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The geometric average size of Selmer groups over function fields

Algebraic Geometry

Speaker: Aaron Landesman, Stanford
Location: 2112 msb
Start time: Wed, Jan 30 2019, 1:10PM

We explain why, as q tends to infinity, 100% of elliptic curves of a given height over $\mathbb F_q(t)$ have rank 0 or 1. We deduce this from a computation of the average size of n-Selmer groups of elliptic curves of fixed height over $\mathbb F_q(t)$ in the large q limit. The idea of the proof is to create a moduli space for Selmer elements, relate the average size to the number of components of this moduli space, and compute the number of components via a monodromy calculation. Our proof reveals alternate heuristics for Selmer group distributions: the averages size is the number of balanced scrolls, and is also the number orbits of certain orthogonal group actions.