2019/03/29 by Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver +1 · 1 citation
Computer Science · Mathematics · #Abelian group #Abelian variety #Algebraic Geometry and Number Theory #Algebraic number field #Coding theory and cryptography #Combinatorics #Cryptography and Residue Arithmetic #Elliptic curve #Geometry #Group (periodic table) #Isogeny #Mathematics #Order (exchange) #Prime (order theory) #Pure mathematics #Quadratic equation #Quadratic field #Quadratic function #Quarter period #Schoof's algorithm #Twist #Twists of curves #math.NT
paper · pdf · doi:10.2140/ant.2021.15.627
published as Alg. Number Th. 15 (2021) 627-655
arxiv created 2019/03/29 · openalex created_date 2019/04/11 · openalex publication_date 2021/05/20 · arxiv updated 2021/05/26 · openalex updated_date 2026/08/05
Let A be an abelian variety over a number field F and let p be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of As should contain an element of order p for a positive proportion of quadratic twists As of A. We give a general method to prove instances of this conjecture by exploiting independent isogenies of A. For each prime p, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial p-torsion in their Tate-Shafarevich groups. In particular, when the modular curve X0(3p) has infinitely many F-rational points the method applies to ``most'' elliptic curves E having a cyclic 3p-isogeny. It also applies in certain cases when X0(3p) has only finitely many points. For example, we find an elliptic curve over ℚ for which a positive proportion of quadratic twists have an element of order 5 in their Tate-Shafarevich groups. The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime p ≡ 1 \pmod 9, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order p in their Tate-Shafarevich groups.