Ambrus Pàl (London)
The Manin constant of elliptic curves over function fields
We study the p-adic valuation of the values of normalised Hecke eigenforms
attached to non-isotrivial 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 also 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.