Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture. We also prove the existence of a discrete probability distribution on the set of isomorphism classes of finite abelian pp-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over ZpZp, and we prove that the two probability distributions are compatible with each other and with Delaunay’s predicted distribution for ШШ. BSc Computer Science Discrete Mathematics Handwritten Notes PDF Free Download Computer Science is the subject of study which incorporates discrete mathematics to the fullest. 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. ![]() Using maximal isotropic submodules in a quadratic module over Z, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of cofinite type Z-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0→E(k)⊗Q/Z p]→SelE→Ш]→0 is that one.
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