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

Understand

Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 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 of 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 Z_p, and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for Sha.
• Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.