Cambridge Journal of Mathematics

Volume 3 (2015)

Number 3

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Pages: 275 – 321

DOI: https://dx.doi.org/10.4310/CJM.2015.v3.n3.a1

Authors

Manjul Bhargava (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Daniel M. Kane (Dept. of Mathematics and Dept. of Computer Science and Engineering, University of California at San Diego)

Hendrik W. Lenstra, Jr. (Mathematisch Instituut, Universiteit Leiden, The Netherlands)

Bjorn Poonen (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Eric Rains (Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)

Abstract

Using maximal isotropic submodules in a quadratic module over $\mathbb{Z}_p$, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of cofinite type $\mathbb{Z}_p$-modules, and then conjecture that as $E$ varies over elliptic curves over a fixed global field $k$, the distribution of\[0 \to E(k) \otimes \mathbb{Q}_p / \mathbb{Z}_p \to \mathrm{Sel}_{p^\infty} \; E \to Ш [ p^{\infty} ] \to 0\]is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution on the set of isomorphism classes of finite abelian $p$-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over $\mathbb{Z}_p$, and we prove that the two probability distributions are compatible with each other and with Delaunay’s predicted distribution for $Ш$. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

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Published 25 August 2015