Reductions of an elliptic curve and their Tate-Shafarevich groups
Let \(E\) be an elliptic curve defined over \({\mathbb{Q}}\) with conductor \(N\). For a prime \(p\nmid N\), let \(E_p\) be the reduction modulo \(p\) of \(E\) defined over the finite field \(k:={\mathbb F}_p\) with \(p\) elements. Let \(K\) be the function field of \(E_p\) and consider \(E_p\) as a constant elliptic curve over \(K\). This paper is concerned with the Tate-Shafarevich group Ш\(_p:=\text{Ш}(E_p/K)\) of \(E_p/K\). The main results are: Theorem 1: The group Ш\(_p\) contains elements of any fixed prime order \(\ell\) for a positive proportion of primes \(p\). Theorem 2: Suppose that \(E\) has an irrational point of order two. If \(E\) does not have CM assume the Generalized Riemann Hypothesis. Then the group Ш\(_p\) is trivial for a positive proportion of primes \(p\). Using the Birch and Swinnerton-Dyer conjecture (BSD), which is a theorem in this situation, the authors derive a formula for the cardinality \(|\)Ш\(_p|\) of Ш\(_p\) in terms of \(|E_p(k)|\) and the endomorphism algebra End\(_k(E_p)\). This formula provides an upper bound for \(|\)Ш\(_p|\). A second application of BSD gives the characterization of those squares \(n^2\) dividing \(|\)Ш\(_p|\) in terms of the decomposition of \(p\) in a certain Galois extension \(J_n\) of \({\mathbb{Q}}\), built as a subfield of the \(n\)-th division field of \(E\). Using the Chebotarev density theorem, this characterization gives Theorem 1. Theorem 2 is proven using a variant of Serre's sieve method: the fact that \(J_n\) does not contain the full \(n\)-th cyclotomic field implies that the Brun-Titchmarsh theorem cannot be applied. The essential point is that \(|\)Ш\(_p|\) is a square, so that a ``square sieve'' can be used to estimate the remainder term. A nice result included in the paper is the determination of the group structure of \(E_p(k)\) in terms of its cardinality and its endomorphism algebra: Proposition 3.5: For a prime \(p\nmid N\), let \(|E_p(k)|:=p+1-a_p\) and let \(\Delta_p\) be the discriminant of the order End\(_k(E_p)\). Take \(\delta_p\) equal to 0 or 1 according to \(\Delta_p\) being congruent to 0 or 1 modulo 4. Let \(b_p\) be the unique positive integer such that \(a_p^2-4p=b_p^2\Delta_p\). Then \(E_p(k)=({\mathbb{Z}}/d_p{\mathbb{Z}})\oplus({\mathbb{Z}}/e_p{\mathbb{Z}})\) where: \[ d_p= \gcd(b_p,\tfrac12(a_p+b_p\delta_p-2)), \qquad e_p=(p+1-a_p)/d_p. \]
scientific article; zbMATH DE number 2120966