LARGE TORSION SUBGROUPS OF SPLIT JACOBIANS OF CURVES OF GENUS TWO OR THREE

EVERETT W. HOWE, FRANCK LEPRÉVOST, AND BJORN POONEN

Abstract. We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^{1} of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve

15625(X^{4} + Y^{4} + Z^{4}) − 96914(X^{2}Y^{2} + X^{2}Z^{2} + Y^{2}Z^{2}) = 0,

whose Jacobian has 864 rational torsion points.

1. Introduction

Nearly twenty years ago Mazur settled the question of which groups can occur as the group of rational torsion points on an elliptic curve over Q, but the analogous question for Jacobian varieties of curves over Q of genus greater than 1 remains open. Most of the work that has been done on this question has centered on the problem of finding groups that do occur as rational torsion subgroups of Jacobians. Several researchers have produced families of genus-2 curves whose Jacobians contain various given groups in their rational torsion (see [21], [22], [25], [26], [33], and the summary in [29]) while others have constructed families of curves in which the size of the rational torsion subgroup of the Jacobian increases as the genus of the curve increases (see [8], [9], [23], [24], [27], [28]). The largest group of rational torsion heretofore known to exist on the Jacobian of a curve of genus 2 was a group of order 30; for genus-3 curves, the largest group had order 64.

In this paper we present many explicit families of curves of genus 2 and 3 whose Jacobians possess large rational torsion subgroups. The strategy behind our constructions is to take a product of elliptic curves, each with large rational torsion, and to find a curve whose Jacobian is isogenous to the given product. Thus it is no surprise that the groups we list occur as torsion groups of abelian varieties; rather, the point of interest is that they occur as torsion groups of Jacobian varieties.

For curves of genus 2, we have the following result:

Theorem 1. For every abstract group G listed in the first column of Table 1, there exists a family of curves over Q of genus 2, parameterized by the rational points on a non-empty Zariski-open subset of a variety of the type listed in the third column, whose Jacobians contain a group of rational points isomorphic to G.

When we say that a family is parameterized by the rational points in a non-empty Zariski open subset U of a variety X, we mean in particular that the closure of the image of U in the moduli space of genus-2 curves is of the same dimension as X. Also, by “positive rank elliptic surface” we mean an elliptic surface over P^{1} with positive rank. Note that a family parameterized by the

Date: 13 February 1997; revised 19 September 1998.

1991 Mathematics Subject Classification. Primary 14G05; Secondary 11G30, 14H25, 14H45.

Key words and phrases. Torsion, Jacobian, elliptic differential.

The first author is supported partially by NSA Young Investigator Grant MDA904-95-H-1044. The second author thanks the Max-Planck-Institut für Mathematik of Bonn for its hospitality. The third author is supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. The final version of this paper has appeared in Forum Math. 12 (2000), 315–364.

G  |G|  Parameterizing variety
Z/20Z  20  P^{2}
Z/21Z  21  P^{2}
Z/3Z × Z/9Z  27  P^{2}
Z/30Z  30  P^{2}
Z/35Z  35 positive rank elliptic curve
Z/6Z × Z/6Z  36  P^{2}
Z/3Z × Z/12Z  36  P^{2}
Z/40Z  40 positive rank elliptic surface
Z/45Z  45 positive rank elliptic curve
Z/2Z × Z/24Z  48  P^{2}
Z/7Z × Z/7Z  49  P^{0}
Z/5Z × Z/10Z  50 positive rank elliptic surface
Z/60Z  60 positive rank elliptic curve
Z/63Z  63  P^{0}
Z/8Z × Z/8Z  64  P^{2}
Z/2Z × Z/4Z × Z/8Z  64  P^{2}
Z/6Z × Z/12Z  72 positive rank elliptic surface
Z/2Z × Z/6Z × Z/6Z  72 positive rank elliptic surface
Z/2Z × Z/2Z × Z/24Z  96 positive rank elliptic curve
Z/2Z × Z/2Z × Z/4Z × Z/8Z 128 positive rank elliptic surface

Table 1. Families of curves over Q of genus 2 such that G is contained in the torsion subgroup of the Jacobian.

rational points in a non-empty open subset of P^{0} consists of a single curve. We will often refer to families parameterized by P^{1} and P^{2} as 1- and 2-parameter families.

A similar table expresses our results for curves of genus 3.

Theorem 2. For every abstract group G listed in the first column of Table 2, there exists a family of curves over Q of genus 3, parameterized by the rational points on a non-empty Zariski-open subset of a variety of the type listed in the third column, whose Jacobians contain a group of rational points isomorphic to G. The fourth column of the table indicates whether or not the family consists entirely of hyperelliptic curves.

In Part 2 of the paper we review the results on elliptic curves that we will need to prove these theorems. In Part 3 we show how, given a pair of non-isomorphic elliptic curves whose Galois modules of 2-torsion points are isomorphic, one can construct explicitly a curve of genus 2 whose Jacobian is isogenous to the product of the given elliptic curves. After giving some quick applications of the construction to the problems of finding genus-2 curves of low conductor and of high rank, we give a modular interpretation of our construction in Section 3.3. The rest of Part 3 is taken up with the proof of Theorem 1. In Part 4 we begin with another explicit construction: We show in Section 4.1 how one can construct a curve of genus 3 whose Jacobian is isogenous to a product of three given elliptic curves, provided that each of the elliptic curves has a rational 2-torsion point, that the product of their discriminants is a square, and that a certain explicitly calculable number depending on the curves is a square. The remainder of Part 4 contains the proof of Theorem 2. The reader should note that our verifications of the many entries in Tables 1 and 2 are organized

G  |G|  Parameterizing variety  All hyp.?
Z/2Z × Z/30Z  60 positive rank elliptic curve  yes
Z/10Z × Z/10Z  100  P^{1}  yes
Z/2Z × Z/8Z × Z/8Z  128 positive rank elliptic surface  yes
Z/4Z × Z/4Z × Z/8Z  128  P^{1}  yes
Z/4Z × Z/40Z  160 positive rank elliptic curve  no
Z/2Z × Z/4Z × Z/24Z  192 positive rank elliptic curve  no
Z/2Z × Z/2Z × Z/2Z × Z/24Z  192 positive rank elliptic surface  yes
Z/10Z × Z/20Z  200  P^{2}  no
Z/6Z × Z/6Z × Z/6Z  216 positive rank elliptic curve  no
Z/4Z × Z/60Z  240 positive rank elliptic curve  no
Z/4Z × Z/8Z × Z/8Z  256 positive rank elliptic curve  no
Z/2Z × Z/2Z × Z/8Z × Z/8Z  256  P^{2}  no
Z/2Z × Z/4Z × Z/4Z × Z/8Z  256  P^{2}  no
Z/2Z × Z/2Z × Z/2Z × Z/4Z × Z/8Z  256  P^{2}  yes
Z/2Z × Z/12Z × Z/12Z  288  P^{2}  no
Z/2Z × Z/2Z × Z/6Z × Z/12Z  288 positive rank elliptic surface  yes
Z/2Z × Z/2Z × Z/4Z × Z/4Z × Z/8Z  512 positive rank elliptic curve  no
Z/2Z × Z/2Z × Z/2Z × Z/2Z × Z/4Z × Z/8Z 512  P^{1}  yes
Z/6Z × Z/12Z × Z/12Z  864  P^{0}  no

Table 2. Families of curves over Q of genus 3 such that G is contained in the torsion subgroup of the Jacobian. The final column indicates whether or not the family consists entirely of hyperelliptic curves.

not by the sequence of the entries in the tables but rather by the type of argument the verifications require. Consequently, the proofs of the theorems are distributed among several sections.

Throughout the paper, and often without further mention, we will make use of van Hoeij’s Maple package IntBasis for computing Weierstrass models for genus-1 curves with a rational point; Cremona’s programs findinf and mwrank for finding points on, and computing ranks of, elliptic curves over Q; Mathematica; and especially PARI.

2. Genus one

In this section we record facts about torsion of elliptic curves over Q that we will need later. Mazur’s theorem [30] states that if E is an elliptic curve over Q, then the group of rational torsion points on E is isomorphic to Z/NZ with N ≤ 10 or N = 12, or isomorphic to Z/2Z × Z/2NZ with N ≤ 4. For each possibility where the group is not killed by 2, the elliptic curves having that group as torsion subgroup form a 1-parameter family. We will need to have an explicit equation for the universal curve for each family. For N = 3, this universal elliptic curve is y^{2} = x^{3} + (x + t)^{2}/4 and a 3-torsion point is (0, t/2). For the other cases, we copy^{1} Table 3 in [19] to our Table 3.

Let E_{N}^{t} denote the elliptic curve with a rational N-torsion point with parameter t, and similarly define E_{2,2N}^{t}. We will need to know something about the field of definition of the 2-torsion points on the curves E_{N}^{t}. Therefore we record the discriminant Δ_{N}(t) of E_{N}^{t} modulo squares in Q(t) in Table 4. If N is odd, the discriminant Δ_{N}(t) is equal (modulo squares) to the discriminant of the

1Actually, we have done a tiny bit more than copy: we have expanded the implicit expressions for the parameters b and c in [19] to express b and c in terms of a single parameter t.

\begin{matrix}N or (2, 2N) & b & c \\ 4 & t & 0 \\ 5 & t & t \\ 6 & t^{2} + t & t \\ 7 & t^{3} − t^{2} & t^{2} − t \\ 8 & 2t^{2} − 3t + 1 & \frac{2t^{2} − 3t + 1}{t} \\ 9 & t^{5} − 2t^{4} + 2t^{3} − t^{2} & t^{3} − t^{2} \\ 10 & \frac{2t^{5} − 3t^{4} + t^{3}}{(t^{2} − 3t + 1)^{2}} & \frac{−2t^{3} + 3t^{2} − t}{t^{2} − 3t + 1} \\ 12 & \frac{12t^{6} − 30t^{5} + 34t^{4} − 21t^{3} + 7t^{2} − t}{(t − 1)^{4}} & \frac{−6t^{4} + 9t^{3} − 5t^{2} + t}{(t − 1)^{3}} \\ (2,4) & t^{2} − \frac{1}{16} & 0 \\ (2,6) & \frac{−2t^{3} + 14t^{2} − 22t + 10}{(t + 3)^{2}(t − 3)^{2}} & \frac{−2t + 10}{(t + 3)(t − 3)} \\ (2,8) & \frac{16t^{3} + 16t^{2} + 6t + 1}{(8t^{2} − 1)^{2}} & \frac{16t^{3} + 16t^{2} + 6t + 1}{2t(4t + 1)(8t^{2} − 1)} \end{matrix}

Table 3. Parameters b, c for the universal elliptic curve y^{2}+(1−c)xy−by = x^{3}−bx^{2} over X_{1}(N) or X_{1}(2, 2N). In each case, (0, 0) is a torsion point of maximal order.

N Discriminant Δ_{N}(t) modulo squares
3  t(1 − 27t)
4  16t + 1
5  t(t^{2} − 11t − 1)
6  (t + 1)(9t + 1)
7  t(t − 1)(t^{3} − 8t^{2} + 5t + 1)
8  8t^{2} − 8t + 1
9 t(t − 1)(t^{2} − t + 1)(t^{3} − 6t^{2} + 3t + 1)
10  (2t − 1)(4t^{2} − 2t − 1)
12  (2t^{2} − 2t + 1)(6t^{2} − 6t + 1)

Table 4. Discriminant (modulo squares) of the elliptic curve E_{N}^{t}.

cubic field obtained by adjoining the coordinates of one 2-torsion point; if N is even, Δ_{N}(t) is equal (modulo squares) to the discriminant of the quadratic field obtained by adjoining the coordinates of a non-rational 2-torsion point.

We will need to know the x-coordinates of the nonzero 2-torsion points on E_{2,2N}^{t}, at least for N = 3 and N = 4. These are given in Table 5. For N = 4, the point T_{1} is the one that is 4 times a rational 8-torsion point. Note that these x-coordinates are also valid for the model

y^{2} = x^{3} − bx^{2} + [(1 − c)x − b]^{2}/4

obtained by completing the square in y.

\begin{matrix}Curve & x(T_{1}) & x(T_{2}) & x(T_{3}) \\ E_{2,6}^{t} & \frac{−2t + 10}{(t + 3)(t − 3)} & \frac{−t^{3} + 7t^{2} − 11t + 5}{4(t + 3)(t − 3)^{2}} & \frac{−2t^{2} + 4t − 2}{(t + 3)^{2}(t − 3)} \\ E_{2,8}^{t} & \frac{16t^{3} + 12t^{2} + 2t}{(8t^{2} − 1)^{2}} & \frac{32t^{3} + 24t^{2} + 8t + 1}{16t^{2}(8t^{2} − 1)} & \frac{−32t^{4} − 32t^{3} − 12t^{2} − 2t}{(4t + 1)^{2}(8t^{2} − 1)} \end{matrix}

Table 5. The x-coordinates of the 2-torsion points on E_{2,2N}^{t}.

For our work with genus-3 curves, we will require different models of the universal elliptic curves E_{N}^{t} for N = 4, 6, 8, 10, 12 and E_{2,2N}^{t} for N = 2, 3, 4; in particular, we will want to have each curve written in the form y^{2} = x(x^{2} + Ax + B), where (x, y) = (0, 0) is a specified 2-torsion point. Table 6 lists the values of A and B for the curves we will need, as well as the value of the number Δ = A^{2} − 4B, which differs from the discriminant of the elliptic curve by a factor of 16B^{2}. The entries in the table were calculated by completing the square in y for the models of the E_{2N}^{t} and E_{2,2N}^{t} given above, moving a rational 2-torsion point to x = 0, scaling with respect to x to clear denominators, and making a linear change of variables in t so as to simplify the resulting polynomials. We will refer to these models as F_{2N}^{t} and F_{2,2N}^{t} depending on which universal elliptic curve they model. However, there are two essentially different ways of putting the curves E_{2,4}^{t} and E_{2,8}^{t} into the desired form, because one of the 2-torsion points on these curves is a multiple of a point of order 4 while the others are not. We denote the models in which the 2-torsion point at x = 0 is a multiple of a 4-torsion point by F_{2,4}^{t} and F_{2,8}^{t}, and we denote the other models by F_{4,2}^{t} and F_{8,2}^{t}.

For convenience, we list in Table 7 the coordinates of a torsion point of maximal order on the curves F_{2N}^{t}, F_{2,2N}^{t}, and F_{2N,2}^{t}. Also, in Table 8 we give the x-coordinates of the 2-torsion points other than (0, 0) on the curves whose 2-torsion points are all rational. In the entries for F_{4,2}^{t} and F_{8,2}^{t}, the point labeled T_{1} is twice a rational 4-torsion point.

Finally, we note that while there is no universal elliptic curve over the modular curve X(2) there is a replacement that will suffice for our purposes. If k is a field of characteristic different from 2, then every elliptic curve over k that has all of its 2-torsion defined over k is isomorphic to a twist of a specialization of the curve F_{2,2}^{t} over k(t) defined by y^{2} = x(x^{2} + Ax + B), where A = −1 − t and B = t.

3. Genus two

3.1. Conventions. All curves are supposed to be nonsingular and irreducible unless we specifically mention that they might not be. The modular curves we consider in section 3.3 are possibly singular. If A is a variety over a field k and if K is an extension field of k, we will denote by A_{K} the K-scheme A ×_{Spec k} Spec K. If A is an abelian variety over a field k and N is a positive integer, we will denote by A[N] the k-group scheme that is the kernel of the multiplication-by-N map on A.

3.2. Jacobians (2, 2)-isogenous to a product of elliptic curves. In this section we will show how one can construct a curve of genus 2 whose Jacobian is (2, 2)-isogenous to a product of two given elliptic curves, provided one has an isomorphism of their 2-torsion groups that does not come from an isomorphism of elliptic curves. Related results, some of them constructive, have appeared in the recent literature — see for example [10], [11], [13], [15], [20] — and in fact the first examples of curves like the ones we construct go back to Jacobi. Chapter XXII of [1] provides inroads to the older literature on the subject.

Suppose E and F are elliptic curves over a separably closed field K, and let N be a positive integer not divisible by the characteristic of K. The product of the canonical polarizations on E

2N, (2, 2N), or (2N, 2) A, B, and Δ = A^{2} − 4B

A = 2t + 1

4  B = t^{2}

Δ = 4t + 1 A = 2t^{2} + 2

(2,4)  B = (t − 1)^{2}(t + 1)^{2}

Δ = 16t^{2} A = −t^{2} − 6t − 1

(4,2)  B = 4t(t + 1)^{2}

Δ = (t − 1)^{4} A = −3t^{2} + 6t + 1

6  B = −16t^{3}

Δ = (9t + 1)(t + 1)^{3} A = −2t^{4} + 12t^{2} + 6

(2,6)  B = (t + 3)(t − 3)(t + 1)^{3}(t − 1)^{3}

Δ = 256t^{2} A = 2t^{4} + 4t^{2} − 2

8  B = (t + 1)^{4}(t − 1)^{4}

Δ = 16(2t^{2} − 1)t^{4} A = t^{8} − 4t^{6} + 22t^{4} − 4t^{2} + 1

(2,8)  B = 16t^{4}(t + 1)^{4}(t − 1)^{4}

Δ = (t^{2} − 2t − 1)^{2}(t^{2} + 2t − 1)^{2}(t^{2} + 1)^{4} A = −2t^{8} + 8t^{6} + 4t^{4} + 8t^{2} − 2

(8,2)  B = (t^{2} − 2t − 1)(t^{2} + 2t − 1)(t^{2} + 1)^{2}(t + 1)^{4}(t − 1)^{4}

Δ = 256t^{8} A = −(2t^{2} − 2t + 1)(4t^{4} − 12t^{3} + 6t^{2} + 2t − 1)

10  B = 16(t^{2} − 3t + 1)(t − 1)^{5}t^{5}

Δ = (4t^{2} − 2t − 1)(2t − 1)^{5} A = 24t^{8} − 96t^{7} + 216t^{6} − 312t^{5} + 288t^{4} − 168t^{3} + 60t^{2} − 12t + 1

12  B = 16(3t^{2} − 3t + 1)^{2}(t − 1)^{6}t^{6}

Δ = (6t^{2} − 6t + 1)(2t^{2} − 2t + 1)^{3}(2t − 1)^{6}

Table 6. Parameters A, B for the universal elliptic curve y^{2} = x(x^{2} +Ax+B) over X_{1}(2N) or X_{1}(2, 2N). The 2-torsion point (0, 0) is twice a rational 4-torsion point for the entries marked (2, 4) and (2, 8), and is not for the entries marked (4, 2) and (8, 2).

and F is a principal polarization λ on the product variety A = E × F, and by combining the Weil pairings on E[N] and F [N] we get a non-degenerate alternating pairing e_{N} from the N-torsion of A to the group-scheme of Nth roots of unity over K. Suppose G is a sub-group-scheme of A[N] that is isotropic with respect to the pairing e_{N} and that is maximal with respect to this property. Then the polarization Nλ of A reduces to a principal polarization μ on the quotient abelian variety B = A/G (see [32], Proposition 16.8, p. 135). The polarized variety (B, μ) will be either the polarized Jacobian of a curve over K or the product of two polarized elliptic curves over K. Suppose N = 2; in this case it is easy to show that if (B, μ) is a Jacobian then G must be

N, (2, 2N), or (2N, 2) (x, y)-coordinates
4  x = −t

y = t

(2, 4)  x = −(t + 1)(t − 1)

y = 2(t + 1)(t − 1)

(4, 2)  x = 2(t + 1)

y = 2(t + 1)(t − 1)

6  x = −4t

y = 4t(t + 1)

(2, 6)  x = (t − 3)(t + 3)(t − 1)(t + 1)

y = 4(t − 3)(t + 3)(t − 1)(t + 1)

8  x = −(t + 1)^{3}(t − 1)

y = 2(t + 1)^{3}(t − 1)t

(2, 8)  x = −4(t − 1)t(t + 1)^{3}

y = 4(t − 1)t(t + 1)^{3}(t^{2} + 1)(t^{2} − 2t − 1)

(8, 2)  x = (t^{2} + 1)(t^{2} − 2t − 1)(t − 1)(t + 1)^{3}

y = 4(t^{2} + 1)(t^{2} − 2t − 1)(t − 1)(t + 1)^{3}t

10  x = 4(t − 1)(t^{2} − 3t + 1)t^{3}

y = 4(t − 1)(t^{2} − 3t + 1)t^{3}(2t − 1)

12  x = −4(t − 1)(3t^{2} − 3t + 1)t^{5}

y = 4(t − 1)(3t^{2} − 3t + 1)t^{5}(2t^{2} − 2t + 1)(2t − 1)

Table 7. Coordinates of a torsion point of maximal order on the universal curves F_{N}^{t}, F_{2,2N}^{t}, and F_{2N,2}^{t}.

\begin{matrix}(2, 2N) or (2N, 2) & x(T_{1}) & x(T_{2}) \\ (2, 4) & −(t − 1)^{2} & −(t + 1)^{2} \\ (4, 2) & (t + 1)^{2} & 4t \\ (2, 6) & (t + 3)(t − 1)^{3} & (t − 3)(t + 1)^{3} \\ (2, 8) & −16t^{4} & −(t − 1)^{4}(t + 1)^{4} \\ (8, 2) & (t − 1)^{4}(t + 1)^{4} & (t^{2} + 2t − 1)(t^{2} − 2t − 1)(t^{2} + 1)^{2} \end{matrix}

Table 8. The x-coordinates of the 2-torsion points on F_{2,2N}^{t} and F_{2N,2}^{t} other than (0, 0).

the graph of an isomorphism E[N](K) → F [N](K). Our first result is that the converse of this statement is almost true.

Proposition 3. Let E and F be elliptic curves over a field k whose characteristic is not 2, let K be a separable closure of k, let A be the polarized abelian surface E × F, and let G ⊆ A[2](K) be the graph of a group isomorphism ψ: E[2](K) → F [2](K). Then G is a maximal isotropic subgroup of A[2](K). Furthermore, the quotient polarized abelian variety A_{K}/G is isomorphic to the polarized Jacobian of a curve C over K, unless ψ is the restriction to E[2](K) of an isomorphism E_{K} → F_{K}. If ψ gives rise to a curve C, then C and the isomorphism Jac C ≅ A_{K}/G can be defined over k if and only if G can be defined over k, if and only if ψ is an isomorphism of Galois modules.

Proof. All of the proposition except for the final sentence is the special case N = 2 of the results of [15]. The final statement of the proposition follows from standard descent arguments that make use of the fact that the automorphism group of C is naturally isomorphic to that of the polarized variety A_{K}/G. □

Let k and K be as in Proposition 3 and let E and F be the elliptic curves over k defined by the equations y^{2} = f and y^{2} = g, respectively, where f and g are separable monic cubic polynomials in k[x] with discriminants Δ_{f} and Δ_{g}. Suppose ψ is a Galois-module isomorphism E[2](K) → F [2](K) that does not come from an isomorphism E_{K} → F_{K}. Our next proposition shows how we can use f, g, and ψ to find a model for the curve C over k that appears in Proposition 3.

Proposition 4. With notation as above, let α_{1}, α_{2}, and α_{3} be the roots of f in K and let β_{1}, β_{2}, and β_{3} be the roots of g in K. Suppose the roots are indexed so that ψ((α_{i}, 0)) = (β_{i}, 0). The numbers a_{1}, b_{1}, a_{2}, and b_{2} defined by

a_{1} = (α_{3} − α_{2})^{2}/(β_{3} − β_{2}) + (α_{2} − α_{1})^{2}/(β_{2} − β_{1}) + (α_{1} − α_{3})^{2}/(β_{1} − β_{3})
b_{1} = (β_{3} − β_{2})^{2}/(α_{3} − α_{2}) + (β_{2} − β_{1})^{2}/(α_{2} − α_{1}) + (β_{1} − β_{3})^{2}/(α_{1} − α_{3})
a_{2} = α_{1}(β_{3} − β_{2}) + α_{2}(β_{1} − β_{3}) + α_{3}(β_{2} − β_{1})
b_{2} = β_{1}(α_{3} − α_{2}) + β_{2}(α_{1} − α_{3}) + β_{3}(α_{2} − α_{1})

are nonzero, and the ratios a_{1}/a_{2} and b_{1}/b_{2} are in k. Let A = Δ_{g}a_{1}/a_{2} and let B = Δ_{f}b_{1}/b_{2}. Then the polynomial h defined by

h = −(A(α_{2} − α_{1})(α_{1} − α_{3})x^{2} + B(β_{2} − β_{1})(β_{1} − β_{3}))
· (A(α_{3} − α_{2})(α_{2} − α_{1})x^{2} + B(β_{3} − β_{2})(β_{2} − β_{1}))
· (A(α_{1} − α_{3})(α_{3} − α_{2})x^{2} + B(β_{1} − β_{3})(β_{3} − β_{2}))

is a separable sextic in k[x], and the polarized Jacobian of the curve C over k defined by y^{2} = h is isomorphic to the quotient of E × F by the graph of ψ.

Proof. Simple algebra shows that if either a_{1} or a_{2} were zero we would have

β_{3} = α_{3} \frac{β_{2} − β_{1}}{α_{2} − α_{1}} + \frac{β_{1}α_{2} − β_{2}α_{1}}{α_{2} − α_{1}} .

But then the automorphism

Ψ: z ↦ z \frac{β_{2} − β_{1}}{α_{2} − α_{1}} + \frac{β_{1}α_{2} − β_{2}α_{1}}{α_{2} − α_{1}}

of P_{K}^{1} would take α_{i} to β_{i} for i = 1, 2, 3 and would also take ∞ to ∞, and this would mean that ψ came from the isomorphism E_{K} → F_{K} obtained from Ψ, contrary to our hypotheses. Therefore a_{1} and a_{2} are nonzero. It is easy to check that the ratio a_{1}/a_{2} is fixed by the action of S_{3} that permutes the indices of the α’s and β’s. But the Galois equivariance of the map ψ shows that the action of Gal(K/k) on a_{1}/a_{2} factors through this action of S_{3}, so a_{1}/a_{2} is an element of k. A similar argument shows that b_{1} and b_{2} are nonzero and that b_{1}/b_{2} ∈ k.

The group Gal(K/k) acts on h by permuting its factors, so h is an element of k[x]. The coefficient of x^{2} in each factor is nonzero, so h is a sextic. To show that h is separable it will be enough to show that the polynomial \overline{g} ∈ k[u] defined by

\overline{g} = −(A(α_{2} − α_{1})(α_{1} − α_{3})u + B(β_{2} − β_{1})(β_{1} − β_{3}))
· (A(α_{3} − α_{2})(α_{2} − α_{1})u + B(β_{3} − β_{2})(β_{2} − β_{1}))
· (A(α_{1} − α_{3})(α_{3} − α_{2})u + B(β_{1} − β_{3})(β_{3} − β_{2}))

is separable, because h(x) = \overline{g}(x^{2}) and the roots of \overline{g} are nonzero. Let t_{1} = −(A/B)(b_{2}/b_{1}) and let t_{2} be the element

t_{2} = \frac{1}{b_{1}} (\frac{β_{1}(β_{3} − β_{2})^{2}}{α_{3} − α_{2}} + \frac{β_{2}(β_{1} − β_{3})^{2}}{α_{1} − α_{3}} + \frac{β_{3}(β_{2} − β_{1})^{2}}{α_{2} − α_{1}} )

of k. The reader may verify that the automorphism z ↦ t_{1}z + t_{2} of P_{K}^{1} takes the roots of \overline{g} to the roots of g. The roots of g are distinct by assumption, so the roots of \overline{g} must also be distinct, so \overline{g} is separable.

Now we turn to the final statement of the proposition. Let \overline{F} be the elliptic curve over k defined by v^{2} = \overline{g}. Once one knows that z ↦ t_{1}z +t_{2} takes the roots of \overline{g} to those of g, it is a simple matter to verify that the map

(u, v) ↦ (t_{1}u + t_{2}, (Δ_{f}/B^{3})v)

provides an isomorphism between \overline{F} and F. Thus we can define a morphism χ: C → F of degree 2 by

(x, y) ↦ (t_{1}x^{2} + t_{2}, (Δ_{f}/B^{3})y).

The involution τ of C defined by this double cover is given by (x, y) ↦ (−x, y).

Let \overline{E} be the elliptic curve over k defined by v^{2} = \overline{f}, where \overline{f} ∈ k[u] is given by

\overline{f} = −(A(α_{2} − α_{1})(α_{1} − α_{3}) + B(β_{2} − β_{1})(β_{1} − β_{3})u)
· (A(α_{3} − α_{2})(α_{2} − α_{1}) + B(β_{3} − β_{2})(β_{2} − β_{1})u)
· (A(α_{1} − α_{3})(α_{3} − α_{2}) + B(β_{1} − β_{3})(β_{3} − β_{2})u).

The α–β symmetry in our equations shows that there is an isomorphism \overline{E} → E given by

(u, v) ↦ (s_{1}u + s_{2}, (Δ_{g}/A^{3})v),

where s_{1} and s_{2} are the elements of k defined by exchanging α’s and β’s in the definitions of t_{1} and t_{2}. Thus we get a k-morphism φ: C → E of degree 2 defined by

(x, y) ↦ (s_{1}/x^{2} + s_{2}, (Δ_{g}/A^{3})(y/x^{3})).

The involution σ of C defined by this double cover is given by (x, y) ↦ (−x, −y).

Let A = E × F, let J be the Jacobian of C, and let ω: A → J be the morphism φ^{∗} × χ^{∗}. Note that the image of φ^{∗} in J is fixed by σ^{∗}, while the image of χ^{∗} is fixed by τ^{∗}; since σ^{∗}τ^{∗} = −1, we see that ω is an isogeny. Let μ be the canonical polarization of J. The fact that φ has degree 2 implies that φ̂^{∗}μφ^{∗} is the multiplication-by-2 map on E, and similarly χ̂^{∗}μχ^{∗} is the multiplicationby-2 map on F; here indicates the dual morphism. If we let λ be the product polarization on A obtained from the canonical polarizations on E and F, then we have a commutative diagram

\begin{matrix}A & ⟶^{2λ} & Â \\ ^{↓^{ω}} & & ^{↑ω̂} \\ J & ⟶^{μ} & Ĵ. \end{matrix}

The diagram shows that ω must have degree four, and its kernel lies in the 2-torsion of A. By using the explicit representation of 2-torsion elements of E, F, and J as degree-zero K-divisors on E, F, and C that are supported only on Weierstrass points, one may check easily that the graph G of ψ is contained in ker ω, and since #G = # ker ω, we must have G = ker ω. □

Below we give a few quick applications of Proposition 4. First, we exhibit a curve of genus 2 over Q whose Jacobian has a very small conductor. Mestre [31] proved under standard conjectures that the conductor of a g-dimensional abelian variety over Q must be greater than (10.32)^{g}, so for a 2-dimensional variety a conductor of 121 is close to the minimum of 107.

Corollary 5. The conductor of the Jacobian of the curve y^{2} = −2x^{6} − 10x^{4} + 26x^{2} + 242 is 121.

Proof. Take E and F to be the modular curves X_{1}(11) and X_{0}(11) over Q. The Q-rational 5isogeny E → F gives us a Galois-module isomorphism ψ: E[2](\overline{Q}) → F [2](\overline{Q}), and ψ does not come from an isomorphism E\overline{Q} → F\overline{Q} because E\overline{Q} and F\overline{Q} are not isomorphic to one another. Applying Proposition 4 to convenient models of E and F and simplifying the resulting equation gives us the curve in the statement of the corollary. □

Remark . The curve in Corollary 5 is none other than X_{0}(22). An isomorphism from the model

Y^{2} = (X^{3} + 2X^{2} − 4X + 8)(X^{3} − 2X^{2} + 4X − 4)

for X_{0}(22) given in [12] to our curve

y^{2} = −2x^{6} − 10x^{4} + 26x^{2} + 242

is given by (x, y) = (1 − 4/X, 16Y/X^{3}).

Our next two results give sufficient conditions for there to be a genus-2 curve whose Jacobian is isogenous to the product of a given elliptic curve with itself. (The sufficient conditions given by Kani [16] apply only when k is algebraically-closed.)

Corollary 6. Suppose E is an elliptic curve over a field k of characteristic p > 2. If j(E) is neither 0 nor 1728, then there exists a genus-2 curve over k whose Jacobian is isogenous over k to E × E.

Proof. First suppose that j(E) ∉ F_{p}. Let y^{2} = f be a Weierstrass equation for E, and let F be the elliptic curve over k defined by y^{2} = f^{(p)}, where f^{(p)} is the polynomial obtained from f by replacing every coefficient with its pth power. Note that j(F) = j(E)^{p} ≠ j(E) since j(E) ∉ F_{p}. The Frobenius morphism (x, y) ↦ (x^{p}, y^{p}) from E to F defines a Galois-module isomorphism ψ: E[2](K) → F [2](K). The curves E and F have different j-invariants and so are not isomorphic to one another, so Proposition 3 provides us with the required genus-2 curve.

Next suppose that j(E) ∈ F_{p}. Suppose we could find a Galois-module automorphism ψ of E[2](K) other than the identity. Such an automorphism could not come from an automorphism of E_{K} because our assumption on j(E) implies that Aut E_{K} = {±1}, so Proposition 3 would give us a curve whose Jacobian is (2, 2)-isogenous to E × E. Thus we will be done if we can find such a ψ.

If #E[2](k) = 4 then E[2](K) is a trivial Galois module, and we may take ψ to be any one of the 5 non-trivial automorphisms of E[2](K).

If #E[2](k) = 2, we may take ψ to be the automorphism of E[2](K) that swaps the two nonrational points.

Finally, suppose #E[2](k) = 1. The fact that Aut E_{K} = {±1} implies that E has a model of the form dy^{2} = f, where f ∈ F_{p}[x] and d ∈ k. The 2-torsion points of E are defined over the splitting field of f, which in this case is a cyclic cubic extension of k because f ∈ F_{p}[x]. Therefore the two group automorphisms of E[2](K) that rotate the nonzero points are actually Galois-module isomorphisms, and we have two choices for ψ. □

In characteristic 0 it is a little more difficult to find such curves C.

Corollary 7. Let E be an elliptic curve over a field k of characteristic not 2, and suppose End E_{K} = Z. If E contains a k-rational finite subgroup-scheme G that is not the kernel of multiplication-by-N for any N, then there exists a genus-2 curve over k whose Jacobian is isogenous over k to E × E.

Proof. We may assume that the order of G is a prime q. If q = 2, then there is a nonzero k-rational point of order 2, and we may use in Proposition 3 the isomorphism ψ : E[2](K) → E[2](K) that interchanges the other 2-torsion points. If q is an odd prime, then the natural isogeny from E to the elliptic curve F = E/G over k defines an isomorphism of Galois-modules ψ : E[2](K) → F [2](K).

The condition End E_{K} = Z ensures that E_{K} and F_{K} are not isomorphic, so the result again follows from Proposition 3. □

Remark . The conclusion of Corollary 7 holds for some elliptic curves in characteristic 0 that do not satisfy the corollary’s hypotheses. For example, let E be the elliptic curve

E : y^{2} = x^{3} − 169x + 845

over Q. The cubic on the right is irreducible, and has square discriminant 13^{4}, so its Galois group is A_{3}. Therefore any isomorphism ψ : E[2](\overline{Q}) → E[2](\overline{Q}) that rotates the three non-trivial 2torsion points will be defined over Q. Since j(E) ≠ 0, such a rotation cannot be the restriction of an automorphism of E, so by Proposition 3, we obtain a genus-2 curve over Q whose Jacobian is (2, 2)-isogenous over Q to E × E. On the other hand, E is curve 676D1 in [5], which has no Q-rational cyclic subgroups.

We can use Corollary 7 to construct genus-2 curves over Q whose Jacobians have high rank, as was also noticed by Stéfane Fermigier.

Corollary 8. The Jacobian of the curve

y^{2} = −1707131824107329945 · (x^{2} + 55871769054504519799033274614104129)

· (x^{4} − 1086862437115841494920959046499163042x^{2}

+ 3121654577279888882305769763628790308995888274656243920700573254848641)

has rank 28 over Q.

Proof. According to [7], the elliptic curve

E : y^{2} = x(x^{2} + 2429469980725060x + 275130703388172136833647756388)

has rank 14, and (0, 0) is a rational 2-torsion point on E. The j-invariant is

\frac{483941743120924000812123996730853715578647268051688786879688}{5250870830712351132421548861849566889806152906127048721},

which is not an integer, so E cannot have complex multiplication. Using Corollary 7 and the formulas of Proposition 4, we obtain the desired genus-2 curve over Q whose Jacobian is (2, 2)- isogenous to E × E. □

3.3. A modular interpretation. One of our goals in this paper is to construct curves over Q of genus 2 whose Jacobians have large rational torsion subgroups, and our strategy will be to use Proposition 4 to “tie together” two elliptic curves that each have large torsion subgroups. In particular, every curve C we construct will come equipped with a (2, 2)-isogeny E × F → Jac C, where E and F have some particular rational torsion structure. We would like to construct the moduli space of curves equipped with such isogenies.

Suppose k is a field and K is a separable closure of k. Pick a set of elements {ζ_{M} : M ∈ Z_{>0}} of K such that ζ_{M} generates the group of Mth roots of unity in K and ζ_{M} = ζ_{jM}^{j} for all integers M, j > 0. By a full level-M structure on an elliptic curve E/K we mean a pair of points (P, Q) in E(K) that form a Drinfeld basis for E[M] (see [17], Chapter 1) and such that P and Q pair to ζ_{M} under the Weil pairing on E[M]. This corresponds to the moduli problem denoted in [17] by [Γ(M)]^{can} (see [17], Sections 3.1 and 9.1), but only because we are working over a field — we would have to be more careful with the roots of unity otherwise. There is an obvious right action of the group SL_{2}(Z/MZ) on the set of full level-M structures on a given curve E. Suppose G is a subgroup of SL_{2}(Z/MZ); by a partial level-M structure of type G on a curve E/K we mean a G-orbit of full level-M structures on E. If N is a positive divisor of M, then an (N, M)-structure on an elliptic curve E/K is a pair (P, Q) of points on E(K) such that Q has “exact order M” (see [17], Chapter 1) and such that P and (M/N)Q form a full level-N structure on E; this is an example of a partial level-M structure. If E is an elliptic curve over k, then by a partial level-M structure of type G on E we mean a partial level-M structure of type G on E_{K} that is stable under the action of Gal(K/k).

We let X(M) denote the usual compactified coarse moduli space of elliptic curves with full level-M structure; we view X(M) as a curve over k(ζ_{M}). Note that if char k divides M then X(M) will have several components. For every subgroup G of SL_{2}(Z/MZ) there is also a modular curve, which we will denote by X(M; G), that parameterizes elliptic curves with partial level-M structure of type G. The curve X(M; G) is a k(ζ_{M}^{a})-scheme, where a ∈ (Z/MZ)^{∗} is a generator of the subgroup det G ⊂ (Z/MZ)^{∗}. Finally, we denote by X_{1}(N, M) the modular curve that parameterizes elliptic curves with (N, M)-structure. The curve X_{1}(N, M) is a scheme over k(ζ_{N}).

Suppose the characteristic of the base field k is not 2, and suppose we are given two integers M and N and subgroups G ⊂ SL_{2}(Z/MZ) and H ⊂ SL_{2}(Z/NZ). Let ℓ be the smallest field containing the fields of definition of X(M; G) and X(N; H). We are interested in the functor F from the category of fields over ℓ to the category of sets defined as follows: If r ⊃ ℓ is a field with separable closure R, then F(r) is the set of all R-isomorphism classes of triples ((E, α), (F, β), ψ), where E is an elliptic curve with partial level-M structure α of type G over r, where F is an elliptic curve with partial level-N structure β of type H over r, and where ψ is a Galois-module isomorphism E[2](R) → F [2](R); here we say that ((E, α), (F, β), ψ) and ((E′, α′), (F′, β′), ψ′) are R-isomorphic if there are isomorphisms φ: (E, α)_{R} → (E′, α′)_{R} and χ: (F, β)_{R} → (F′, β′)_{R} such that ψ′ ∘ φ = χ ∘ ψ on E[2](R). We will show that this functor is represented by the ℓ-scheme Y^{0} defined in the next paragraph.

The modular curve X(2) is defined over k, and since char k ≠ 2 it has only one component. The covering X(2) → X(1) is Galois with group S = SL_{2}(Z/2Z), and the action of an element s ∈ S on a point X(2) is determined by its action on the triple (E, P, Q) corresponding to that point. Let Z_{1} = X(2)_{ℓ} ×_{X(1)} X(M; G)_{ℓ} and let Z_{2} = X(2)_{ℓ} ×_{X(1)} X(N; H)_{ℓ}, where ×_{X(1)} means the fiber product over X(1)_{ℓ}. The covers Z_{1} → X(M; G)_{ℓ} and Z_{2} → X(N; H)_{ℓ} are Galois with group S. Let Z be the 2-dimensional ℓ-scheme Z_{1} × Z_{2}, where × means the fiber product over Spec ℓ, let S act on the cover Z → X(M; G)_{ℓ} × X(N; H)_{ℓ} diagonally, and let Y be the quotient surface of Z by this action. Finally, let Y^{0} be the open subvariety of Y that lies over the open subvariety of X(1)_{ℓ} × X(1)_{ℓ} where neither factor is ∞.

Proposition 9. The scheme Y^{0} represents F.

Proof. First let us determine how to describe the r-valued points on Y^{0}. Let y be such a point. Then y corresponds to a Gal(R/r)-stable S-orbit of elements of Z(R) that lie over the finite part of X(1)_{R} × X(1)_{R}. Let z be one of the points in this orbit. According to the modular interpretation of Z, the point z corresponds to the R-isomorphism class of a quadruple

((E, P, Q), (E, α), (F, U, V), (F, β)),

where E and F are elliptic curves over R, where P and Q are independent 2-torsion points in E(R) and U and V are independent 2-torsion points in F (R), where α is a partial level-M structure of type G on E, and where β is a partial level-N structure of type H on F. When an element of S is applied to this quadruple, the only things that get changed are the points P, Q, U, and V, so the fact that the S-orbit containing z is defined over r means that (E, α) is isomorphic to all of its Galois conjugates and (F, β) is isomorphic to all of its Galois conjugates. According to Proposition 3.2 (p. 274) of [6], this means that (E, α) and (F, β) can be defined over r. If we think of E and F as curves over r, then the fact that the S-orbit of z is defined over r means exactly that the group isomorphism ψ: E[2](R) → F [2](R) defined by sending P to U and Q to V is Galois equivariant. Thus, a point y ∈ Y^{0}(r) gives us a triple ((E, α), (F, β), ψ) — but only up to R-isomorphism. And clearly the R-isomorphism class of such a triple will give us a point on Y^{0}. This gives us a bijection between F(r) and Hom(Spec r, Y^{0}) for every r, and this collection of bijections is easily seen to provide a natural equivalence F ↔ Hom( · , Y^{0}). □

Let W be the open subscheme of Y^{0} whose r-valued points correspond to R-isomorphism classes of triples ((E, α), (F, β), ψ) such that ψ does not come from an isomorphism between E_{R} and F_{R}. From Proposition 3 we see that the r-valued points of W correspond to R-isomorphism classes of triples (C, (E, α), (F, β)), where (E, α) is an elliptic curve with partial level-M structure of type G over r, where (F, β) is an elliptic curve with partial level-N structure of type H over r, and where C is a curve of genus 2 over r provided with a (2, 2)-isogeny E × F → Jac C that takes twice the canonical polarization of E × F to the canonical polarization of Jac C. We abbreviate this by saying that W is the moduli space for such triples.

Corollary 10. Let M′ be the least common multiple of 2 and M and let N′ be the least common multiple of 2 and N. Every geometric component of W is an open subvariety of a quotient surface of X(M′)_{K} × X(N′)_{K}.

Proof. Let Z_{1} and Z_{2} be as in the construction of Y above and let φ and ψ be the natural quotient maps from X(M′)_{K} to X(2)_{K} and to X(M; G)_{K}, respectively. For every s in the covering group S of X(2)/X(1) we get a morphism Φ_{s}: X(M′)_{K} → (Z_{1})_{K} by combining the morphisms sφ and ψ. It is clear from the modular interpretation of these schemes that the maps Φ_{s} provide a surjective morphism from the sum of six copies of X(M′)_{K} to (Z_{1})_{K}. Similarly we find a surjective morphism from the sum of six copies of X(N′)_{K} to (Z_{2})_{K}. Therefore every component of Z_{K} = Z_{1K} × Z_{2K} is a quotient surface of X(M′)_{K} × X(N′)_{K}, and every component of W_{K} is an open subvariety of a quotient surface of X(M′)_{K} × X(N′)_{K}. □

We will be interested in finding genus-2 curves over Q whose Jacobians are equipped with (2, 2)- isogenies from a product of elliptic curves with specified rational torsion structures. Thus we will want to look at the Q-rational points on the moduli space W, and it would be particularly nice to find subvarieties of W whose Q-rational points are Zariski dense. In the next few sections we will find such subvarieties for several different choices of torsion structures, although we will not phrase our arguments in terms of moduli spaces.

Example . Suppose we are interested in tying together an elliptic curve with (2, M)-structure and an elliptic curve with (2, N)-structure, where M and N are even integers. It is easy to check that X(2) ×_{X(1)} X_{1}(2, M) is the sum of six copies of X_{1}(2, M), and X(2) ×_{X(1)} X_{1}(2, N) is the sum of six copies of X_{1}(2, N). The group S acts on each of these varieties by permuting the summands, and the quotient surface Y is the sum of six copies of X_{1}(2, M) × X_{1}(2, N). Thus in this case W has a very simple structure. The reader is encouraged to work out the structure of W for other pairs of partial level structures and to keep the results in mind when reading the following sections.

3.4. Building Jacobians from elliptic curves E with #E(Q)[2] = 4. If two elliptic curves over Q are to have 2-torsion subgroups isomorphic as Galois-modules, it is necessary that they have the same number of rational 2-torsion points! In this section we consider the case where this number is 4, so we are concerned with the families of elliptic curves over Q with torsion subgroup containing Z/2Z × Z/2NZ, for N ≤ 4. Any member of the family with N = N_{1} can be paired with any member of the family with N = N_{2}, since the 2-torsion subgroups are automatically isomorphic as (trivial) Galois-modules. Moreover the generic members of each family (choosing a different indeterminate parameter for each) are clearly not isomorphic to each other, so by Proposition 3, we get 2-parameter families of genus-2 curves whose Jacobians map via a (2, 2)-isogeny to E_{1} × E_{2}. In other words, we have shown that the corresponding moduli space is a union of rational surfaces over Q. (This also follows immediately from the example at the end of the preceding section.) That these families really have two parameters can be seen from the fact that over C, one can specify the j-invariants of the two elliptic curves arbitrarily and independently^{2}. Similar arguments apply later in this paper; we leave the details to the reader.

The product of the rational torsion in the two elliptic curves does not map injectively to the rational torsion points of the Jacobian, but only a (2, 2)-subgroup is killed. The group structure of the image of this product in the Jacobian depends on N_{1} and N_{2}, but also on the choice of isomorphism between the 2-torsion of the two curves if N_{1} and N_{2} are even, since in this case each elliptic curve has a 2-torsion point which is distinguished by the property of being N_{i} times another rational torsion point. For instance, if N_{1} = N_{2} = 4, elementary calculations with abelian groups show that this group has structure Z/2Z × Z/4Z × Z/8Z if these special 2-torsion points are identified under ψ, and Z/8Z × Z/8Z otherwise. If N_{1} = 3 and N_{2} = 4, then we obtain Z/2Z × Z/24Z. If N_{1} = N_{2} = 3, then we obtain Z/6Z × Z/6Z. We have not considered the cases where N_{i} ≤ 2, since these cases lead to subgroups of the above.

Although some rational 2-power torsion is lost upon passing from E ×F to the Jacobian, there is also the possibility that some 2-power torsion can be gained: a non-rational point on E × F might map to a rational point on J. This phenomenon will be explored in Section 3.7.

3.5. Building Jacobians from elliptic curves E with #E(Q)[2] = 2. We now consider elliptic curves E and F having torsion subgroups Z/NZ and Z/N′Z with even N, N′ ≤ 12. An isomorphism of Galois-modules from E[2](\overline{Q}) to F [2](\overline{Q}) must map the rational 2-torsion point to the rational 2-torsion point, so we see that such an isomorphism exists if and only if the quadratic field over which the non-rational 2-torsion points of E are defined equals the quadratic field for F, and this holds if and only if the discriminants of E and F are equal modulo squares. We are thus led to the problem of finding the rational solutions to

(1)  Δ_{N}(t)y^{2} = Δ_{N′}(u)

outside the 1-dimensional closed subset corresponding to cases where E or F degenerates or where j(E) = j(F). Each such solution gives rise to a Jacobian of a genus-2 curve over Q whose torsion subgroup contains the quotient of Z/NZ × Z/N′Z by the identification of the points of order 2 in each factor.

If N′ = 4, then (1) is

Δ_{N}(t)y^{2} = 16u + 1,

which is a rational surface over Q, since we can solve for u in terms of t and y. In particular, for N = 10, we obtain a 2-parameter family of Jacobians whose torsion subgroups contain Z/20Z. (The other N will give results which are subsumed in our other results.)

If N′ = 6, then (1) is

(2)  Δ_{N}(t)y^{2} = (u + 1)(9u + 1).

This can be considered as a conic over Q(t) with a Q(t)-rational point, namely (u, y) = (−1, 0), and it is easy to see that this makes (2) a rational surface. In particular, for N = 10 or N = 12, we obtain a 2-parameter family of Jacobians whose torsion subgroups contain Z/30Z or Z/3Z×Z/12Z, respectively.

If N′ = 10 and N = 8, then (1) is

(8t^{2} − 8t + 1)y^{2} = (2u − 1)(4u^{2} − 2u − 1).

If we set t = (s^{2} − 2s + 3)/(4s^{2} + 4), we obtain a split elliptic surface over the s-line, and the Q(s)- rational point (u, y) = (−1/2, (2s^{2} + 2)/(s^{2} − 2s − 1)) is of infinite order, since its specialization at s = 0 is of infinite order on the resulting elliptic curve over Q. Thus we have an elliptic surface over P^{1} of positive rank, and the Q-rational points on this surface outside of the 1-dimensional set of degenerate solutions parameterize Jacobians over Q whose torsion subgroups contain Z/40Z.

2Actually, one should choose the j-invariants to be different, so that the elliptic curves are guaranteed not to be isomorphic, but this is an open condition, so the number of parameters is not reduced by this constraint.

If N′ = N = 10, then (1) is

(2t − 1)(4t^{2} − 2t − 1)y^{2} = (2u − 1)(4u^{2} − 2u − 1)

which is an elliptic surface over the t-line, and (u, y) = (t, 1) is a Q(t)-rational point of infinite order, since under the obvious isomorphism over Q(t)(\sqrt{(2t − 1)(4t^{2} − 2t − 1)}) to the elliptic curve

y^{2} = (2u − 1)(4u^{2} − 2u − 1)

it maps to a point with non-constant u-coordinate. Hence we obtain a positive rank elliptic surface whose points (outside a 1-dimensional set) parameterize Jacobians whose torsion subgroups contain Z/5Z × Z/10Z.

Similarly, if N′ = N = 12, then (1) is

(2t^{2} − 2t + 1)(6t^{2} − 6t + 1)y^{2} = (2u^{2} − 2u + 1)(6u^{2} − 6u + 1),

which again is an elliptic surface over the t-line if we choose (u, y) = (t, 1) as the zero section. We then have the Q(t)-rational point (u, y) = (t, −1), which is of infinite order, for the same reason as in the previous case. Hence we obtain a family of Jacobians, parameterized by the points on an open subset of a positive rank elliptic surface, whose torsion subgroups contain Z/6Z × Z/12Z.

Finally, if N′ = 10 and N = 12, then (1) is

(3)  (2t^{2} − 2t + 1)(6t^{2} − 6t + 1)y^{2} = (2u − 1)(4u^{2} − 2u − 1).

If we choose t = 1/3, the resulting elliptic curve is curve 900A1 of [5], which has rank 1. There are only finitely many rational points on this elliptic curve that give u such that E_{10}^{u} degenerates or is isomorphic to E_{12}^{1/3}, so we obtain a family of Jacobians, parameterized by the points on an open subset of a positive rank elliptic curve, whose torsion subgroups contain Z/60Z.

Remark . In fact, there are infinitely many other specializations of t for which (3) becomes an elliptic curve of positive rank.

3.6. Building Jacobians from elliptic curves E with #E(Q)[2] = 1. Here we consider elliptic curves E and F having torsion subgroups Z/NZ and Z/N′Z, respectively, with N, N′ odd (and at most 9). For an elliptic curve y^{2} = f(x) with trivial rational 2-torsion, each non-trivial 2-torsion point is defined over a cubic extension, namely the extension obtained by adjoining a root of f(x). The Galois-modules E[2](\overline{Q}) and F [2](\overline{Q}) are isomorphic if and only if the corresponding cubic fields are isomorphic. In this case, the discriminants of the elliptic curves must be equal modulo squares. The converse is not quite true (cubic fields having the same discriminant modulo squares are not necessarily isomorphic), but it will turn out that the discriminants often contain enough information for our purposes.

A short search for solutions to Δ_{7}(t) = Δ_{9}(u) modulo squares (and such that the discriminants do not vanish) leads to the solution (t, u) = (−16/3, 4). PARI shows that the corresponding cubic fields are both isomorphic to the unique cubic field of discriminant −2964. (Uniqueness can be seen from

the tables obtainable by ftp at megrez.math.u-bordeaux.fr in directory /pub/numberfields.)

Hence we find a genus-2 curve whose Jacobian has a rational torsion point of order 63. Following the recipe given by Proposition 4 gives an explicit model for this genus-2 curve. After a few simplifying changes of variable, we obtain the model

(4)  C : y^{2} = 897x^{6} − 197570x^{4} + 79136353x^{2} − 146398496.

Let D be the divisor (R) + (R′) − (∞^{+}) − (∞^{−}) on C, where

R = (\frac{−69 + \sqrt{4369}}{2}, 4515015 − 68241\sqrt{4369}) ,

where R′ is the Galois conjugate of R, and where ∞^{+} and ∞^{−} are the two points at infinity on a desingularized model of C. One can check that D maps to a 9-torsion point on one of the elliptic quotients of C and to a 7-torsion point on the other elliptic quotient (see [14]), so D represents a divisor of order at least 63 on C. Since C has good reduction at 5, and since there is only one positive multiple of 63 less than the Hasse-Weil bound (1 + \sqrt{5})^{4} for #(Jac C)(F_{5}), we must have #(Jac C)(F_{5}) = 63, and hence the torsion subgroup of (Jac C)(Q) is isomorphic to Z/63Z and is generated by the class of D. It seems likely that there will be only finitely many genus-2 curves over Q whose Jacobians possess a rational 63-torsion point. It is perhaps even the case that the curve (4) is the only one.

Similarly, we find the solution (t, u) = (7, −14/13) to Δ_{7}(t) = Δ_{7}(u) modulo squares. (We must be careful to exclude solutions where u = t, u = 1/(1 − t), or u = (t − 1)/t, since these correspond to taking the same elliptic curve but with one 7-torsion point a multiple of the other.) For these values of t and u, the corresponding cubic fields turn out to be isomorphic, so we indeed obtain a curve C whose Jacobian contains a subgroup of rational points isomorphic to Z/7Z × Z/7Z. Using Proposition 4, we find the model

C : y^{2} = x^{6} + 3025x^{4} + 3232987x^{2} + 869675859

for this curve. The Jacobian of the reduction of C modulo 5 is isogenous to the product of two elliptic curves each with exactly 7 points (7 being the only multiple of 7 less than the Weil bound), so the Jacobian of the reduction has exactly 49 points. Thus we find that the rational torsion on the Jacobian of C is in fact isomorphic to Z/7Z × Z/7Z.

To handle some of the other cases (in particular those with N = 3) we will use the following lemma. The restrictions on E are not necessary, but we only need the result under these restrictions.

Lemma 11. Let k be a field whose characteristic is neither 2 nor 3. If E is an elliptic curve over k such that E[2](k) is trivial and j(E) ≠ 0, 1728, then there is a 1-parameter family of elliptic curves E′ over k such that E′ has a k-rational 3-torsion point and E′[2] ≅ E[2] as Gal(\overline{k}/k)-modules.

Proof. Write E in Weierstrass form as y^{2} = x^{3} + Ax + B (so A, B ≠ 0), and let r be a root of x^{3}+Ax+B. We claim that specializing t to −B^{2}/A^{3} in the universal elliptic curve y^{2} = x^{3}+(x+t)^{2} over X_{1}(3) gives one E′ with the desired properties. A calculation shows that s = −(B/Ar)^{2} is a root of x^{3} + (x − B^{2}/A^{3})^{2} in k(r), and s cannot be in k, since r is at most quadratic over k(s). Thus k(r) and k(s) are the same cubic extension of k, and hence the curves

y^{2} = x^{3} + Ax + B  and  y^{2} = x^{3} + (x − B^{2}/A^{3})^{2}

have isomorphic 2-torsion as Galois-modules.

The set of all E′ with the desired properties corresponds to the k-rational points of a twist of the modular curve X_{1}(2, 6) classifying elliptic curves with a 3-torsion point and full level-2 structure. But this modular curve is rational, and the previous paragraph shows that our twist of it has a k-rational point, so our twist is a rational curve over k, and we obtain the desired 1-parameter family. □

Applying the lemma with k = Q(t) and E as the universal elliptic curve with a 7-torsion point yields a 2-parameter family of pairs of elliptic curves with a 3-torsion point and a 7-torsion point respectively, and having isomorphic 2-torsion as Galois-modules, so by Proposition 3, we obtain a 2-parameter family of genus-2 curves over Q whose Jacobians possess a rational 21-torsion point. Similarly, if we take E = E_{9}^{t} we obtain a 2-parameter family of genus-2 curves over Q whose Jacobians have torsion subgroup containing Z/3Z × Z/9Z.

Next we construct infinitely many genus-2 curves with a rational 35-torsion point. Let E be the elliptic curve E_{7}^{−1} with a rational 7-torsion point. The elliptic curves E′ over Q equipped with a rational 5-torsion point and a Galois-module isomorphism E′[2] → E[2] correspond to the rational points on a twist X′ of X_{1}(2, 10). Now X_{1}(2, 10) is a covering of X_{1}(5) with Galois group GL_{2}(Z/2Z) ≅ S_{3}, and the subgroup A_{3} corresponds by Galois theory to an intermediate covering whose function field is the quadratic extension of Q(t) (where t is the parameter on X_{1}(5)) obtained by adjoining the square root of the discriminant of the cubic f_{t}(x) if E_{5}^{t} is written in the form y^{2} = f_{t}(x). This function field is of genus 1, since from Table 4, Δ_{5}(t) = t(t^{2} − 11t − 1). But X_{1}(2, 10) is an elliptic curve as well (see [18]), so its map down to the intermediate covering must be an isogeny (in fact, a 3-isogeny). Similarly our twist X′ of X_{1}(2, 10) is a genus-1 curve with a degree-3 map to the unique intermediate covering X′′ of degree 2 over X_{1}(5). The curve X′′ classifies elliptic curves E′ with a 5-torsion point and a Galois-stable A_{3}-orbit of isomorphisms E′[2] → E[2]. There are two A_{3}-orbits, and they are defined over Q(\sqrt{Δ_{7}(−1)Δ_{5}(t)}), since an automorphism of this field over Q(t) is trivial on Q(\sqrt{Δ_{7}(−1)}) if and only if it is trivial on Q(Δ_{5}(t)), which means the signatures of its permutation actions on the nonzero 2-torsion points of E and E_{5}^{t} must be the same. Thus X′′ is the genus-1 curve y^{2} = Δ_{7}(−1)Δ_{5}(t), i.e.,

y^{2} = −26t(t^{2} − 11t − 1).

This is an elliptic curve of conductor 54080, which is too large for it to be listed in the tables of [5], but Cremona’s rank-computing program shows that it has rank 2; the points (−2/13, 22/13) and (−26, 806) are independent of one another and have infinite order. The genus-1 twist X′ of X_{1}(2, 10) has a rational point, because a PARI search finds an elliptic curve E_{5}^{t}, with t = 1/26, such that the cubic field (of discriminant −104) obtained by adjoining a 2-torsion point is the same as that obtained by adjoining a 2-torsion point of E. Thus X′ is an elliptic curve 3-isogenous to X′′. In particular, X′ has rank 2, so it has infinitely many rational points, all but finitely many of which give rise to genus-2 curves whose Jacobians possess a subgroup Z/5Z × Z/7Z ≅ Z/35Z.

Similarly the elliptic curve E = E_{9}^{−5} with a rational 9-torsion point has 2-torsion subgroup isomorphic as Galois-module to that of the elliptic curve E_{5}^{93/10}, and the elliptic curve

y^{2} = Δ_{9}(−5)t(t^{2} − 11t − 1)

of conductor 13838400 (!) has rank 2 again according to Cremona’s program, with (−10/93, 6970/93) and (−640/27, 5860240/81) as independent points of infinite order. Thus we obtain infinitely many genus-2 curves whose Jacobians possess a rational 45-torsion point.

3.7. Gaining 2-power torsion. Let k be a field of characteristic not 2, let K be a separable closure of k, and let G_{k} = Gal(K/k). If E is an elliptic curve over k, then E[2] \ {0} ≅ Spec L where L is a separable k-algebra of dimension 3. Explicitly, if E is in the form y^{2} = f(x) with f(x) ∈ k[x] a cubic polynomial, then we may take L = k[T]/(f(T)). As is well known (see [2], [3], [35]),

H^{1}(G_{k}, E[2]) ≅ ker (L^{∗}/L^{∗2} ⟶^{Norm} k^{∗}/k^{∗2})

and the composition

E(k)/2E(k) ↪ H^{1}(G_{k}, E[2]) ≅ ker (L^{∗}/L^{∗2} ⟶^{Norm} k^{∗}/k^{∗2})

is a map ι sending a rational non-2-torsion point P with x-coordinate x_{P} to the image of x_{P} − T. When P is a non-trivial rational 2-torsion point, x_{P} − T vanishes in exactly one component of L, and ι(P) equals the image of x_{P} − T is all but this component; the image of P in this last component (which is a copy of k) can be deduced up to squares from knowing that ι(P) is in the kernel of the norm.

Proposition 12. Let f(x) and g(x) be cubic polynomials in k[x] such that

E : y^{2} = f(x)  and  F : y^{2} = g(x)

are elliptic curves admitting an isomorphism of G_{k}-modules ψ : E[2](K) → F [2](K). Define L and ι as above for E, and similarly define L′ and ι′ for F. The map ψ induces an isomorphism ψ̃ : L′ → L. Let A be the quotient of E × F by the graph of ψ.

(a) If a point (P_{0}, Q_{0}) of (E × F)(K) maps to a k-rational point on A, then 2P_{0} ∈ E(k) and 2Q_{0} ∈ F (k).

(b) Given P ∈ E(k) and Q ∈ F (k), there exists a point (P_{0}, Q_{0}) of (E × F)(K) that maps to a k-rational point on A and such that 2P_{0} = P and 2Q_{0} = Q, if and only if ι′(Q) corresponds to ι(P) (up to squares) under the isomorphism ψ̃.

Proof. Let λ be the principal polarization of A derived from the principal polarization on E × F. If we compose the isogeny E × F → A with λ and the dual isogeny Â → E × F, the result is multiplication-by-2 on E × F, so part (a) is clear.

Now let P ∈ E(k) and Q ∈ F (k). Suppose that there exists (P_{0}, Q_{0}) ∈ (E × F)(K) that maps to a k-rational point on A and such that 2P_{0} = P and 2Q_{0} = Q. This means that (P_{0}, Q_{0})^{σ} −(P_{0}, Q_{0}) is in the graph of ψ for all σ ∈ G_{k}. In particular, under the map induced by ψ, the class of ξ_{σ} := P_{0}^{σ} − P_{0} in H^{1}(G_{k}, E[2]) is mapped to the class of ξ′_{σ} := Q_{0}^{σ} − Q_{0}. In other words, ψ̃ takes ι′(Q) to ι(P).

Conversely suppose that ψ̃ takes ι′(Q) to ι(P). This means that the map induced by ψ takes the image of P in H^{1}(G_{k}, E[2]) (under the coboundary map) to the image of Q in H^{1}(G_{k}, F [2]). Fix P_{1} ∈ E(K) such that 2P_{1} = P and Q_{1} ∈ F (K) such that 2Q_{1} = Q. Then there exist 2-torsion points R ∈ E[2](K) and S ∈ F [2](K) such that

ψ(P_{1}^{σ} − P_{1} + R^{σ} − R) = Q_{1}^{σ} − Q_{1} + S^{σ} − S

for all σ ∈ G_{k}. Let P_{0} = P_{1} + R and Q_{0} = Q_{1} + R. Then 2P_{0} = P, 2Q_{0} = Q, and

(P_{0}, Q_{0})^{σ} − (P_{0}, Q_{0})

is in the graph of ψ for all σ, so (P_{0}, Q_{0}) maps to a rational point on A. □

For elliptic curves E over Q with all 2-torsion points rational, L is simply Q×Q×Q, the factors corresponding to the non-trivial torsion points T_{1}, T_{2}, T_{3}. Now assume that E = E_{2,8}^{t}. Then ι(T_{2}) ∈ L is (x_{T_{2}} − x_{T_{1}}, ∗, x_{T_{2}} − x_{T_{3}}), where ∗ is determined by the condition that the product of all three components equal 1 (modulo squares). By the formulas in Table 5, we have (modulo squares)

ι(T_{2}) = (−1, −(8t^{2} − 1)(8t^{2} + 8t + 1), (8t^{2} − 1)(8t^{2} + 8t + 1)) ∈ (Q^{∗}/Q^{∗2})^{3} .

Now if F = E_{2,8}^{u} with non-trivial 2-torsion points T′_{1}, T′_{2}, T′_{3} and corresponding map ι′, then

ι′(T′_{2}) = (−1, −(8u^{2} − 1)(8u^{2} + 8u + 1), (8u^{2} − 1)(8u^{2} + 8u + 1)) ∈ (Q^{∗}/Q^{∗2})^{3} .

If ψ is the isomorphism E[2] → F [2] taking T_{i} to T′_{i}, then the map ψ̃ of Proposition 12 is simply the identity Q × Q × Q → Q × Q × Q. Thus by Proposition 12, if there exists y such that

(5)  (8t^{2} − 1)(8t^{2} + 8t + 1)y^{2} = (8u^{2} − 1)(8u^{2} + 8u + 1),

then there exists (P_{0}, Q_{0}) on E × F with double (T_{2}, T′_{2}) such that (P_{0}, Q_{0}) maps to a rational point on the quotient A of E × F by the graph of ψ. In this case, (P_{0}, Q_{0}) maps to a new rational 2-torsion point on A, not in the image of E(Q) × F (Q).

We can consider (5) as a genus-1 curve over Q(t), and we make it an elliptic curve by choosing (u, y) = (t, 1) as the origin. Then the point (u, y) = (t, −1) has infinite order, since the divisor (t, −1) − (t, 1) corresponds to a non-constant point on the Jacobian of

y^{2} = (8u^{2} − 1)(8u^{2} + 8u + 1),

which is isomorphic to (5) over Q(t)(\sqrt{(8t^{2} − 1)(8t^{2} + 8t + 1)}). Hence (5) is a positive rank elliptic surface whose points (outside a 1-dimensional set) parameterize a family of genus-2 curves whose Jacobians have torsion subgroup over Q containing Z/2Z × Z/2Z × Z/4Z × Z/8Z.

Let us now try to do the same for E = E_{2,6}^{t} and F = E_{2,6}^{u}. In this case, from Table 5 we compute

ι(T_{1}) = (2(t − 3)(t + 3)(t − 5), (t + 3)(t − 5), 2(t − 3)) ∈ (Q^{∗}/Q^{∗2})^{3} .

This time in order to get an extra 2-torsion point on A coming from a half of (T_{1}, T′_{1}), we need to find rational solutions to the system

(6)  2(t − 3) = 2(u − 3)y^{2}
(t + 3)(t − 5) = (u + 3)(u − 5)z^{2}.

(Note that the third condition

2(t − 3)(t + 3)(t − 5) = 2(u − 3)(u + 3)(u − 5) (modulo squares)

would then be automatic.) If we solve for t in the first equation and substitute into the second, we obtain the equation

((u − 3)y^{2} + 6) ((u − 3)y^{2} − 2) = (u + 3)(u − 5)z^{2},

which defines a genus-1 curve over Q(u). We make it an elliptic curve by choosing (y, z) = (1, 1) as origin, and then note that (y, z) = (−1, 1) is a point of infinite order, since it is of infinite order for the specialization u = 0. Thus the system (6) provides us with a positive rank elliptic surface whose points parameterize a family of genus-2 curves whose Jacobians have torsion subgroup over Q containing Z/2Z × Z/6Z × Z/6Z.

Next we investigate the possibility of gaining 2-power torsion when E = E_{2,6}^{t} and F = E_{2,8}^{u}. Let T_{1}, T_{2}, T_{3} and T′_{1}, T′_{2}, T′_{3} be the nontrivial 2-torsion points on E and F, respectively, as in Table 5. We have

ι(T_{1}) = (2(t − 3)(t + 3)(t − 5), (t + 3)(t − 5), 2(t − 3)) ∈ (Q^{∗}/Q^{∗2})^{3} ,
ι′(T′_{2}) = (−1, −(8u^{2} − 1)(8u^{2} + 8u + 1), (8u^{2} − 1)(8u^{2} + 8u + 1)) ∈ (Q^{∗}/Q^{∗2})^{3} .

In an attempt to obtain simpler equations than we would by mapping T_{i} to T′_{i} for each i, we let ψ : E[2] → F [2] be the isomorphism such that ψ(T_{1}) = T′_{2}, ψ(T_{2}) = T′_{3}, and ψ(T_{3}) = T′_{1}. Hence ψ̃ is the isomorphism Q × Q × Q → Q × Q × Q acting on the factors as the permutation (1 3 2). By Proposition 12, a point on E × F with double (T_{1}, T′_{2}) maps to a new rational point on the quotient A if and only if we can find rational numbers y and z such that

2(t − 3) = (−1)y^{2},
(t + 3)(t − 5) = (8u^{2} − 1)(8u^{2} + 8u + 1)z^{2}.

If we solve the first equation for t, substitute into the second, and multiply both sides by 4, we obtain

(y^{2} − 12)(y^{2} + 4) = 4(8u^{2} − 1)(8u^{2} + 8u + 1)z^{2}.

For y = 2/9, the resulting genus-1 curve has a rational point (u, z) = (1/3, 44/9), and is birational to the elliptic curve

Y^{2} = X^{3} − 1681X

of conductor 53792 and rank 2. The new rational point on A is a 2-torsion point, since its double is the image of (T_{1}, T′_{2}), which is in the graph of ψ. Hence we have produced a family of genus-2 curves, parameterized by the points on a positive rank elliptic curve, whose Jacobians have torsion subgroup over Q containing Z/2Z × Z/2Z × Z/24Z.

4. Genus three

4.1. Jacobians (2, 2, 2)-isogenous to a product of elliptic curves. In this section we will show how one can find a curve of genus 3 whose Jacobian is isogenous over a quadratic extension of the base field to a product of three given elliptic curves. Genus-3 curves of the sort we will see were used in [4].

We maintain the conventions of Section 3.1.

Suppose E_{1}, E_{2}, and E_{3} are elliptic curves over a separably closed field K, and let N be a positive integer not divisible by the characteristic of K. The product of the canonical polarizations on the E_{i} is a principal polarization λ on the abelian variety A = E_{1} × E_{2} × E_{3}, and the Weil pairings on the N-torsion subgroups of the E_{i} combine to give us a non-degenerate alternating pairing e_{N} from A[N] to the group scheme of Nth roots of unity over K. Suppose G is a sub-group-scheme of A[N] that is maximal isotropic with respect to the pairing e_{N}. As in the similar situation we saw in Section 3.2, the polarization Nλ on A reduces to a principal polarization μ on the quotient variety B = A/G. A result of Oort and Ueno [34] shows that the polarized variety (B, μ) either breaks up as a product of lower-dimensional polarized varieties or is the canonically polarized Jacobian of a curve C over K of genus 3.^{3} We would like to see which group-schemes G lead to curves in the case where N = 2. Since we will be working over a separably closed field, we will identify sub-group-schemes of A[2] with subgroups of A[2](K).

Lemma 13. Let A = E_{1} × E_{2} × E_{3} and e_{2} be as above. There are exactly 135 maximal isotropic subgroups G of A[2](K). Exactly 81 of these group-schemes are of the form G_{1} × G_{2}, where G_{1} is a maximal isotropic subgroup of E_{i}[2](K) for some i and G_{2} is a maximal isotropic subgroup of ∏_{j≠i} E_{j}[2](K); for these G, the polarized variety (B, μ) splits into a product of lower-dimensional polarized varieties. If G is one of the remaining 54 groups, then for each i we may label the nonzero elements of E_{i}[2](K) by the symbols P_{i}, Q_{i}, and R_{i} in such a way so that G is the group

{(0, 0, 0),(0, Q_{2}, Q_{3}), (Q_{1}, 0, Q_{3}), (Q_{1}, Q_{2}, 0),
(P_{1}, P_{2}, P_{3}), (P_{1}, R_{2}, R_{3}), (R_{1}, P_{2}, R_{3}), (R_{1}, R_{2}, P_{3})}.

Remark . In fact, our constructions below will show that every group of the last type gives rise to a curve. This fact can also be proven by assuming that a G of the given type is the kernel of a map A → A_{1} × A_{2} of polarized varieties and obtaining a contradiction. In anticipation of this result, we will call maximal isotropic subgroups of A[2](K) (or sub-group-schemes of A[2]) non-split if they are of the latter type.

Proof. In any group isomorphic to (Z/2Z)^{6} with a non-degenerate alternating pairing, there are (2^{6} − 1)(2^{5} − 2)(2^{4} − 2^{2}) ways of choosing an ordered triple (v_{1}, v_{2}, v_{3}) that generate a maximal isotropic subgroup, and each maximal isotropic subgroup has (2^{3} − 1)(2^{3} − 2)(2^{3} − 2^{2}) such bases, so there are

\frac{(2^{6} − 1)(2^{5} − 2)(2^{4} − 2^{2})}{(2^{3} − 1)(2^{3} − 2)(2^{3} − 2^{2})} = 135

such subgroups.

For i = 1, 2, 3 let S_{i} denote the set of maximal isotropic subgroups of A[2](K) that can be written G_{1} × G_{2}, with G_{1} ⊂ E_{i}[2](K) and G_{2} ⊂ ∏_{j≠i} E_{j}[2](K). There are 3 choices for G_{1} and (2^{4} − 1)(2^{3} − 2)/(2^{2} − 1)(2^{2} − 2) = 15 choices for G_{2}, so #S_{i} = 45. The intersection of any two of the S_{i} is the set S of subgroups of A[2](K) that can be written G_{1} × G_{2} × G_{3}, with G_{i} ⊂ E_{i}[2](K); clearly #S = 27. Thus there are #S_{1} + #S_{2} + #S_{3} − 2#S = 81 subgroups that split as in the statement of the lemma.

We are left with 135 − 81 = 54 subgroups to account for, and it is easy to see that there are exactly this many subgroups of the form described in the final sentence of the lemma: There are 3 choices for each of the Q_{i}, and given P_{1} and P_{2}, there are 2 choices for P_{3}. □

Suppose now that k is a field of characteristic not 2 with separable closure K, and let E_{1}, E_{2}, and E_{3} be elliptic curves over k. It is clear that a non-split sub-group-scheme of E_{1K} × E_{2K} × E_{3K} will come from a sub-group-scheme of E_{1} × E_{2} × E_{3} if and only if we have both that all of the points Q_{i} are defined over k and that every k-automorphism of K that moves any of the P_{i} moves exactly two of them. Given the first condition, the second condition will hold if and only if the product of the discriminants of the curves E_{i} is a square in k.

3The analogous statement is not necessarily true over a field that is not separably closed. See the remark following the proof of Proposition 15.

So suppose E_{i} is the elliptic curve over k given by the equation y^{2} = x(x^{2} + A_{i}x + B_{i}) where B_{i} ≠ 0, and let Q_{i} be the rational 2-torsion point (0, 0). The discriminant of E_{i} differs by a square factor from the number Δ_{i} = A_{i}^{2} − 4B_{i}, so let us assume that Δ_{1}Δ_{2}Δ_{3} is a square in k. For each i let P_{i} be a nonzero element of E_{i}[2](K) different from Q_{i}. Let G be the non-split sub-group-scheme of the product A = E_{1} × E_{2} × E_{3} corresponding to this choice of P ’s and Q’s. We are led to the question: Is the quotient polarized variety A/G the Jacobian of a curve over k, and if so, what equations define the curve?

Before we answer this question, we must define some numbers. For each i, we let d_{i} = −(A_{i} + 2x_{P_{i}}), where x_{P_{i}} denotes the x-coordinate of the point P_{i}. Note that d_{i}^{2} = Δ_{i}. The product R = d_{1}d_{2}d_{3} is an element of k because Δ_{1}Δ_{2}Δ_{3} was assumed to be a square. We let λ_{i} denote A_{i}/d_{i}, and we define the twisting factor (associated to the given E_{i}, P_{i}, and Q_{i}) to be the number

T = R (\frac{A_{1}^{2}}{Δ_{1}} + \frac{A_{2}^{2}}{Δ_{2}} + \frac{A_{3}^{2}}{Δ_{3}} − 1) − 2A_{1}A_{2}A_{3}
= d_{1}d_{2}d_{3} (λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2} − 2λ_{1}λ_{2}λ_{3} − 1) .

(The twisting factor is so named because it determines a quadratic extension of k over which A/G becomes isomorphic to a Jacobian; see Proposition 15.)

Proposition 14. With notation as above, suppose T = 0. Then each of the products B_{1}B_{2}, B_{1}B_{3}, and B_{2}B_{3} is a square, and A/G is isomorphic (over k) to the polarized Jacobian of the hyperelliptic curve C over k defined by the homogeneous equations

W^{2}Z^{2} = aX^{4} + bY^{4} + cZ^{4}
0 = dX^{2} + eY^{2} + fZ^{2},

where a, b, and c are given by

a = (\frac{RB_{1}}{2} ) (−\frac{B_{1}}{Δ_{1}} + \frac{B_{2}}{Δ_{2}} + \frac{B_{3}}{Δ_{3}} )
b = (\frac{RB_{2}}{2} ) ( \frac{B_{1}}{Δ_{1}} − \frac{B_{2}}{Δ_{2}} + \frac{B_{3}}{Δ_{3}} )
c = (\frac{RB_{3}}{2} ) ( \frac{B_{1}}{Δ_{1}} + \frac{B_{2}}{Δ_{2}} − \frac{B_{3}}{Δ_{3}} ) ,

where d, e, and f are determined up to sign by the relations

B_{2}B_{3}d^{2} = 1
B_{1}B_{3}e^{2} = 1
B_{1}B_{2}f^{2} = 1,

and where the signs of d, e, and f are chosen so that we have A_{1} = −aef and A_{2} = −bdf and A_{3} = −cde.

Proof. The statement that T = 0 is equivalent to the statement that λ_{1}^{2} +λ_{2}^{2} +λ_{3}^{2} −2λ_{1}λ_{2}λ_{3} −1 = 0. Solving this equation for λ_{3} in terms of λ_{1} and λ_{2} leads to

λ_{3} = λ_{1}λ_{2} ± \sqrt{(λ_{1}^{2} − 1)(λ_{2}^{2} − 1)},

and dividing this last equality by d_{3} gives

\frac{A_{3}}{Δ_{3}} = \frac{A_{1}A_{2} ± 4\sqrt{B_{1}B_{2}}}{R} .

Thus B_{1}B_{2} is a square in k. By symmetry, the numbers B_{1}B_{3} and B_{2}B_{3} are squares as well.

We leave it to the reader to verify that the signs of d, e, and f can be chosen so that the relations A_{1} = −aef and A_{2} = −bdf and A_{3} = −cde hold; this can be seen by noting that the squares of the desired relations, as well as the product of the desired relations, follow from the formulas given and the condition that T = 0.

Note that the coefficients d, e, and f are all nonzero, so the single equation dX^{2} +eY^{2} +fZ^{2} = 0 defines a nonsingular curve C′ of genus 0. The fact that none of the B_{i} is zero implies that none of the numbers a/d^{2} + b/e^{2}, a/d^{2} + c/f^{2}, and b/e^{2} + c/f^{2} is zero. Furthermore, the nonvanishing of the B_{i} together with the fact that T is zero can be used to show that

(\frac{a}{d^{2}} )(\frac{b}{e^{2}} ) + (\frac{a}{d^{2}} )(\frac{c}{f^{2}} ) + (\frac{b}{e^{2}} )(\frac{c}{f^{2}} )

is not zero. We leave it to the reader to show that the nonvanishing of the several expressions just mentioned guarantees that the element a(X/Z)^{4} + b(Y/Z)^{4} + c of the function field of C′ has exactly 8 zeroes, each simple, and exactly 2 poles, each of order 4. Since the total quotient ring of C is obtained from the function field of C′ by adjoining a square root of this function, we see that C is a hyperelliptic curve of genus 3.

It will suffice to prove that Jac C ≅ A/G in the special case where k has characteristic 0, for if k has positive characteristic we can simply lift all of the coefficients A_{i} and B_{i} up to the ring W of Witt vectors over k; to see that this can be done in such a way that T lifts to 0, we argue as follows. First we lift each Δ_{i} up to W in such a way that the product of the lifted values is a square in W, and we lift R to a square root of this product. Now we view T as a function of the three variables A_{i}. We will be able to use Hensel’s lemma to lift the A_{i} up to W so as to make T = 0 if any one of the partial derivatives ∂T/∂A_{i} is nonzero. We claim that at least one of these derivatives is nonzero. To prove this, let us assume that all three of the partial derivatives are zero and obtain a contradiction. From our assumption we find that RA_{i} = Δ_{i} ∏_{j≠i} A_{j} for each i. If any one of the A_{i} were zero, these three equalities would imply that all of the A_{i} were zero, which would contradict the assumption that T = 0. But if all of the A_{i} were nonzero, then by multiplying the three equalities together we would find that R = A_{1}A_{2}A_{3}, and this formula for R, combined with RA_{i} = Δ_{i} ∏_{j≠i} A_{j}, would show that A_{i}^{2} = Δ_{i}, which would lead to the impossibility B_{i} = 0. This proves our claim.

To prove that Jac C ≅ A/G in characteristic zero we need only consider the universal case, in which we let the d_{i} and the λ_{i} be indeterminates, we let ℓ be the quotient field of the domain

Q[d_{1}, d_{2}, d_{3}, λ_{1}, λ_{2}, λ_{3}]/(λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2} − 2λ_{1}λ_{2}λ_{3} − 1),

we take

A_{i} = λ_{i}d_{i}
B_{i} = (A_{i}^{2} − d_{i}^{2})/4
Q_{i} = (0, 0)
P_{i} = (−(d_{i} + A_{i})/2, 0)
R = d_{1}d_{2}d_{3},

and we let k be the subfield Q(A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}, R) of ℓ. (Note that k is fixed by the involution of ℓ that acts on λ_{1}, λ_{2}, d_{1}, and d_{2} by multiplication by −1, so k is a proper subfield of ℓ and contains (by symmetry) none of the d_{i}.) So let us assume that we are in the universal case.

Let ι_{X} be the involution on C that fixes W, Y, and Z and that sends X to −X. The involution ι_{X} gives us a double cover φ_{X}: C → F_{X} of curves over k, and we would like to find equations for the curve F_{X}. If we dehomogenize the equations for C with respect to Z by letting w = W/Z, x = X/Z, and y = Y/Z, and if we then divide by ι_{X} by defining u = x^{2}, we find that the quotient curve F_{X} is given by

w^{2} = au^{2} + by^{4} + c
0 = du + ey^{2} + f.

This pair of equations can be combined to get the single equation

(7)  v^{2} = (ae^{2} + bd^{2})y^{4} + 2aefy^{2} + (af^{2} + cd^{2}),

where v = dw. Using Example 3.7 (pp. 293–294) in Section X of [36], we see that the Jacobian of the genus-1 curve F_{X} is the elliptic curve E_{X} over k defined by

(8)  y^{2} = x(x^{2} + A_{X}x + B_{X})

where A_{X} = −aef and 4B_{X} = (aef)^{2} − (ae^{2} + bd^{2})(af^{2} + cd^{2}). Clearly we have A_{X} = A_{1}, and by using a little algebra and the fact that T = 0 we can see that B_{X} = B_{1}. Thus the double cover φ_{X} gives us a map φ_{X}^{∗} from E_{1} to the Jacobian J of C.

Similarly, the involution ι_{Y} on C that sends Y to −Y and that fixes the other variables gives us a degree-2 cover φ_{Y} from C to the curve F_{Y} over k given by the equation

v^{2} = (ae^{2} + bd^{2})x^{4} + 2bdfx^{2} + (bf^{2} + ce^{2}).

The Jacobian of F_{Y} is isomorphic to E_{2}, so we get a map φ_{Y}^{∗} from E_{2} to J.

Lastly, the involution ι_{Z} = ι_{X} ∘ ι_{Y} us a degree-2 cover φ_{Z}: C → F_{Z} to a curve F_{Z} over k whose Jacobian is isomorphic to E_{3}, so we get a map φ_{Z}^{∗} from E_{3} to J. (When dividing C by ι_{Z}, the reader may find it helpful to recast the first defining equation of C into the form V^{2}Y^{2} = aX^{4} +bY^{4} +CZ^{4} by letting V = W Z/Y; this will make it possible to dehomogenize with respect to Y and get equations similar to the ones obtained when dividing by ι_{X}.)

Let I denote the subgroup of the automorphism group of J generated by ι_{X}^{∗}, ι_{Y}^{∗}, and ι_{Z}^{∗}, and let C denote the category of abelian varieties over the separable closure K of k up to isogeny. The semisimple group ring Q[I] acts on the class [J] of J in C, and [J] splits into the direct sum of its eigenspaces. The class of the image of φ_{X}^{∗} consists of the sum of the eigenspaces on which ι_{X}^{∗} acts as 1, while the class of the image of φ_{Y}^{∗} consists of the sum of the eigenspaces on which ι_{Y}^{∗} acts as 1. However, the eigenspaces on which both ι_{X}^{∗} and ι_{Y}^{∗} act as 1 are trivial, because their sum is the class of the Jacobian of the quotient of C by the group ⟨ι_{X}, ι_{Y} ⟩, and this quotient has genus 0. Thus the classes of the images of φ_{X}^{∗} and φ_{Y}^{∗} have trivial intersection. Similarly, we find that the class of the image of φ_{Z}^{∗} shares no nonzero eigenspaces with the classes of φ_{X}^{∗} or φ_{Y}^{∗}. It follows that the morphism Φ^{∗} = φ_{X}^{∗} × φ_{Y}^{∗} × φ_{Z}^{∗} from A to J is an isogeny.

Let μ denote the canonical polarization of J. The fact that φ_{X} has degree 2 implies that φ̂_{X}^{∗}μφ_{X}^{∗} is the multiplication-by-2 map on E_{1}, and similarly φ̂_{Y}^{∗} μφ_{Y}^{∗} and φ̂_{Z}^{∗}μφ_{Z}^{∗} are the multiplication-by-2 maps on E_{2} and E_{3}; here indicates the dual morphism. We find that we have a commutative diagram

A ⟶^{2λ} Â
↓_{Φ^{∗}}  ^{↑Φ̂^{∗}}
J ⟶^{μ} Ĵ.

We see that Φ^{∗} has degree eight and its kernel G′ is a maximal isotropic sub-group-scheme of A[2]. The group-scheme G′ must be non-split. To complete the proof of the proposition we must show that it is equal to the given group-scheme G.

Since G′ is non-split, there must be nonzero elements Q′_{i} ∈ E_{i}[2](K) such that G′(K) contains the three elements (0, Q′_{2}, Q′_{3}), (Q′_{1}, 0, Q′_{3}), and (Q′_{1}, Q′_{2}, 0). As we noted before the statement of the proposition, the points Q_{i} must actually be defined over k. However, because d_{i} is not in k, the curve E_{i} has only one nonzero k-defined 2-torsion point, namely Q_{i}. Therefore Q′_{i} = Q_{i}. There are exactly two non-split maximal isotropic subgroups of A[2](K) that contain the subgroup

H = {(0, 0, 0), (0, Q_{2}, Q_{3}), (Q_{1}, 0, Q_{3}), (Q_{1}, Q_{2}, 0)},

and the intersection of these two groups is H. So to show that G′(K) = G(K), all we must do is show that the two groups contain a common element that is not in H. To show this, we can specialize our universal example to a particular case, and show that the specialized groups G and G′ contain a common element not in H.

Consider the specialization map ℓ → C that takes each d_{i} to −4 and each λ_{i} to −1/2; we will abuse notation by saying that d_{1} = −4, and so on. We see that each A_{i} = 2, each B_{i} = −3, each Δ_{i} = 16, each P_{i} = (1, 0), and the curve C is defined by the two equations W^{2}Z^{2} = −18(X^{4} + Y^{4} + Z^{4}) and 0 = (1/3)(X^{2} + Y^{2} + Z^{2}).

Let Φ_{∗} denote the map φ_{X∗} × φ_{Y ∗} × φ_{Z∗} from J to A and recall that Φ^{∗} denotes the map φ_{X}^{∗} × φ_{Y}^{∗} × φ_{Z}^{∗} from A to J. An easy computation shows that Φ^{∗}Φ_{∗} is multiplication by 2 on J. Thus the image under Φ_{∗} of a 2-torsion element of J is in the kernel of Φ^{∗}. To complete the proof, we will show that (P_{1}, P_{2}, P_{3}) ∈ A(K) is the image under Φ_{∗} of a 2-torsion point of J.

Let U_{1} and U_{2} be the Weierstrass points on C given in homogeneous coordinates [W : X : Y : Z] by [0 : ζ : ζ^{2} : 1] and [0 : ζ^{2} : ζ : 1], respectively, where ζ = e^{2πi/3}. Note that U_{1} − U_{2} represents a 2-torsion element of J. Under the specialization we have made, equation (7), which defines F_{X}, becomes v^{2} = −4y^{4} − 4y^{2} − 4, and in these (y, v) coordinates we have φ_{X}(U_{1}) = (ζ^{2}, 0) and φ_{X}(U_{2}) = (ζ, 0). The curve E_{X}, defined by equation (8), is given by y^{2} = x^{3} + 2x^{2} − 3x, and under the isomorphism

(y, v) ↦ (\frac{iv − y^{2} − 2}{y^{2}}, \frac{−2v − 2iy^{2} − 4i}{y^{3}})

from F_{X} to E_{X} these two points on F_{X} map to (i\sqrt{3}, 2ζ\sqrt{3}) and (−i\sqrt{3}, 2ζ^{2}\sqrt{3}), respectively. The difference of these two points on the elliptic curve E_{X} is equal to the 2-torsion point (1, 0). Thus φ_{X∗}(U_{1} − U_{2}) = P_{1}. By symmetry, we find that φ_{Y ∗}(U_{1} − U_{2}) = P_{2} and φ_{Z∗}(U_{1} − U_{2}) = P_{3} as well, so (P_{1}, P_{2}, P_{3}) is in the image of J[2](K) under Φ_{∗} and hence in G′(K). It is in G(K) as well, so G′ = G and the proposition is proved. □

Proposition 15. Let notation be as before Proposition 14, and suppose T ≠ 0. Let C be the plane quartic over k defined by

B_{1}X^{4} + B_{2}Y^{4} + B_{3}Z^{4} + dX^{2}Y^{2} + eX^{2}Z^{2} + fY^{2}Z^{2} = 0,

where

d = \frac{1}{2} (−A_{1}A_{2} + \frac{A_{3}R}{Δ_{3}})
e = \frac{1}{2} (−A_{1}A_{3} + \frac{A_{2}R}{Δ_{2}})
f = \frac{1}{2} (−A_{2}A_{3} + \frac{A_{1}R}{Δ_{1}}) .

Let k′ be the field k(\sqrt{T}). Then the polarized Jacobian of C_{k′} is isomorphic to the polarized variety A_{k′}/G_{k′}.

Proof. We leave it to the reader to show that an equation of the form

aX^{4} + bY^{4} + cZ^{4} + dX^{2}Y^{2} + eX^{2}Z^{2} + fY^{2}Z^{2} = 0

defines a non-singular curve if and only if the seven numbers a, b, c, d^{2} − 4ab, e^{2} − 4ac, f^{2} − 4bc, and af^{2} + be^{2} + cd^{2} − 4abc − def are nonzero. In our case, these numbers are B_{1}, B_{2}, B_{3}, T R/4Δ_{3}, T R/4Δ_{2}, T R/4Δ_{1}, and T^{2}/16, which are all nonzero. Thus our C is a non-singular curve of genus 3.

As in the proof of Proposition 14, we quickly reduce to the universal case. This time, that means that the d_{i} and the λ_{i} are indeterminates, that ℓ is the field

Q(d_{1}, d_{2}, d_{3}, λ_{1}, λ_{2}, λ_{3}),

that

A_{i} = λ_{i}d_{i}
B_{i} = (A_{i}^{2} − d_{i}^{2})/4
Q_{i} = (0, 0)
P_{i} = (−(d_{i} + A_{i})/2, 0)
R = d_{1}d_{2}d_{3},

and that k is the proper subfield Q(A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}, R) of ℓ. Let ℓ′ = ℓ(\sqrt{T}); note that k′ = k(\sqrt{T}) is a proper subfield of ℓ′ because it contains none of the d_{i}.

Let ι_{X} be the involution X ↦ −X of C_{k′} and let φ_{X}: C_{k′} → F_{X} be the double cover induced by ι_{X}. To find a model for the curve F_{X} over k′, we dehomogenize the equation for C by letting x = X/Z and y = Y/Z; then, setting u = x^{2}, we find the model

B_{1}u^{2} + B_{2}y^{4} + B_{3} + duy^{2} + eu + fy^{2} = 0

for F_{X}. If we let v = 2B_{1}u + dy^{2} + e and simplify, we get the model

v^{2} = (d^{2} − 4B_{1}B_{2})y^{4} + (2de − 4B_{1}f)y^{2} + (e^{2} − 4B_{1}B_{3}).

Example 3.7 (pp. 293–294) of [36] shows that the Jacobian E_{X} of F_{X} is the elliptic curve over k′ defined by y^{2} = x^{3}+A_{X}x^{2}+B_{X}x, where A_{X} = 2B_{1}f −de and B_{X} = B_{1}(B_{1}f^{2}+B_{2}e^{2}+B_{3}d^{2}−def − 4B_{1}B_{2}B_{3}). Using the formulas for d, e, and f given in the proposition, we find that A_{X} = A_{1}T/4 and B_{X} = B_{1}T^{2}/16. Thus we see that E_{X} ≅ E_{1k′}, and the double cover φ_{X}: C_{k′} → F_{X} gives us a map φ_{X}^{∗} from E_{1k′} to the Jacobian J of C_{k′}.

If we define two more involutions ι_{Y} and ι_{Z} of C_{k′} in the obvious way, we get double covers φ_{Y} : C_{k′} → F_{Y} and φ_{Z}: C_{k′} → F_{Z} that give rise to homomorphisms φ_{Y}^{∗} : E_{2k′} → J and φ_{3}^{∗}: E_{3k′} → J.

Let μ be the canonical polarization of J and let λ be the product polarization on A = E_{1}×E_{2}×E_{3}. As in the proof of Proposition 14, we get a commutative diagram

A_{k′} ⟶^{2λ} Â_{k′}
↓_{Φ^{∗}}  ^{↑Φ̂^{∗}}
J ⟶^{μ} Ĵ

where Φ^{∗} = φ_{X}^{∗} ×φ_{Y}^{∗} ×φ_{Z}^{∗} is an isogeny of degree 8 whose kernel G′ is a non-split maximal isotropic sub-group-scheme of A_{k′}[2]. Our task is to show that G′ = G_{k′}.

Let K be an algebraic closure of ℓ′. Rationality arguments as in the proof of Proposition 14 show that G′(K) contains the subgroup

H = {(0, 0, 0), (0, Q_{2}, Q_{3}), (Q_{1}, 0, Q_{3}), (Q_{1}, Q_{2}, 0)},

and, as before, to show that G′(K) = G(K) all we must do is show that the two groups contain a common element that is not in H. To show this, we once again specialize our universal example to a particular example.

Consider the specialization map ℓ → C that takes each d_{i} to 4 and each λ_{i} to −1/2; we will abuse notation by saying that d_{1} = 4, and so on. We see that each A_{i} = 2, each B_{i} = −3, each Δ_{i} = 16, each P_{i} = (−3, 0), and T = −32. For each i, let R_{i} be the 2-torsion point (1, 0) of E_{i}. Note that the E_{i} are the same as in the specialization at the end of the proof of Proposition 14. From that proof, we know that the polarized quotient of A by the subgroup generated by H and (R_{1}, R_{2}, R_{3}) is the Jacobian of a hyperelliptic curve. Since A/G′ is the Jacobian of a plane quartic, Torelli’s theorem shows that G′(K) cannot possibly contain (R_{1}, R_{2}, R_{3}). The only possibility remaining is that G′(K) contains (P_{1}, P_{2}, P_{3}), which shows that G′ = G and completes the proof. □

Remark . One might ask whether the base field extension to k(\sqrt{T}) is necessary for the proposition to be true. Indeed it is necessary. To see this, consider an arbitrary plane quartic C over k, let J be its polarized Jacobian, and let K be a separable closure of k. Since C is not hyperelliptic, we have an isomorphism Aut J ≅ Aut C × {±1} of Galois modules (where the Galois action on {±1} is trivial). Taking Galois cohomology, we find

H^{1}(Aut J) ≅ H^{1}(Aut C) × Hom(Gal(K/k), {±1}).

The two H^{1}’s catalog the twists of J and C, respectively, and the Hom catalogs field extensions of k of degree at most 2. Suppose J′ is a quadratic twist of J corresponding to an element of H^{1}(Aut J) that is trivial in H^{1}(Aut C) but nontrivial in the Hom. The curve over k that one obtains from J′ is none other than C, and it takes a quadratic extension to make Jac C isomorphic to J′.

We can use Proposition 15 to give an example of a Jacobian of a curve over Q whose conductor, while not exactly small, is at least not so big. Recall that Mestre’s result [31] implies that under standard conjectures the conductor of a 3-dimensional abelian variety over Q is at least 1100.

Corollary 16. The conductor of the Jacobian of the curve

2X^{4} + 2Y^{4} + 15Z^{4} + 3X^{2}Y^{2} − 11X^{2}Z^{2} − 11Y^{2}Z^{2} = 0

is 2940.

Proof. Take E_{1} and E_{2} to be the curve y^{2} = x^{3} −11x^{2} +32x, which is isomorphic to the curve 14A4 of [5] and has conductor 14. Take E_{3} to be the curve y^{2} = x^{3} − 31x^{2} + 240x, which is isomorphic to the curve 15A3 of [5] and has conductor 15. If we take P_{1} = P_{2} to be a nonzero 2-torsion point on E_{1} other than (0, 0), and if we take P_{3} to be (15, 0), then we find that the twisting factor is T = 32^{2}. Applying Proposition 15 to these curves gives the curve in the statement of the corollary. □

4.2. Building hyperelliptic Jacobians — introduction. In the next few sections we will find triples (E_{1}, E_{2}, E_{3}) of elliptic curves over Q that have large rational torsion subgroups and for which we can choose 2-torsion points P_{i} and Q_{i} that make the twisting factor equal to zero. Our strategy will be to specify the rational torsion structure on E_{3} and determine the corresponding conditions on E_{1} and E_{2} that will make the twisting factor zero. We will not exhaust the possible combinations of torsion structures; the equations that arise become very messy very quickly, so we will only look at the cases where it seems likely that the solutions to the equations will be easy to find.

Suppose we have three elliptic curves E_{1}, E_{2}, and E_{3} over a field k with each E_{i} defined by an equation y^{2} = x(x^{2} + A_{i}x + B_{i}) and such that the product Δ_{1}Δ_{2}Δ_{3} is a square in k. For each i let Q_{i} be the point (0, 0) on E_{i} and let P_{i} be some other 2-torsion point on E_{i}, corresponding as in Section 4.1 to a square root d_{i} of Δ_{i}. We noted in the proof of Proposition 14 that the condition that the twisting factor be 0 is equivalent to the condition that

(9)  λ_{3} = λ_{1}λ_{2} ± \sqrt{(λ_{1}^{2} − 1)(λ_{2}^{2} − 1)},

where λ_{i} = A_{i}/d_{i}. We can rewrite this equation in the equally useful form

(10)  \frac{A_{3}}{d_{3}} = \frac{A_{1}A_{2} ± 4\sqrt{B_{1}B_{2}}}{d_{1}d_{2}} .

The fact that these equations hold precisely when T = 0 will be the basis of all of our constructions of genus-3 hyperelliptic curves with large torsion subgroups.

4.3. Building hyperelliptic Jacobians with E_{3} of type (2, 2). Suppose E_{3} = F_{2,2}^{t}, so that A_{3} = −t − 1 and B_{3} = t. We have Δ_{3} = (t − 1)^{2}, so let us choose d_{3} to be 1 − t. If E_{1} and E_{2} are any elliptic curves over Q such that B_{1}B_{2} and Δ_{1}Δ_{2} are both squares, say Δ_{1}Δ_{2} = r^{2} and B_{1}B_{2} = s^{2}, then equation (10) becomes

\frac{t + 1}{t − 1} = \frac{A_{1}A_{2} ± 4s}{r}.

This will have a rational nonzero solution for t as long as the right-hand side is neither 1 nor −1. Thus, we need only search for pairs (E_{1}, E_{2}) such that B_{1}B_{2} and Δ_{1}Δ_{2} are both squares.

Take E_{1} = F_{10}^{2} and E_{2} = F_{6}^{u}. Then B_{1} = −2^{9} and Δ_{1} = 11 · 3^{5}, and up to squares in Q(u) we have

B_{1}B_{2} = 2u
Δ_{1}Δ_{2} = 33(9u + 1)/(u + 1).

Thus we would like to find rational solutions to the pair of equations u = 2v^{2} and (9u+1)/(u+1) = 33w^{2}. Solving the second equation for u gives u = (33w^{2} − 1)/(9 − 33w^{2}), and inserting this in the first equation and setting x = 2(9 − 33w^{2})v gives us

x^{2} = 2(33w^{2} − 1)(9 − 33w^{2}).

This curve of genus 1 has a rational point (w, x) = (1/3, 16/3), and a calculation shows that it is birational with the elliptic curve

y^{2} = x(x + 66)(x − 198).

This elliptic curve has rank 2; its group of rational points is generated by its 2-torsion and the points (−44, 484) and (−2, 160). Suppose C is the curve associated to one of these rational points via Proposition 14 and our choice of the curves E_{i}. We leave it to the reader to show that the image of the rational torsion of E_{1} × E_{2} × E_{3} in the Jacobian of C is a group of the form Z/2Z × Z/30Z. Thus we have a family of hyperelliptic curves of genus 3, parameterized by the points on a positive rank elliptic curve, whose Jacobians contain a rational subgroup of the form Z/2Z × Z/30Z.

Suppose we take E_{1} to be equal to E_{2}. Then B_{1}B_{2} and Δ_{1}Δ_{2} are automatically squares. In particular, if we take E_{1} = F_{10}^{t}, we find a 1-parameter family of hyperelliptic genus-3 curves whose Jacobians contain Z/10Z × Z/10Z.

Suppose we take E_{1} = F_{8,2}^{u} and E_{2} = F_{8,2}^{v}. Then Δ_{1} and Δ_{2} are each squares in Q(u, v). To find values of u and v that make the product B_{1}B_{2} a square, we must find rational solutions to the equation

(u^{2} − 2u − 1)(u^{2} + 2u − 1)w^{2} = (v^{2} − 2v − 1)(v^{2} + 2v − 1).

This equation defines an elliptic surface over Q(u) if we take the zero section to be (v, w) = (u, 1). Then the section (v, w) = (−u, 1) has infinite order. Thus we find a family of hyperelliptic genus-3 curves, parameterized by the points on a positive rank elliptic surface, whose Jacobians contain a rational subgroup of the form Z/2Z × Z/8Z × Z/8Z.

The groups we can obtain from other choices of E_{1} and E_{2} are subgroups of the groups we build in the next few sections.

4.4. Building hyperelliptic Jacobians with E_{3} of type (2, 4). Suppose now we take E_{3} = F_{2,4}^{t}, so that A_{3} = 2t^{2} + 2, B_{3} = (t − 1)^{2}(t + 1)^{2}, and Δ_{3} = 16t^{2}. If we take d_{3} = 4t, then equation (10) becomes

\frac{t^{2} + 1}{2t} = \frac{A_{1}A_{2} ± 4\sqrt{B_{1}B_{2}}}{\sqrt{Δ_{1}Δ_{2}}} .

Solving this quadratic equation for t, we find

t = \frac{(A_{1} ± 2\sqrt{B_{1}})(A_{2} ± 2\sqrt{B_{2}})}{\sqrt{Δ_{1}Δ_{2}}}.

Thus, if B_{1}, B_{2}, and Δ_{1}Δ_{2} are all squares in Q, we can find a specialization of F_{2,4}^{t} that will give us a twisting factor of 0. (Note that A_{i} ≠ ±2\sqrt{B_{i}} since E_{i} is nonsingular, so the bad value t = 0 in F_{2,4}^{t} is automatically avoided.)

Suppose we take E_{1} = F_{8}^{u} and E_{2} = F_{12}^{v}. We check from Table 6 that B_{1} and B_{2} are squares in Q(u, v), so all we must do is find values of u and v such that Δ_{1}Δ_{2} is a square. Finding such u and v reduces to finding rational points on the surface S defined by

(2u^{2} − 1)w^{2} = (6v^{2} − 6v + 1)(2v^{2} − 2v + 1).

Let Y be the genus-0 curve 2t^{2} − 1 = 41z^{2}, and let E be the genus-1 curve

41y^{2} = (6v^{2} − 6v + 1)(2v^{2} − 2v + 1).

We have a rational map Y × E → S over Q mapping (t, z), (v, y) to (u, v, w) = (t, v, y/z). Since Y has the rational point (t, z) = (9/11, 1/11), Y is isomorphic to P^{1} over Q. Since E has the rational point (v, y) = (5, 11), it is an elliptic curve over Q, and in fact it is isomorphic to y^{2} = x^{3} − 41x^{2} + 1681x. Moreover E has positive rank, since x = 81/121 gives a point of infinite order. Hence Y × E is a split elliptic surface over P_{Q}^{1} of positive rank, and the points on this surface parameterize a family of hyperelliptic curves of genus 3 whose Jacobians contain groups isomorphic to Z/2Z × Z/2Z × Z/2Z × Z/24Z.

Next we take E_{1} = F_{2,8}^{u} and E_{2} = F_{2,8}^{v}. We check that B_{1}, B_{2}, Δ_{1}, and Δ_{2} are all squares in Q(u, v), so every pair of rational values of u and v will give us a rational values of t. This gives us a 2-parameter family of hyperelliptic curves of genus 3 whose Jacobians contain groups isomorphic to Z/2Z × Z/2Z × Z/2Z × Z/4Z × Z/8Z.

Finally, if we take E_{1} = F_{12}^{u} and E_{2} = F_{12}^{v}, we see that B_{1} and B_{2} are squares. The condition that Δ_{1}Δ_{2} be a square leads us to find rational solutions to the equation

(6u^{2} − 6u + 1)(2u^{2} − 2u + 1)w^{2} = (6v^{2} − 6v + 1)(2v^{2} − 2v + 1).

This equation defines an elliptic surface over Q(u) if we take the zero section to be (v, w) = (u, 1), and then the section (v, w) = (−u + 1, 1) has infinite order. Thus we find a positive rank elliptic surface whose points parameterize a family of hyperelliptic curves of genus 3 whose Jacobians contain groups isomorphic to Z/2Z × Z/2Z × Z/6Z × Z/12Z.

4.5. Building hyperelliptic Jacobians with E_{3} of type (4, 2). Building hyperelliptic Jacobians by taking E_{3} = F_{4,2}^{t} is much more difficult than doing so by taking E_{3} = F_{2,4}^{t}, but it is possible. Here we have A_{3} = −t^{2} − 6t − 1 and we may take d_{3} = (t − 1)^{2}, so equation (10) becomes

(11)  \frac{−t^{2} − 6t − 1}{t^{2} − 2t + 1} = \frac{A_{1}A_{2} ± 4\sqrt{B_{1}B_{2}}}{d_{1}d_{2}} .

Suppose we have chosen elliptic curves E_{1} and E_{2}, and have calculated the right hand side of equation (11) to get a number r. Solving for t, we find that we must have (r+1)t^{2}+(2−6r)t+(r+1) = 0, and for t to be a rational number the discriminant of this quadratic must be a square, which reduces to the condition that 2(1 − r) be a square.

If we set E_{1} and E_{2} equal to some of our universal elliptic curves, the condition that 2(1 − r) be a square turns into an absolute mess that the reader should be thankful we do not go into here. However, in the special case where E_{1} = E_{2} and we take the plus sign in equation (11), we have r = 2λ_{1}^{2} − 1, so 2(1 − r) is a square precisely when 1 − λ_{1}^{2} is a square. A little algebra shows that this will be the case when −B_{1}Δ_{1} is a square.

Suppose we take E_{1} = F_{8}^{t}. Finding t such that −B_{1}Δ_{1} is a square is equivalent to solving the equation w^{2} = 1 − 2t^{2}. This equation defines a rational curve, so we obtain a 1-parameter family of hyperelliptic curves, and we compute that the rational torsion subgroup of the Jacobian of each of these curves contains a group isomorphic to Z/4Z × Z/4Z × Z/8Z.

We can take E_{1} to be any of the other universal curves, but for most of the choices it is not possible to have −B_{1}Δ_{1} be a square, and for the others the groups we get are subgroups of groups that we have already obtained.

4.6. Gaining 2-power torsion. We noted in Section 3.7 that the rational torsion subgroup of a quotient of an abelian variety is sometimes larger than the image of the rational torsion of the original variety. As in the genus-2 case, we can use this fact to increase the size of the torsion subgroups we can make. In order to do this, it will be useful to have some of the ideas used in Section 3.7 spelled out in more detail.

Let k be a field with separable closure K and suppose E is an elliptic curve over k with #E[2](k) = 4. Let y^{2} = f(x) = (x−x_{S})(x−x_{T})(x−x_{U}) be a model for E, and let S, T, and U be the 2-torsion points on E with x-coordinates x_{S}, x_{T}, and x_{U}, respectively.

Lemma 17. Let notation be as above, and suppose W is an element of E(K) such that 2W = S. Then W can be defined over the field ℓ = k(\sqrt{x_{S} − x_{T}} , \sqrt{x_{S} − x_{U}}). Furthermore, the action of an element σ of Gal(K/k) on W can be determined by its action on \sqrt{x_{S} − x_{T}} and \sqrt{x_{S} − x_{U}} as follows:

(a) If σ fixes neither \sqrt{x_{S} − x_{T}} nor \sqrt{x_{S} − x_{U}}, then W^{σ} − W = S.

(b) If σ fixes \sqrt{x_{S} − x_{T}} but not \sqrt{x_{S} − x_{U}}, then W^{σ} − W = T.

(c) If σ fixes \sqrt{x_{S} − x_{U}} but not \sqrt{x_{S} − x_{T}}, then W^{σ} − W = U.

Proof. Under our assumptions, the k-algebra L of Section 3.7 is isomorphic to k×k×k. The lemma then follows from the fact that the isomorphism

H^{1}(G_{k}, E[2]) ≅ ker (L^{∗}/L^{∗2} ⟶^{Norm} k^{∗}/k^{∗2})

from [35] sends the image of S in H^{1}(G_{k}, E[2]) to the class of the element

((x_{S} − x_{T})(x_{S} − x_{U}), (x_{S} − x_{T}), (x_{S} − x_{U})).
□

Using Lemma 17 we can write down families of elliptic curves with specific rational torsion subgroups and extra 4-torsion over an extension that is at most quadratic. For instance, suppose we take E to be the universal curve F_{2,4}^{t}, with x_{S} = −(1 + t)^{2} and x_{T} = 0 and x_{U} = −(1 − t)^{2}. We see that x_{S} − x_{T} = −(1 + t)^{2} and x_{S} − x_{U} = −4t. If we take t = −s^{2} then x_{S} − x_{U} will be a square and x_{S} − x_{T} will be −1, up to squares, so for this choice of t there will be a point W of E defined over ℓ = k(\sqrt{−1}) such that 2W = S and such that W^{σ} − W = U for every non-trivial k-automorphism σ of ℓ. With a little calculation we can find the coordinates for W. Putting this together with the information we have about F_{2,4}^{t} from Tables 6, 7, and 8 gives us the information summarized in Table 9. We will refer to the universal curve we have thus constructed as F_{2,4a}^{s}.

We can use the curve F_{2,4a}^{s} to build Jacobians. Suppose we take E_{1} = E_{2}, E_{3} = F_{2,4a}^{s}, d_{1} = d_{2}, and d_{3} = 4s^{2}. Then equation (9) gives us two possibilities: either λ_{3} = 1, or λ_{3} = 2λ_{1}^{2} − 1. The

Coefficients of model  A = 2(s^{4} + 1)
y^{2} = x(x^{2} + Ax + B)  B = (s^{2} + 1)^{2}(s + 1)^{2}(s − 1)^{2}
of universal curve  Δ = 16s^{4}
x-coordinates of  x_{S} = −(s − 1)^{2}(s + 1)^{2}
2-torsion points  x_{T} = 0
S, T, U  x_{U} = −(s^{2} + 1)^{2}
x- and y-coordinates of a  x_{V} = −(s^{2} + 1)(s + 1)(s − 1)
4-torsion point V with 2V = T y_{V} = 2(s^{2} + 1)(s + 1)(s − 1)
x- and y-coordinates of a  x_{W} = −(s − 1)(s + 1)(s − i)^{2}
4-torsion point W with 2W = S y_{W} = −2s(s − 1)(s + 1)(s − i)^{2}

Table 9. Data for the universal elliptic curve F_{2,4a}^{s}. Here i denotes a square root of −1. If σ is a non-trivial element of Gal(k(i)/k), then W^{σ} − W = U.

former is impossible, because in that case x(x^{2} + A_{3}x + B_{3}) would have a double root at 0. The latter becomes

\frac{s^{4} + 1}{2s^{2}} = 2λ_{1}^{2} − 1,

which can be solved to obtain s = ±λ_{1} ± \sqrt{λ_{1}^{2} − 1}, or

s = \frac{±A_{1} ± 2\sqrt{B_{1}}}{\sqrt{Δ_{1}}} .

So suppose we take E_{1} = E_{2} = F_{2,8}^{u} over k = Q(u). For this curve the numbers B_{1} and Δ_{1} are both squares in k, so we can set s to be an element of k that makes the twisting factor equal to zero. Thus we find a hyperelliptic curve C over k = Q(u) whose Jacobian is (2, 2, 2)-isogenous to E_{1} ×E_{1} ×E_{3}. Given the choice of points Q_{i} implicit in the above expressions, it is easy to calculate that the image of the known k-rational torsion of E_{1} × E_{1} × E_{3} in the Jacobian J of C is a group of the form Z/2Z × Z/2Z × Z/2Z × Z/4Z × Z/8Z. Now we will show that in fact J(k) contains a torsion group larger than this.

Let P_{1} be the point on E_{1} with x = −16u^{4} and let R_{1} be the point with x = −(u − 1)^{4}(u + 1)^{4} (see Table 8). If we apply Lemma 17 to the curve E_{1}, with S = P_{1} and T = Q_{1} and U = R_{1}, we find that there is a point W_{1} ∈ E_{1}(K) defined over an (at worst) biquadratic extension ℓ of k with 2W_{1} = P_{1}. Note that x_{S} − x_{T} = −16u^{4} differs from −1 by a square, so k(i) ⊆ ℓ and the action of a σ ∈ Gal(ℓ/k) on \sqrt{x_{P_{1}} − x_{Q_{1}}} is the same as its action on i.

Let P_{3} be the point on E_{3} with x = −(s^{2} + 1)^{2} (see Table 9) and let W_{3} be the point labeled W in Table 9. Let us consider how an element σ of Gal(ℓ/k) acts upon the element (W_{1}, W_{1}, W_{3}) of (E_{1} × E_{1} × E_{3})(ℓ). If σ is not the identity and yet fixes i, then (W_{1}, W_{1}, W_{3})^{σ} − (W_{1}, W_{1}, W_{3}) = (Q_{1}, Q_{1}, 0). If σ does not fix i, then W_{3}^{σ} − W_{3} = P_{3} and we see that (W_{1}, W_{1}, W_{3})^{σ} − (W_{1}, W_{1}, W_{3}) is either (P_{1}, P_{1}, P_{3}) or (R_{1}, R_{1}, P_{3}). Thus we see that (W_{1}, W_{1}, W_{3})^{σ} − (W_{1}, W_{1}, W_{3}) is an element of the kernel G of E_{1} × E_{1} × E_{3} → J for every σ, so the image Z of (W_{1}, W_{1}, W_{3}) in J is defined over k. Since 2(W_{1}, W_{1}, W_{3}) = (P_{1}, P_{1}, P_{3}) is in G, the point Z is a 2-torsion point. Finally, we note that (W_{1}, W_{1}, W_{3}) does not differ by an element of G from any of the known rational torsion points of E_{1} × E_{1} × E_{3}, so Z is not in the image of the known rational torsion of E_{1} × E_{1} × E_{3}. Thus, the k-rational torsion of the Jacobian of C contains a group isomorphic to Z/2Z × Z/2Z × Z/2Z × Z/2Z × Z/4Z × Z/8Z. Since k is a rational function field over Q, we get a 1-parameter family of hyperelliptic curves of genus 3 having this group in their rational torsion subgroups.

4.7. Building Jacobians of plane quartics — introduction. Now we turn our attention to the task of building plane quartics whose Jacobians have large rational torsion subgroups. Most of the families we will construct will be produced by fixing the torsion structure of the curve E_{3} and analyzing the twisting factor as a function of the coefficients of E_{1} and E_{2}, but one interesting class of examples will arise by setting E_{1} = E_{2} = E_{3}. Instead of trying to get the twisting factor to be zero, as we did in the last few sections, we will try to get the twisting factor to be a square, so that (by the final statement of Proposition 15) the product of the E_{i} will be isogenous over Q to the Jacobian of a curve. It will be convenient to use the second expression for the twisting factor, namely

(12)  T = d_{1}d_{2}d_{3}(λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2} − 2λ_{1}λ_{2}λ_{3} − 1),

where λ_{i} = A_{i}/d_{i}.

As in the hyperelliptic case, we will not examine all possible combinations of torsion structures on the curves E_{1}, E_{2}, and E_{3} here because of the complexity of the equations that arise.

4.8. Building Jacobians of plane quartics with E_{3} of type (4, 2). In this section we will take E_{3} = F_{4,2}^{t}, so that A_{3} = −t^{2} − 6t − 1 and Δ_{3} = (t − 1)^{4}. We will take d_{3} = (t − 1)^{2}.

Suppose we take E_{1} = E_{2} and d_{1} = d_{2}. Then λ_{1} = λ_{2}, and the twisting factor is

T = d_{1}^{2}d_{3}(2λ_{1}^{2} + λ_{3}^{2} − 2λ_{1}^{2}λ_{3} − 1)
= Δ_{1}d_{3}(λ_{3} − 1)(λ_{3} − 2λ_{1}^{2} + 1)
= Δ_{1}(λ_{3} − 1)(A_{3} − (2λ_{1}^{2} − 1)d_{3})
= (λ_{3} − 1)(Δ_{1}A_{3} − (2A_{1}^{2} − Δ_{1})d_{3}).

A quick calculation shows that λ_{3} − 1 = −2(t + 1)^{2}/(t − 1)^{2}, so up to squares in Q(t) the twisting factor is

4A_{1}^{2}d_{3} − 2Δ_{1}(A_{3} + d_{3}) = 4(t − 1)^{2}A_{1}^{2} + 16tΔ_{1}.

To get the twisting factor to be a square, we need to find rational solutions to the equation

w^{2} = 4(t − 1)^{2}A_{1}^{2} + 16tΔ_{1}.

For fixed A_{1} and Δ_{1}, this last equation defines a curve of genus 0 in the (t, w)-plane, and since it has a rational point (namely (t, w) = (0, 2A_{1})) it is isomorphic to P^{1}. We can parameterize the curve by setting

t = (z + 4B_{1})(z − Δ_{1})/(A_{1}^{2}z),
w = 2(z^{2} + 4B_{1}Δ_{1})/(A_{1}z).

Suppose in particular we take E_{1} = F_{10}^{u}. Then we get a 2-parameter family (the parameters being u and z) of plane quartics whose Jacobians are isogenous to E_{1} × E_{1} × E_{3}, and a simple computation with abelian groups shows that these Jacobians have a rational subgroup isomorphic to Z/10Z × Z/20Z.

If we take E_{1} = F_{2,8}^{u} we get a 2-parameter family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/2Z × Z/4Z × Z/4Z × Z/8Z. If we take E_{1} = F_{8,2}^{u} we get a 2-parameter family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/2Z × Z/2Z × Z/8Z × Z/8Z. And if we take E_{1} = F_{12}^{u} we get a 2-parameter family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/2Z × Z/12Z × Z/12Z.

Without the assumption that E_{1} = E_{2} it is not as easy to make the twisting factor a square. But suppose we take E_{1} = F_{10}^{−1/2} and E_{2} = F_{8}^{1/2}. Then λ_{1}^{2} = −625/2048 and λ_{2}^{2} = −49/32, and we can choose d_{1} and d_{2} so that d_{1}d_{2} = 4 and λ_{1}λ_{2} = 175/256. Then the twisting factor is

T = 4d_{3} (−\frac{625}{2048} − \frac{49}{32} + \frac{A_{3}^{2}}{Δ_{3}} − \frac{175}{128} \frac{A_{3}}{d_{3}} − 1)

= \frac{−1922t^{4} + 118024t^{3} + 29940t^{2} + 118024t − 1922}{2^{10}(t − 1)^{2}},

so in order to make the twisting factor a square we must find rational solutions to the equation

w^{2} = −1922t^{4} + 118024t^{3} + 29940t^{2} + 118024t − 1922.

This last equation defines a curve of genus 1, and it has a rational point, namely (t, w) = (1, 512). A calculation then shows that the curve is birational with the elliptic curve defined by y^{2} = x^{3} + 2565x − 15606. This happens to be the curve 528A2 in [5], which has rank 1. (The point (33, 324) is of infinite order.) We see that there is a family of plane quartics, parameterized by the points on a positive rank elliptic curve, whose Jacobians contain a rational subgroup isomorphic to Z/4Z × Z/40Z.

Now let us try taking E_{1} = F_{8,2}^{2} and E_{2} = F_{6,2}^{1/4}. If we take d_{1} = 256 and d_{2} = 4 then we have λ_{1} = 47/128 and λ_{2} = 863/512. The twisting factor is

T = 1024d_{3} (\frac{2209}{16384} + \frac{744769}{262144} + \frac{A_{3}^{2}}{Δ_{3}} − \frac{40561}{32768} \frac{A_{3}}{d_{3}} − 1)

= \frac{1104601t^{4} + 2371804t^{3} + 9824406t^{2} + 2371804t + 1104601}{2^{8}(t − 1)^{2}},

so in order to make the twisting factor a square we must find rational solutions to the equation

w^{2} = 1104601t^{4} + 2371804t^{3} + 9824406t^{2} + 2371804t + 1104601.

The genus-1 curve defined by this equation has a rational point — namely, (t, w) = (1, 4096) — so it is an elliptic curve. A calculation shows that it is birational with the elliptic curve y^{2} = x^{3} − 151563x + 10810438. Cremona’s rank-finding program calculates that this elliptic curve has rank 2, and provides the two independent rational points (59, 1440) and (−157, 5544). Thus we have a positive rank elliptic curve whose points parameterize a family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/2Z × Z/4Z × Z/24Z.

Finally, let us take E_{1} = F_{10}^{−1/3} and E_{2} = F_{12}^{1/3}. Then λ_{1}^{2} = −485809/759375 and λ_{2}^{2} = −3721/375, and we can choose d_{1} and d_{2} so that d_{1}d_{2} = 625/59049 and λ_{1}λ_{2} = −42517/16875. The twisting factor is

T = \frac{625}{59049}d_{3} (−\frac{485809}{759375} − \frac{3721}{375} + \frac{A_{3}^{2}}{Δ_{3}} + \frac{85034}{16875} \frac{A_{3}}{d_{3}} − 1)

= \frac{−177710460t^{4} + 433908240t^{3} + 216604440t^{2} + 433908240t − 177710460}{3^{16}5^{2}(t − 1)^{2}},

so in order to make the twisting factor a square we must find rational solutions to the equation

w^{2} = −177710460t^{4} + 433908240t^{3} + 216604440t^{2} + 433908240t − 177710460.

This genus-1 curve has rational points — for instance, (t, w) = (1, 27000) — so it can be made into an elliptic curve. A calculation shows that the elliptic curve we get is isomorphic to the curve y^{2} = x^{3} +213x−30566. Cremona’s rank-finding program says that this curve has rank 1, and gives the non-torsion point (53, 360). Thus we have found a positive rank elliptic curve whose points parameterize a family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/4Z × Z/60Z.

4.9. Building Jacobians of plane quartics with E_{3} of type (2, 4). In this section we will take E_{3} = F_{2,4}^{t}, so that A_{3} = 2t^{2} + 2 and Δ_{3} = 16t^{2}. We will take d_{3} = 4t.

As we saw in the preceding section, if we take E_{1} = E_{2} and d_{1} = d_{2} then the twisting factor is

T = (λ_{3} − 1)(Δ_{1}A_{3} − (2A_{1}^{2} − Δ_{1})d_{3}).

For our choice of E_{3} we have λ_{3} − 1 = (t − 1)^{2}/(2t), so up to squares in Q(t) the twisting factor is

(2t)(Δ_{1}(2t^{2} + 2) − (2A_{1}^{2} − Δ_{1})(4t)) = 4t(Δ_{1}(t + 1)^{2} − 4A_{1}^{2}t).

Thus, we would like to find rational solutions to the equation

(13)  w^{2} = 4t(Δ_{1}(t + 1)^{2} − 4A_{1}^{2}t).

Suppose we take E_{1} = F_{8,2}^{2}, so that A_{1} = 2 · 47 and Δ_{1} = 2^{16}. Equation (13) becomes w^{2} = 4t(2^{16}(t + 1)^{2} − 2^{4}47^{2}t), and by setting s = 4096t and z = 512w we get

z^{2} = s^{3} + 5983s^{2} + 16777216s.

A search for points on this curve using Cremona’s programs comes up with the non-torsion point (s, z) = (5929/64, 20520885/512). Thus we find a family of plane quartics, parameterized by the points on a positive rank elliptic curve, whose Jacobians contain a rational subgroup isomorphic to Z/4Z × Z/8Z × Z/8Z.

Other choices for E_{1} lead to groups we have already constructed, most of them in the sections on hyperelliptic curves.

4.10. Building Jacobians of plane quartics with E_{1} = E_{2} = E_{3}. Suppose we take E_{1} = E_{2} = E_{3} and d_{1} = d_{2} = d_{3}. Since Δ_{1}Δ_{2}Δ_{3} is supposed to be a square, we see that Δ_{1} must be a square and d_{1} must an element of the base field. If we write λ and d for λ_{1} and d_{1}, we find that the twisting factor is

T = d^{3}(3λ^{2} − 2λ^{3} − 1) = −d^{3}(λ − 1)^{2}(2λ + 1).

Up to squares, then, the twisting factor is

−d(2λ + 1) = −(2A + d),

where we write A for A_{1}.

Suppose we take E_{1} = F_{2,6}^{t}. Then A = −2t^{4} + 12t^{2} + 6 and we can take d = 16t. The twisting factor, up to squares, is t^{4} − 6t^{2} − 4t − 3, so we would like to find rational solutions to

w^{2} = t^{4} − 6t^{2} − 4t − 3.

The desingularization of this genus-1 curve has rational points at infinity, so it is an elliptic curve. A calculation shows that it is isomorphic as an elliptic curve to y^{2} = x^{3} − 48. This is the curve 243A1 in [5]; it has rank 1, and its group of rational points is generated by the point (4, 4). Thus we find a positive rank elliptic curve whose points parameterize a family of plane quartics whose Jacobians contain a rational subgroup isomorphic to Z/6Z × Z/6Z × Z/6Z.

Taking E_{1} to be F_{2,4}^{t} or F_{4,2}^{t} gives us rational subgroups that we can get in other ways, and taking E_{1} to be E_{2,8}^{t} or E_{8,2}^{t} leads to equations with no rational solutions.

4.11. Gaining 2-power torsion. In this section we will show how to further specialize two of the families we wrote down in the last few sections to obtain larger rational torsion subgroups. For the first example, we will need to write down the universal curve F_{2,4a}^{s} in a slightly different form. All we would like to do is make a change of variables in the equation for F_{2,4a}^{s} so that the point labeled S in Table 9 will have x-coordinate 0. After translating x by the proper amount to do this, we obtain a curve that we will call F_{4,2a}^{s}. All the information we will need about this curve is listed in Table 10. Note that F_{4,2a}^{s} is a specialization of F_{4,2}^{t}, as the notation suggests. In fact, F_{4,2a}^{s} = F_{4,2}^{−s^{2}}.

Coefficients of model  A = −(s^{2} − 2s − 1)(s^{2} + 2s − 1)
y^{2} = x(x^{2} + Ax + B)  B = −4s^{2}(s − 1)^{2}(s + 1)^{2}
of universal curve  Δ = (s^{2} + 1)^{4}
x-coordinates of  x_{S} = 0
2-torsion points  x_{T} = (s + 1)^{2}(s − 1)^{2}
S, T, U  x_{U} = −4s^{2}
x- and y-coordinates of a  x_{V} = −2(s + 1)(s − 1)
4-torsion point V with 2V = T y_{V} = 2(s^{2} + 1)(s + 1)(s − 1)
x- and y-coordinates of a  x_{W} = 2is(s + 1)(s − 1)
4-torsion point W with 2W = S y_{W} = −2s(s + 1)(s − 1)(s − i)^{2}

Table 10. Data for the universal elliptic curve F_{4,2a}^{s}. Here i denotes a square root of −1. If σ is a non-trivial element of Gal(k(i)/k), then W^{σ} − W = U.

Now suppose we try to build a plane quartic by taking E_{1} = E_{2} = F_{2,8}^{u} and E_{3} = F_{4,2a}^{s}. For notational convenience, we denote by t the number −s^{2}, so that E_{3} = F_{4,2}^{t} as well. Let P_{1} be the 2-torsion point on E_{1} with x = −16u^{4} and let P_{3} be the point labeled U in Table 10, so that d_{3} = (t − 1)^{2} = (s^{2} + 1)^{2}.

As we noted in Section 4.8, the twisting factor in this situation is equal to

4(t − 1)^{2}A_{1}^{2} + 16tΔ_{1} = 4(s^{2} + 1)^{2}A_{1}^{2} − 16s^{2}Δ_{1},

up to squares, so we would like to find solutions to

w^{2} = (s^{2} + 1)^{2}A_{1}^{2} − 4s^{2}Δ_{1}.

For fixed A_{1} and Δ_{1}, this is a genus-1 curve in the (s, w)-plane whose desingularization has rational points at infinity, so it is isomorphic to its Jacobian, which (according to the formulas in Example 3.7 (pp. 293–294) of [36]) is given by

y^{2} = x(x + Δ_{1})(x − 4B_{1}).

If we take u = 2 then this curve is given by y^{2} = x(x + 30625)(x − 82944) and has a non-torsion point (−21600, 4514400). Thus, for this choice of u there are infinitely many values of s ∈ Q that make the twisting factor a square.

Let C be the plane quartic associated to one of these choices. Let A be the abelian variety E_{1} × E_{1} × E_{3} and let G be the kernel of the homomorphism ψ from A to the Jacobian J of C. The image under ψ of the known rational torsion of A is a rational torsion subgroup of J isomorphic to Z/2Z × Z/4Z × Z/4Z × Z/8Z. Note that the number of independent (Z/4Z)-factors contained in this group is as large as is allowed by the restrictions imposed by the Galois-equivariance of the Weil pairing. However, there is more rational torsion on J than just this.

Let R_{1} be the 2-torsion point on E_{1} with x = −(u−1)^{4}(u+1)^{4} (see Table 8). Applying Lemma 17 to the curve E_{1}, taking S = P_{1} and T = Q_{1} and U = R_{1}, we find that there is a point W_{1} ∈ E_{1}(K) defined over a Galois extension ℓ of Q with 2W_{1} = P_{1}. Since x_{S} − x_{T} = −16u^{4} we see that ℓ contains Q(i); furthermore, if σ is any automorphism of ℓ that fixes i then W_{1}^{σ} − W_{1} is either 0 or Q_{1}, while if σ does not fix i then W_{1}^{σ} − W_{1} is either P_{1} or R_{1}.

Let W_{3} be the point on E_{3} labeled W in Table 10. The point W_{3} is defined over Q(i), and if σ is the non-trivial automorphism of Q(i) then W_{3}^{σ} − W_{3} = P_{3}. Knowing the Galois action on W_{1} and W_{3}, one can check quite simply that (W_{1}, W_{1}, W_{3})^{σ} − (W_{1}, W_{1}, W_{3}) is an element of G for every σ ∈ Gal(K/Q). Thus the image Z of (W_{1}, W_{1}, W_{3}) in J is a rational point. It is certainly not one of the points we already calculated because (W_{1}, W_{1}, W_{3}) does not differ from one of the known rational points of A by an element of G. However, 2Z is one of the points we previously calculated, and indeed 2Z is a 2-torsion point. As remarked before, the torsion subgroup cannot contain (Z/4Z)^{n} for n > 3, so the only possibility is that J contains a rational torsion subgroup isomorphic to Z/2Z × Z/2Z × Z/4Z × Z/4Z × Z/8Z.

We end by trying to add 2-power torsion to the family of curves we obtained in the preceding section by taking E_{1} = E_{2} = E_{3} = F_{2,6}^{t}. Since the three elliptic curves are isomorphic to one another, we drop the subscripts. Recall that we chose d = 16t. This corresponds to choosing P to be the 2-torsion point on E with x-coordinate equal to (t − 3)(t + 1)^{3}. Let R be the 2-torsion point on E with x-coordinate (t + 3)(t − 1)^{3}. Of course, Q is the point (0, 0) on E.

Suppose we can find a value of t ∈ Q that makes the twisting factor a square and that equals (3 + s^{2})/(1 − s^{2}) for some s ∈ Q. Then the x-coordinate of P will be a square, and if we apply Lemma 17 to E, taking S = P and T = Q and U = R, we find that there will be point W on E defined over a quadratic extension ℓ of Q such that 2W = P and W^{σ} − W = Q for the non-trivial element of Gal(ℓ/Q). Let C be the plane quartic corresponding to this hypothetical t, and let ψ: E ×E ×E → J be the map to the Jacobian of C with kernel G generated by (0, Q, Q), (Q, 0, Q), and (P, P, P). The image under ψ of the rational torsion of E × E × E is a group isomorphic to Z/6Z × Z/6Z × Z/6Z, but there is still more rational torsion on J. For consider the element Y_{1} = (W, W, 0) of E × E × E: since Y_{1}^{σ} − Y_{1} = (Q, Q, 0) is in G, we see the image of Y_{1} in J is a rational point Z_{1}. Similarly, the image Z_{2} of Y_{2} = (W, 0, W) is a rational point. Since neither 2Y_{1} nor 2Y_{2} is in G, while both 4Y_{1} and 4Y_{2} are zero, we see that Z_{1} and Z_{2} are 4-torsion points. Moreover, they are independent 4-torsion points, because neither 2(Y_{1} + Y_{2}) nor 2(Y_{1} + 3Y_{2}) is in G. Thus J contains a rational torsion subgroup isomorphic to Z/6Z × Z/12Z × Z/12Z.

To find such a curve C, we must find a value of s such that t = (3 + s^{2})/(1 − s^{2}) will make the twisting factor a square. From the preceding section, we know that the twisting factor will be a square if there is a w such that

w^{2} = t^{4} − 6t^{2} − 4t − 3.

Inserting our formula for t into this equation and clearing denominators shows that we want to find solutions to

y^{2} = −s^{8} + 6s^{4} + 56s^{2} + 3,

with s ≠ ±1. Amazingly enough, there are such solutions to this equation: we can take (s, y) = (±1/5, 1432/625). These solutions give t = 19/6. Inserting the corresponding values for A, B, and Δ into the formulas of Proposition 15 leads us to the plane quartic C defined by

15625(X^{4} + Y^{4} + Z^{4}) − 96914(X^{2}Y^{2} + X^{2}Z^{2} + Y^{2}Z^{2}) = 0.

The Jacobian of this curve contains a rational torsion subgroup of order 6 · 12 · 12 = 864. Both E and C have good reduction modulo 7; since the reduction E′ of E modulo 7 has 12 points (12 being the only multiple of 12 lying within the Weil bounds), and since the reduction J′ of the Jacobian of C is isogenous to E′ × E′ × E′, we see that J′ has 12^{3} = 1728 points. Therefore the rational torsion subgroup of J is isomorphic either to Z/6Z × Z/12Z × Z/12Z, to Z/12Z × Z/12Z × Z/12Z, to Z/2Z × Z/6Z × Z/12Z × Z/12Z, or to Z/6Z × Z/12Z × Z/24Z. The second and third possibilities can be ruled out by looking at the action of Galois on the 4-torsion of E × E × E. The fourth can also be ruled out: the dual isogeny Ψ̂^{∗} : J → E × E × E would take a 24-torsion point on J to a point of order at least 12 on E × E × E, since Ψ^{∗}Ψ̂^{∗} is multiplication-by-2 on J, but the torsion subgroup of E × E × E is of exponent only 6. Thus, we have found a single plane quartic C such that the rational torsion subgroup of Jac C is isomorphic to Z/6Z × Z/12Z × Z/12Z.

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Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967, USA.

   E-mail address: however@alumni.caltech.edu

Université Pierre et Marie Curie, CNRS- UMR 9994 Arithmetique, Tour 46-56, 5ème étage Case 247; 4, place Jussieu; F-75252 Paris Cedex 05; France. and Technische Universität Berlin, Fachbereich Mathematik MA 8-1; Straße des 17. Juni 136; D-10623 Berlin; Germany.

   E-mail address: leprevot@math.tu-berlin.de, leprevot@math.jussieu.fr

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

   E-mail address: poonen@math.berkeley.edu