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The geometric average size of Selmer groups over function fields
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-05-20, DOI: 10.2140/ant.2021.15.673
Aaron Landesman

We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over ${\mathbb{𝔽}}_{q}\left(t\right)$ is the sum of the divisors of $n$. As a corollary, again in the large $q$ limit, we deduce that $100%$ of elliptic curves of bounded height over ${\mathbb{𝔽}}_{q}\left(t\right)$ have rank $0$ or $1$.