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Using maximal isotropic submodules in a quadratic module over Z_p, we provethe existence of a natural discrete probability distribution on the set ofisomorphism classes of short exact sequences of co-finite type Z_p-modules, andthen conjecture that as E varies over elliptic curves over a fixed global fieldk, the distribution of  0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is thatone. We show that this single conjecture would explain many of the knowntheorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groupsof elliptic curves. We also prove the existence of a discrete probabilitydistribution of the set of isomorphism classes of finite abelian p-groupsequipped with a nondegenerate alternating pairing, defined in terms of thecokernel of a random alternating matrix over Z_p, and we prove that the twoprobability distributions are compatible with each other and with Delaunay'spredicted distribution for Sha. Finally, we prove new theorems on the fppfcohomology of elliptic curves in order to give further evidence for ourconjecture.

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