Selmer groups of abelian varieties in extensions of function fields

Abstract

Let $k$ be a field of characteristic $q$, $\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p\ne q$ be a prime number and $\cac'\to\cac$ a finite geometrically \textsc{Galois} and \'etale cover defined over $k$ with function field $K':=k(\cac')$. Let $(\tau',B')$ be the $K'/k$-trace of $A/K$. We give an upper bound for the $\bbz_p$-corank of the \textsc{Selmer} group $\text{Sel}_p(A\times_KK')$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $\bbz$-rank of the \textsc{Lang-N\'eron} group $A(K')/\tau'B'(k)$. In the case of a geometric tower of curves whose \textsc{Galois} group is isomorphic to $\bbz_p$, we give sufficient conditions for the \textsc{Lang-N\'eron} group of $A$ to be uniformly bounded along the tower.