Modeling the distribution of Selmer groups, Shafarevich-Tate groups, and ranks of elliptic curves
speaker: Bjorn Poonen (Massachusetts Institute of Technology)abstract:
Using only linear algebra over \(Z_p\), we define a discrete probability
distribution on the set of isomorphism classes of short exact sequences
of \(Z_p\)-modules, and then conjecture that as \(E\) varies over elliptic curves
over a fixed global field, the distribution of
\(0 \to E(k) \otimes Q_p/Z_p \to Sel_{p^\infty} E \to Sha[p^\infty] \to 0\)
is that one. This one conjecture would have the following consequences:
1) Asymptotically, 50% of elliptic curves have rank 0 and 50% have rank 1.
2) \(Sha[p^\infty]\) is finite for 100% of elliptic curves.
3) The Poonen-Rains conjecture on the distribution of \(Sel_p E\) holds.
4) Delaunay's conjecture a la Cohen-Lenstra on the distribution of Sha holds.
(This is joint work with M. Bhargava, D. Kane, H. Lenstra, and E. Rains.)
timetable:
Tue 16 Oct, 16:30 - 17:10, Sala Conferenze Centro De Giorgi
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