The geometric distribution of Selmer groups of Elliptic curves over function fields

Abstract: Bhargava, Kane, Lenstra, Poonen, and Rains proposed heuristics for the distribution of arithmetic data relating to elliptic curves, such as their ranks, Selmer groups, and Tate-Shafarevich groups. As a special case of their heuristics, they obtain the minimalist conjecture, which predicts that $50\%$ of elliptic curves have rank $0$ and $50\%$ of elliptic curves have rank $1$. After surveying these conjectures, we will explain joint work with Tony Feng and Eric Rains, verifying a variant of these conjectures over function fields of the form $\mathbb F_q(t)$, after taking a certain large $q$ limit.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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MIT number theory seminar

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