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# 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: http://dx.doi.org/10.4310/CJM.2015.v3.n3.a1

#### Authors

#### 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.