Growth of Selmer Groups over function fields

TL;DR

This paper investigates the growth of the p-Selmer group ranks of abelian varieties over function fields, establishing lower bounds in dihedral extensions using Mazur-Rubin's local constants theory.

Contribution

It extends the understanding of Selmer group growth to function fields, applying local constants theory to relate ranks over quadratic and dihedral extensions.

Findings

If the Z_p-corank of Sel_p(A/K) is odd, then Z_p-corank of Sel_p(A/F) is at least [F:K].

The results connect Selmer group growth with extension degrees in function fields.

Uses Mazur-Rubin local constants theory for elliptic curves over number fields, adapted to function fields.

Abstract

We study the rank of the $p$-Selmer group $Sel_p(A/k)$ of an abelian variety $A/k$, where $k$ is a function field. If $K/k$ is a quadratic extension and $F/k$ is a dihedral extension and the $\mathbb{Z}_p$-corank of $Sel_p (A/K)$ is odd, we show that the $\mathbb{Z}_p$-corank of $Sel_p(A/F) \geq [F:K]$. The result uses the theory of local constants developed by Mazur-Rubin for elliptic curves over number fields.