  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.
