The Manin constant of elliptic curves over function fields

TL;DR

This paper investigates the p-adic valuations of Hecke eigenform values associated with non-isotrivial elliptic curves over function fields, establishing bounds, optimality, and applications to the degree conjecture.

Contribution

It provides new bounds on valuations, demonstrates their optimality, and proves the degree conjecture for certain elliptic curves over function fields.

Findings

Derived upper bounds on p-adic valuations.

Showed bounds are optimal with examples.

Proved the degree conjecture unconditionally for specific curves.

Abstract

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.