Growth of Selmer Groups over 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.