Watkins' conjecture for elliptic curves over function fields

2022/03/21 by Caro, Jerson
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2203.10932

Abstract

In 2002 Watkins conjectured that given an elliptic curve defined over ℚ, its Mordell-Weil rank is at most the 2-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over \mathbbF_q(T) after extending constant scalars, and every quadratic twist of a modular elliptic curve over \mathbbF_q(T) by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins' conjecture.

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