CURVES OVER EVERY GLOBAL FIELD VIOLATING THE LOCAL-GLOBAL PRINCIPLE

BJORN POONEN

Abstract. There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n.

1. Introduction

Let k be a global field, by which we mean a finite extension of either ℚ or 𝔽_{p}(t) for some prime p. Let Ω_{k} be the set of nontrivial places of k. For each v ∈ Ω_{k}, let k_{v} be the completion of k at v. By variety, we mean a separated scheme of finite type over a field. A curve is a variety of dimension 1. Call a variety nice if it is smooth, projective, and geometrically integral. Say that a k-variety X satisfies the local-global principle if the implication

X(k_{v}) ≠ ∅ for all v ∈ Ω_{k} ⟹ X(k) ≠ ∅

holds.

Nice genus-0 curves (and more generally, quadrics in ℙ^{n}) satisfy the local-global principle: this follows from the Hasse-Minkowski theorem for quadratic forms. The first examples of varieties violating the local-global principle were genus-1 curves, such as the smooth projective model of 2y^{2} = 1 − 17x^{4}, over ℚ, discovered by Lind [Lin40] and Reichardt [Rei42].

Our goal is to prove that there exist curves over every global field violating the local-global principle. We can also produce curves having a prescribed positive number of k-rational points. In fact, such examples can be constructed effectively:

Theorem 1.1. There is an algorithm that takes as input a global field k and a nonnegative integer n, and outputs a nice curve X over k such that #X(k) = n and X(k_{v}) ≠ ∅ for all v ∈ Ω_{k}.

Remark 1.2. For the sake of definiteness, let us assume that k is presented by giving the minimal polynomial for a generator of k as an extension of ℚ or 𝔽_{p}(t). The output can be described by giving a finite list of homogeneous polynomials that cut out X in some ℙ^{n}. For more details on representation of number-theoretic and algebraic-geometric objects, see [Len92, §2] and [BGJGP05, §5].

Date: May 14, 2010.

2000 Mathematics Subject Classification. Primary 11G30; Secondary 14H25.

Key words and phrases. Hasse principle, local-global principle, Dem’janenko-Manin method.

This research was supported by NSF grant DMS-0841321.

2. Proof

Lemma 2.1. Given a global field k, one can effectively construct a nice curve Z over k such that Z(k) is finite, nonempty, and computable.

Proof. First suppose that char k = 0. Let E be the elliptic curve X_{1}(11) over k. By computing a Selmer group, compute an integer r strictly greater than the rank of the finitely generated abelian group E(k). Let Z = X_{1}(11^{r}) over k. By [DS05, Theorem 6.6.6], the Jacobian J_{Z} of Z is isogenous to a product of E^{r} with another abelian variety over k (geometrically, these r copies of E in J_{Z} arise from the degeneracy maps Z → E indexed by s ∈ {1, …, r} that in moduli terms send (A, P) to (A/⟨11^{s}P ⟩, 11^{s−1}P) where A is an elliptic curve and P is a point on A of exact order 11^{r}). So the Dem’janenko-Manin method [Dem66, Man69] yields an upper bound on the height of points in Z(k). In particular, Z(k) is finite and computable. It is also nonempty, since the cusp ∞ on X_{1}(11^{r}) is a rational point.

If char k > 0, let Z be any nonisotrivial curve of genus greater than 1 such that Z(k) is nonempty: for instance, let a be a transcendental element of k, and use the curve C_{a} in the first paragraph of the proof of Theorem 1.4 in [PP08]. Then Z(k) is finite by [Sam66, Théorème 4], and computable because of the height bound proved in [Szp81, §8, Corollaire 2].

□

Lemma 2.2. Given a global field k and a nonnegative integer n, one can effectively construct a nice curve Y over k such that Y (k) is finite, computable, and of size at least n.

Proof. Construct Z as in Lemma 2.1. Let κ(Z) denote the function field of Z. Find a closed point P ∈ Z − Z(k) whose residue field is separable over k.

If char k = 0, the Riemann-Roch theorem, which can be made constructive, together with a little linear algebra, lets us find f ∈ κ(Z) taking the value 1 at each point of Z(k), and having a simple pole at P. If char k = p > 2, instead find t ∈ κ(Z) such that t has a pole at P and nowhere else, and such that t takes the value 1 at each point of Z(k); then let f = t + g^{p} for some g ∈ κ(Z) such that g has a pole at P of odd order greater than the order of the pole of t at P and no other poles, such that g is zero at each point of Z(k), and such that t + g^{p} is nonzero at each zero of dt; this ensures that f has an odd order pole at P and no other poles, and is 1 at each point of Z(k), and has only simple zeros (since f and df = dt do not simultaneously vanish). In either case, f has an odd order pole at P, so κ(Z)(\sqrt{f}) is ramified over κ(Z) at P, so the regular projective curve Y with κ(Y) = κ(Z)(\sqrt{f}) is geometrically integral. A local calculation shows that Y is also smooth, so Y is nice. Equations for Y can be computed by resolving singularities of an initial birational model. The points in Z(k) split in Y, so #Y (k) = 2#Z(k), and Y (k) is computable. Iterating this paragraph eventually produces a curve Y with enough points.

If char k = 2, use the same argument, but instead adjoin to κ(Z) a solution α to α^{2}−α = f, where f ∈ κ(Z) has a pole of high odd order at P, no other poles, and a zero at each point of Z(k). □

Proof of Theorem 1.1. Given k and n, apply Lemma 2.2 to find Y over k with Y (k) finite, computable, and of size at least n + 4. Write Y (k) = {y_{1}, …, y_{m}}. Find a closed point P ∈ Y − Y (k) with residue field separable over k.

Suppose that char k ≠ 2. Compute a, b ∈ k^{×} whose images in k^{×}/k^{×2} are 𝔽_{2}-independent. Let S be the set of places v ∈ k such that a, b, and ab are all nonsquares in k_{v}. By Hensel’s lemma, if v ∤ 2, ∞ and v(a) = v(b) = 0, then v ∉ S. So S is finite and computable. Let w ∈ Ω_{k} − S. Weak approximation [AW45, Theorem 1], whose proof is constructive, lets us find c ∈ k^{×} such that c is a square in k_{v} for all v ∈ S and w(c) is odd. The purpose of w is to ensure that c is not a square in k. Find f ∈ κ(Y)^{×} such that f has an odd order pole at P and a simple zero at each of y_{1}, …, y_{n}, and such that f(y_{n+1}) = a, f(y_{n+2}) = b, f(y_{n+3}) = ab, and f(y_{n+4}) = ⋯ = f(y_{m}) = c. If char k = p > 2, the same argument as in the proof of Lemma 2.2 lets us arrange in addition that f has no poles other than P, and that all zeros of f are simple. Construct the nice curve X whose function field is κ(Y)(\sqrt{f}). Then X → Y maps X(k) bijectively to {y_{1}, …, y_{n}}, so X(k) is computable and of size n. Also, for each v ∈ Ω_{k}, at least one of a, b, ab, c is a square in k_{v}, so X(k_{v}) ≠ ∅.

If char k = 2, use the same argument, with the following modifications. For any extension L of k, define the additive homomorphism ℘: L → L by ℘(t) = t^{2} − t. Construct a, b ∈ k such that the images of a and b in k/℘(k) are 𝔽_{2}-independent. Let S be the set of places v ∈ k such that a, b, and a + b are all outside ℘(k_{v}). As before, S is finite and computable. Choose w ∈ Ω_{k} − S. Use weak approximation to find c ∈ k such that c ∈ ℘(k_{v}) for all v ∈ S but c ∉ ℘(k_{w}). Find f ∈ κ(Y) such that f has a pole of high odd order at P, a simple pole at y_{1}, …, y_{n}, and no other poles, and such that f(y_{n+1}) = a, f(y_{n+2}) = b, f(y_{n+3}) = a + b, and f(y_{n+4}) = ⋯ = f(y_{m}) = c. Construct the nice curve X whose function field is obtained by adjoining to κ(Y) a solution α to α^{2} − α = f. □

3. Other constructions of curves violating the local-global principle

3.1. Lefschetz pencils in a Châtelet surface. J.-L. Colliot-Thélène has suggested another approach to constructing curves violating the local-global principle, which we now sketch. For any global field k, there exists a Châtelet surface over k violating the localglobal principle: see [Poo09, Proposition 5.1] and [Vir09, Theorem 1.1]. Let V be such a surface. Choose a projective embedding of V. By [Kat73, Théorème 2.5], after replacing V by a d-uple embedding for some d ≥ 1, there is a Lefschetz pencil of hyperplane sections of V, fitting together into a family Ṽ → ℙ^{1}, where Ṽ is the blowup of V along the intersection of V with the axis of the pencil. Since Ṽ → V is a birational morphism, the Lang-Nishimura theorem (see [Nis55], [Lan54, Theorem 3], and also [CTCS80, Lemme 3.1.1]) shows that Ṽ has a k-point if and only if V does, and the same holds with k replaced by any completion k_{v}. By definition of Lefschetz pencil, each geometric fiber of the pencil is either an integral curve or a union of two nice curves intersecting transversely in a single point. By requiring d ≥ 3 above, we can ensure that each geometric fiber is also 2-connected, which means that whenever it decomposed as a sum D_{1} + D_{2} of two nonzero effective divisors, the intersection number D_{1}.D_{2} is at least 2 (the 2-connectedness follows from [VdV79, Theorem I]; that paper is over ℂ, but the argument works in arbitrary characteristic). This rules out the possibility of a geometric fiber with two components, so every geometric fiber is integral. The “fibration method” (see, e.g., [CTSSD87], [CT98, 2.1], [CTP00, Lemma 3.1]) shows that there is a finite set of places S such that for every place v ∉ S and every point t ∈ ℙ^{1}(k), the fiber of Ṽ → ℙ^{1} above t has a k_{v}-point. For v ∈ S, the set Ṽ (k_{v}) is nonempty, and its image in ℙ^{1} contains a nonempty open subset U_{v} of ℙ^{1}(k_{v}). By weak approximation, we can find t ∈ ℙ^{1}(k) such that t ∈ U_{v} for all v ∈ S, and such that the fiber of Ṽ → ℙ^{1} above t is smooth. That fiber violates the local-global principle.

With a little work, this construction can be made effective. On the other hand, this approach does not seem to let one construct curves with a prescribed positive number of points.

3.2. Atkin-Lehner twists of modular curves. Theorem 1 of [Cla08] constructs a natural family of curves over ℚ violating the local-global principle: namely, for any squarefree integer N with N > 131 and N ≠ 163, there is a positive-density set of primes p such that the twist of X_{0}(N) by the main Atkin-Lehner involution w_{N} and the quadratic extension ℚ(\sqrt{p})/ℚ violates the local-global principle over ℚ. See [Cla08] for details, and for a connection to the inverse Galois problem. The proof involves Faltings’ theorem [Fal83], so it does not yield an effective construction of a suitable pair (N, p).

On the other hand, as P. Clark explained to me, a variant of this construction is effective, and works over an arbitrary global field k. His idea is to replace X_{0}(N) above with a modular curve X having both Γ_{0}(N) and Γ_{1}(M) level structures, for suitable M and N depending on k, and to apply Merel’s theorem (or a characteristic p analogue) to X_{1}(M) to control X(k). See [Cla09] for details.

Remark 3.1. One can also find counterexamples to the local-global principle over ℚ among Atkin-Lehner quotients of Shimura curves: see [RSY05] and [PY07].

Acknowledgements

I thank Pierre Dèbes for the suggestion to use the Dem’janenko-Manin method. I thank Pete L. Clark and Jean-Louis Colliot-Thélène for sharing their ideas sketched in Section 3. I also thank Clark for a correction, and Izzet Coskun for suggesting the reference [VdV79]. Finally I thank the referee for a few suggestions.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

E-mail address: poonen@math.mit.edu
URL: http://math.mit.edu/~poonen