The Manin constant of elliptic curves over function fields

Abstract

We study the $p$-adic valuation of the values of normalised Hecke eigenforms attached to nonisotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic $p$. We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field, and provide examples to show that our estimates are optimal. As an application of our results, we prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields satisfying the assumption mentioned above.