − 12x + 4
C_{66}^{A,B} : y^{2} = x^{6} + 2x^{5} − 5x^{4} − 22x^{3} − 31x^{2} − 24x − 8
C_{80}^{A,B} : y^{2} = x^{5} + 2x^{4} − 26x^{3} − 132x^{2} − 231x − 142
C_{84}^{A,B} : y^{2} = x^{5} + 4x^{4} − 25x^{3} − 172x^{2} − 339x − 222
C_{90}^{A,B} : y^{2} = x^{6} − 18x^{3} + 1
C_{91}^{A,B} : y^{2} = x^{6} + 2x^{5} − x^{4} − 8x^{3} − x^{2} + 2x + 1
C_{96}^{A,B} : y^{2} = x^{5} − 34x^{3} + x
C_{112}^{A,C} : y^{2} = x^{5} − 2x^{4} + 10x^{3} − 16x^{2} + 21x − 14
C_{112}^{A,B} : y^{2} = x^{5} − 6x^{4} − 45x^{3} + 490x^{2} − 1503x + 1564
C_{128}^{B,D} : y^{2} = x^{5} − 24x^{3} + 16x
C_{128}^{A,C} : y^{2} = x^{5} + 24x^{3} + 16x
C_{138}^{A,C} : y^{2} = x^{6} + 8x^{4} + 6x^{3} + 8x^{2} + 1
C_{142}^{B,D} : y^{2} = x^{6} − 2x^{5} − 5x^{4} + 18x^{3} − 19x^{2} + 12x − 4
C_{160}^{A,B} : y^{2} = x^{5} + 12x^{3} + 16x

C : y^{2} = F (x)^{C_{162}^{A,D} : y^{2} = x^{6} + 14x^{3} + 1}^{C_{162}^{B,C} : y^{2} = x^{6} − 10x^{3} + 1}^{C_{184}^{C,D} : y^{2} = x^{5} − 10x^{3} − 15x^{2} − 9x − 7}^{C_{189}^{A,C} : y^{2} = x^{6} − 12x^{4} + 36x^{3} − 48x^{2} + 36x − 12}^{C_{189}^{A,B} : y^{2} = x^{6} − 12x^{4} + 12x^{3} + 24x^{2} − 36x + 12}^{C_{192}^{C,D} : y^{2} = x^{5} − 14x^{3} + x}^{C_{192}^{B,D} : y^{2} = x^{5} + 4x^{4} − 6x^{3} − 58x^{2} − 111x − 70}^{C_{192}^{A,C} : y^{2} = x^{5} − 4x^{4} − 6x^{3} + 58x^{2} − 111x + 70}^{C_{192}^{A,B} : y^{2} = x^{5} + 34x^{3} + x}^{C_{200}^{C,E} : y^{2} = x^{5} − 10x^{3} − 15x^{2} + 8}^{C_{240}^{C,D} : y^{2} = x^{5} − 2x^{4} + 6x^{3} − 13x^{2} + 12x − 4}^{C_{256}^{A,D} : y^{2} = x^{5} + 16x}^{C_{264}^{A,B} : y^{2} = x^{5} + 2x^{4} − 6x^{3} − 23x^{2} − 24x − 8}^{C_{312}^{B,C} : y^{2} = x^{5} − 2x^{4} − x^{3} + 8x^{2} − 9x + 3}^{C_{320}^{A,C} : y^{2} = x^{5} − 2x^{4} − 2x^{3} − 2x^{2} + x}^{C_{320}^{D,E} : y^{2} = x^{5} − 12x^{3} + 16x}^{C_{320}^{B,F} : y^{2} = x^{5} + 2x^{4} − 2x^{3} + 2x^{2} + x}^{C_{336}^{A,F} : y^{2} = x^{5} − 4x^{4} − 25x^{3} + 172x^{2} − 339x + 222}^{C_{368}^{A,G} : y^{2} = x^{5} − 10x^{3} + 15x^{2} − 9x + 7}^{C_{384}^{A,D} : y^{2} = x^{5} + 10x^{3} + x}^{C_{384}^{B,C} : y^{2} = x^{5} − 10x^{3} + x}^{C_{400}^{B,E} : y^{2} = x^{5} − 25x^{2} + 20x − 4}^{C_{400}^{A,H} : y^{2} = x^{5} − 10x^{3} + 15x^{2} − 8}^{C_{405}^{A,F} : y^{2} = x^{6} − 12x^{4} + 28x^{3} − 24x^{2} + 12x − 4}^{C_{405}^{B,F} : y^{2} = x^{6} − 12x^{4} + 20x^{3} − 12x + 4}^{C_{448}^{A,D} : y^{2} = x^{5} − 2x^{4} + 10x^{3} − 2x^{2} + x}^{C_{448}^{B,G} : y^{2} = x^{5} + 2x^{4} + 10x^{3} + 2x^{2} + x}^{C_{480}^{B,C} : y^{2} = x^{5} + 2x^{4} − 4x^{3} − 17x^{2} − 18x − 6}^{C_{480}^{B,G} : y^{2} = x^{5} − 7x^{3} + x}^{C_{480}^{A,G} : y^{2} = x^{5} − 2x^{4} − 4x^{3} + 17x^{2} − 18x + 6}^{C_{528}^{A,D} : y^{2} = x^{5} − 2x^{4} − 6x^{3} + 23x^{2} − 24x + 8}^{C_{544}^{B,C} : y^{2} = x^{5} − 9x^{3} + 16x}^{C_{624}^{C,D} : y^{2} = x^{5} + 2x^{4} − x^{3} − 8x^{2} − 9x − 3}^{C_{672}^{A,G} : y^{2} = x^{5} + 5x^{3} + x} C : y^{2} = F (x)^{C_{760}^{A,E} : y^{2} = x^{5} + 3x^{3} + 14x^{2} + 15x + 5}^{C_{768}^{D,F} : y^{2} = x^{5} − 4x^{3} + x}^{C_{768}^{B,H} : y^{2} = x^{5} + 4x^{3} + x}^{C_{960}^{A,F} : y^{2} = x^{5} + 7x^{3} + x}^{C_{1088}^{L,N} : y^{2} = x^{5} + 9x^{3} + 16x}^{C_{1344}^{C,F} : y^{2} = x^{5} − 5x^{3} + x}^{C_{1520}^{B,D} : y^{2} = x^{5} + 3x^{3} − 14x^{2} + 15x − 5}^{C_{1664}^{F,G} : y^{2} = x^{5} − 2x^{4} + x^{3} + 2x − 4}^{C_{1664}^{O,S} : y^{2} = x^{5} + 2x^{4} + x^{3} + 2x + 4}^{C_{2208}^{A,E} : y^{2} = x^{5} + 2x^{4} + 8x^{3} + 19x^{2} + 18x + 6}^{C_{2208}^{G,I} : y^{2} = x^{5} − 2x^{4} + 8x^{3} − 19x^{2} + 18x − 6}

Table 5: ℚ-simple, g = 3, N ≤ 3000

C  : y^{2} = F (x)
C_{41}^{A}  : y^{2} = x^{8} + 4x^{7} − 8x^{6} − 66x^{5} − 120x^{4} − 56x^{3} + 53x^{2} + 36x − 16
C_{95}^{A}  : y^{2} = (x^{4} + x^{3} − 6x^{2} − 10x − 5)(x^{4} + x^{3} − 2x^{2} + 2x − 1)
C_{152}^{C}  : y^{2} = x(x^{3} − 2x^{2} − 7x − 8)(x^{3} + 4x^{2} + 4x + 4)
C_{248}^{E}  : y^{2} = (x^{3} + x − 1)(x^{4} − 2x^{3} − 3x^{2} − 4x + 4)
C_{284}^{A}  : y^{2} = x^{7} + 4x^{6} + 5x^{5} + x^{4} − 3x^{3} − 2x^{2} + 1
C_{284}^{B}  : y^{2} = x^{7} − 7x^{5} − 11x^{4} + 5x^{3} + 18x^{2} + 4x − 11
C_{304}^{G}  : y^{2} = x(x^{3} − 4x^{2} + 4x − 4)(x^{3} + 2x^{2} − 7x + 8)
C_{496}^{J}  : y^{2} = (x^{3} + x + 1)(x^{4} + 2x^{3} − 3x^{2} + 4x + 4)
C_{544}^{I}  : y^{2} = (x + 1)(x^{2} + x − 4)(x^{4} − x^{2} − 4)
C_{544}^{J}  : y^{2} = (x − 1)(x^{2} − x − 4)(x^{4} − x^{2} − 4)
C_{896}^{I}  : y^{2} = (x − 2)(x^{2} + 2x − 1)(x^{4} − 2x^{2} − 7)
C_{896}^{K}  : y^{2} = (x + 2)(x^{2} − 2x − 1)(x^{4} − 2x^{2} − 7)
C_{1136}^{G} : y^{2} = x^{7} − 7x^{5} + 11x^{4} + 5x^{3} − 18x^{2} + 4x + 11
C_{1136}^{J} : y^{2} = x^{7} − 4x^{6} + 5x^{5} − x^{4} − 3x^{3} + 2x^{2} − 1

Table 6: ℚ-simple, g = 4, #D = 1, N ≤ 3000

C  : y^{2} = F (x)
C_{47}^{A}  : y^{2} = (x^{5} − 5x^{3} − 20x^{2} − 24x − 19)(x^{5} + 4x^{4} + 7x^{3} + 8x^{2} + 4x + 1)
C_{119}^{A}  : y^{2} = (x^{5} − 2x^{4} + 3x^{3} − 6x^{2} − 7)(x^{5} + 2x^{4} + 3x^{3} + 6x^{2} + 4x + 1)

C  : y^{2} = F (x)^{C_{164}^{A}  : y^{2} = x(x^{8} + 4x^{7} − 8x^{6} − 66x^{5} − 120x^{4} − 56x^{3} + 53x^{2} + 36x − 16)}^{C_{376}^{C}  : y^{2} = (x^{4} − 2x^{3} − 3x^{2} + 4x − 4)(x^{5} + 4x^{4} + 3x^{3} − 2x^{2} + 2x + 5)}^{C_{376}^{D}  : y^{2} = (x^{4} − 2x^{3} − 3x^{2} + 4x − 4)(x^{5} − x^{3} + 2x^{2} − 2x + 1)}^{C_{416}^{F}  : y^{2} = x(x^{2} + 4)(x^{3} − 2x^{2} + x − 4)(x^{3} + 2x^{2} + x + 4)}^{C_{512}^{G}  : y^{2} = x(x^{4} − 4x^{2} − 4)(x^{4} + 4x^{2} − 4)}^{C_{656}^{I}  : y^{2} = x(x^{8} − 4x^{7} − 8x^{6} + 66x^{5} − 120x^{4} + 56x^{3} + 53x^{2} − 36x − 16)}^{C_{752}^{G}  : y^{2} = (x^{4} + 2x^{3} − 3x^{2} − 4x − 4)(x^{5} − x^{3} − 2x^{2} − 2x − 1)}^{C_{752}^{I}  : y^{2} = (x^{4} + 2x^{3} − 3x^{2} − 4x − 4)(x^{5} − 4x^{4} + 3x^{3} + 2x^{2} + 2x − 5)}^{C_{832}^{P}  : y^{2} = x(x + 2)(x − 2)(x^{6} + 2x^{4} − 15x^{2} + 16)}^{C_{1216}^{W} : y^{2} = (x^{3} − 2x + 2)(x^{6} + 2x^{4} − 7x^{2} + 8)}^{C_{1216}^{X} : y^{2} = (x^{3} − 2x − 2)(x^{6} + 2x^{4} − 7x^{2} + 8)}

Table 7: ℚ-simple, g = 5, N ≤ 3000

C  : y^{2} = F (x)
C_{59}^{A}  : y^{2} = (x^{9} + 2 x^{8} − 4 x^{7} − 21 x^{6} − 44 x^{5} − 60 x^{4} − 61 x^{3} − 46 x^{2} − 24 x − 11)
(x^{3} + 2 x^{2} + 1)
C_{1664}^{Y} : y^{2} = (x^{2} + 2x + 2)(x^{3} − x + 2)(x^{6} + 2x^{4} + x^{2} + 4)
C_{1664}^{AA} : y^{2} = (x^{2} − 2x + 2)(x^{3} − x − 2)(x^{6} + 2x^{4} + x^{2} + 4)

Table 8: Not ℚ-simple, g = 3, #D = 1, N ≤ 2000

C  : y^{2} = F (x)
C_{35}^{A,B} : y^{2} = (x^{2} + 3x + 1)(x^{6} + x^{5} − 10x^{4} − 39x^{3} − 62x^{2} − 51x − 19)
C_{39}^{A,B} : y^{2} = (x^{4} − 3x^{3} − 4x^{2} − 2x − 1)(x^{4} + 5x^{3} + 8x^{2} + 6x + 3)
C_{88}^{A,B} : y^{2} = (x − 2)(x^{3} − 2x^{2} + 4x − 4)(x^{3} + 2x^{2} − 4x + 8)
C_{104}^{A,B} : y^{2} = (x + 2)(x^{6} + 4x^{5} − 12x^{4} − 114x^{3} − 308x^{2} − 384x − 191)
C_{116}^{A,B,C} : y^{2} = (x + 2)(x^{6} + 2x^{5} − 17x^{4} − 66x^{3} − 83x^{2} − 32x − 4)
C_{128}^{A,B,D} : y^{2} = (x − 2)(x^{2} − 2x + 2)(x^{4} − 12x^{2} + 32x − 28)
C_{128}^{B,C,D} : y^{2} = (x + 2)(x^{2} + 2x + 2)(x^{4} − 12x^{2} − 32x − 28)
C_{160}^{A,C} : y^{2} = (x − 2)(x^{2} + 2x − 7)(x^{4} − 4x^{3} + 10x^{2} − 20x + 17)
C_{160}^{B,C} : y^{2} = (x + 2)(x^{2} − 2x − 7)(x^{4} + 4x^{3} + 10x^{2} + 20x + 17)
C_{176}^{A,D} : y^{2} = (x + 2)(x^{3} − 2x^{2} − 4x − 8)(x^{3} + 2x^{2} + 4x + 4)
C_{184}^{B,E} : y^{2} = (x − 1)(x^{3} − 2x^{2} + 3x − 1)(x^{3} + x^{2} − x + 7)
C_{184}^{A,C,D} : y^{2} = (x − 1)(x^{6} − x^{5} + 4x^{4} − x^{3} + 2x^{2} + 2x + 1)

C  : y^{2} = F (x)^{C_{196}^{B,C} : y^{2} = (x^{3} + 2x^{2} − x − 1)(x^{4} − 2x^{3} − 9x^{2} + 10x − 3)}^{C_{208}^{B,E} : y^{2} = (x − 2)(x^{6} − 4x^{5} − 12x^{4} + 114x^{3} − 308x^{2} + 384x − 191)}^{C_{224}^{A,D} : y^{2} = x(x − 1)(x + 1)(x^{4} − 6x^{2} + 16x − 7)}^{C_{224}^{B,C} : y^{2} = x(x − 1)(x + 1)(x^{4} − 6x^{2} − 16x − 7)}^{C_{248}^{B,D} : y^{2} = (x^{3} + 4x^{2} + 5x + 3)(x^{4} − 2x^{3} − 3x^{2} − 4x + 4)}^{C_{256}^{B,E} : y^{2} = x(x^{2} + 2)(x^{4} + 12x^{2} + 4)}^{C_{256}^{C,E} : y^{2} = x(x^{2} − 4x + 2)(x^{2} − 2)(x^{2} + 4x + 2)}^{C_{280}^{A,D} : y^{2} = (x − 1)(x^{6} − x^{5} + 7x^{3} − 16x^{2} + 15x − 5)}^{C_{368}^{C,I}  : y^{2} = (x + 1)(x^{3} − x^{2} − x − 7)(x^{3} + 2x^{2} + 3x + 1)}^{C_{368}^{A,D,G} : y^{2} = (x + 1)(x^{6} + x^{5} + 4x^{4} + x^{3} + 2x^{2} − 2x + 1)}^{C_{416}^{A,E} : y^{2} = x(x^{6} − 2x^{5} − 2x^{4} + 2x^{2} − 2x − 1)}^{C_{416}^{B,C} : y^{2} = x(x^{6} + 2x^{5} − 2x^{4} + 2x^{2} + 2x − 1)}^{C_{464}^{D,E,F} : y^{2} = (x − 2)(x^{6} − 2x^{5} − 17x^{4} + 66x^{3} − 83x^{2} + 32x − 4)}^{C_{496}^{C,G} : y^{2} = (x^{3} − 4x^{2} + 5x − 3)(x^{4} + 2x^{3} − 3x^{2} + 4x + 4)}^{C_{560}^{A,G} : y^{2} = (x + 1)(x^{6} + x^{5} − 7x^{3} − 16x^{2} − 15x − 5)}^{C_{640}^{C,K} : y^{2} = x(x^{2} − 2x − 1)(x^{4} + 2x^{3} − 2x + 1)}^{C_{640}^{G,I}  : y^{2} = x(x^{2} + 2x − 1)(x^{4} − 2x^{3} + 2x + 1)}^{C_{704}^{B,N} : y^{2} = x(x^{3} − 4x + 4)(x^{3} + 2x^{2} − 2)}^{C_{704}^{C,O} : y^{2} = x(x^{3} − 2x^{2} + 2)(x^{3} − 4x − 4)}^{C_{784}^{G,M} : y^{2} = (x^{3} − 2x^{2} − x + 1)(x^{4} + 2x^{3} − 9x^{2} − 10x − 3)}

Table 9: Not ℚ-simple, g = 4, #D = 1, N ≤ 2000

C  : y^{2} = F (x)
C_{224}^{C,D} : y^{2} = x(x^{4} − 2x^{3} − 5x^{2} − 2x + 1)(x^{4} + 2x^{3} − 5x^{2} + 2x + 1)
C_{236}^{B,C} : y^{2} = x^{9} + 2x^{8} − 4x^{7} − 21x^{6} − 44x^{5} − 60x^{4} − 61x^{3} − 46x^{2} − 24x − 11
C_{368}^{B,F,H} : y^{2} = (x^{3} − x^{2} − x − 7)(x^{6} + x^{5} + 4x^{4} + x^{3} + 2x^{2} − 2x + 1)
C_{448}^{I,J}  : y^{2} = x(x^{8} + 14x^{6} + 19x^{4} + 14x^{2} + 1)
C_{704}^{D,E,M} : y^{2} = (x^{3} − 4x − 4)(x^{3} − 4x + 4)(x^{3} + 2x^{2} − 2)
C_{704}^{F,I,P} : y^{2} = (x^{3} − 2x^{2} + 2)(x^{3} − 4x − 4)(x^{3} − 4x + 4)
C_{944}^{J,L}  : y^{2} = x^{9} − 2x^{8} − 4x^{7} + 21x^{6} − 44x^{5} + 60x^{4} − 61x^{3} + 46x^{2} − 24x + 11

MODULAR CURVES OF GENUS AT LEAST 2 51

Table 10: Not ℚ-simple, g = 6, #D = 1, N ≤ 2000

C  : y^{2} = F (x)
C_{71}^{A,B} : y^{2} = (x^{7} − 7x^{5} − 11x^{4} + 5x^{3} + 18x^{2} + 4x − 11)
(x^{7} + 4x^{6} + 5x^{5} + x^{4} − 3x^{3} − 2x^{2} + 1)

Table 11: g = 3, #D = 3^{C : y^{2} = F (x)} C_{21A_{{0,2}}}^{A}  : y^{2} = (x^{2} − x + 1)(x^{6} + x^{5} − 6x^{4} − 3x^{3} + 14x^{2} − 7x + 1)

C_{36A_{{0,2}}}^{A}  : y^{2} = (x + 1)(x + 2)(x^{2} + 3x + 3)(x^{3} − 9x − 9)
C_{72A_{{{0,0},2}}}^{A} : y^{2} = x(x + 1)(x^{2} + x + 1)(x^{3} − 3x − 1)
C_{144A_{{{0,0},2}}}^{A} : y^{2} = (x − 2)(x − 1)(x^{2} − 3x + 3)(x^{3} − 9x + 9)
C_{144B_{{{0,0},2}}}^{B} : y^{2} = (x − 1)x(x^{2} − x + 1)(x^{3} − 3x + 1)

Table 12: g = 3, 2 | #D, w ∉ D^{C : y^{2} = F (x)} C_{40A_{{{0,0},2}}}^{A}: y^{2} = x(x + 1)(x + 2)(x^{2} − 2x − 4)(x^{2} + 3x + 1)

C_{48A_{{{1,0},1}}}^{A}: y^{2} = (x + 1)(x^{2} − 2x − 2)(x^{2} + x + 1)(x^{2} + 2x + 2)
C_{64A_{{{0,8}}}}^{A} : y^{2} = x(x − 1)(x + 1)(x^{2} − 2 x − 1)(x^{2} + 2 x − 1)
C_{80A_{{{0,0},2}}}^{A}: y^{2} = x(x − 1)(x − 2)(x^{2} − 3x + 1)(x^{2} + 2x − 4)
C_{80A_{{{0,0},2}}}^{B}: y^{2} = (x + 1)(x^{2} − x − 1)(x^{4} + 4 x^{2} + 8 x + 4)
C_{128A_{{{0,16}}}}^{B}: y^{2} = (x − 2)(x^{2} − 2x − 1)(x^{4} − 6x^{2} − 16x + 41)
C_{128A_{{{0,16}}}}^{D}: y^{2} = (x + 2)(x^{2} + 2x − 1)(x^{4} − 6x^{2} + 16x + 41)

Table 13: g ≥ 4, #D > 1, N ≤ 569^{C : y^{2} = F (x)} C_{52A,B_{{0,4}}}^{A} : y^{2} = x(x + 1)(x^{3} − x^{2} − 4x − 1)(x^{6} + 4x^{5} + 6x^{4} + 2x^{3} + x^{2} + 2x + 1)

C_{160A,E_{{{1,0},1}}}: y^{2} = (x − 1)(x^{2} − 2x + 2)(x^{2} − x − 1)(x^{4} − 8x + 8)
C_{160B,F_{{{1,0},1}}}: y^{2} = (x + 1)(x^{2} + 2x + 2)(x^{2} + x − 1)(x^{4} + 8x + 8)
C_{208A,D_{{{0,0},4}}}^{C}: y^{2} = x(x − 1)(x^{3} + x^{2} − 4x + 1)(x^{6} − 4x^{5} + 6x^{4} − 2x^{3} + x^{2} − 2x + 1)
C_{512D_{{{0,64}}}} : y^{2} = x(x^{8} + 24x^{4} + 16)

Acknowledgements

We thank Dinakar Ramakrishnan and Ken Ribet for discussing Proposition 3.2 with us. The proof of Theorem 5.7 grew out of discussions with Matthias Aschenbrenner, Brian Conrad, Tom Graber, Tom Scanlon, and Jason Starr. We thank also Paul Gunnells, Sorin Popescu, Xavier-François Roblot, William Stein and Tonghai Yang for suggesting references. Finally we thank a referee for an especially helpful report.

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Department of Mathematics, University of Georgia, Athens, GA 30602-7403, USA Current address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

E-mail address: mbaker@math.gatech.edu

School of Mathematical Sciences. University of Nottingham. University Park. Nottingham, NG7 2RD, UK

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

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