The geometric average size of Selmer groups over function fields

Abstract

We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over ${\mathbb{F}}_q(t)$ 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 ${\mathbb{F}}_q(t)$ have rank $0$ or $1$.

References in corpus (3)

• Surjectivity Criteria for p-adic Representations, Part I
• Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields
• Average size of Selmer group in large q limit

Cited by in corpus (4)

• Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity
• Average size of Selmer group in large q limit
• The geometric distribution of Selmer groups of elliptic curves over function fields
• Invariance of the tame fundamental group under base change between algebraically closed fields