Growth of Selmer Groups over function fields

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