Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves
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Author(s)
Bhargava, Manjul
Kane, Daniel M.
Lenstra, Hendrik W.
Poonen, Bjorn
Rains, Eric
Date Issued
August 2015
Journal
Cambridge Journal of Mathematics
Publisher
International Press of Boston
Citation
Bhargava, Manjul et al. “Modeling the Distribution of Ranks, Selmer Groups, and Shafarevich–Tate Groups of Elliptic Curves.” Cambridge Journal of Mathematics 3.3 (2015): 275–321.
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Author's final manuscript
Abstract
Using maximal isotropic submodules in a quadratic module over Z[subscript p], we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of cofinite type Z[superscript p]-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0→E(k)⊗Q[subscript p]/Z[subscript p]→Sel[subscript p∞]E→Ш[p[superscript ∞]]→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 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. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4310/CJM.2015.v3.n3.a1
