Frey-Mazur theorem for semistable elliptic curves over function fields

Abstract

Let $k$ be a field, let $B/k$ be a smooth projective geometrically integral curve, and put $K=k(B)$. We prove a Frey--Mazur type theorem for nonisotrivial semistable elliptic curves over $K$. More precisely, after fixing the genus of $B$ and the characteristic of $k$, a sufficiently large prime-to-characteristic torsion representation determines the $K$-isogeny class.