  The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is
determined up to isogeny by its $p$-torsion Galois representation for $p\geq
17$. We study a geometric analog of this conjecture, and show that the map from
isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic
multiplication---to their $p$-torsion Galois representations is one-to-one over
function fields of small genus complex curves for sufficiently large $p$
relative to the genus.
