Uniformization of modular elliptic curves via p-adic periods

TL;DR

This paper develops algorithms to compute explicit equations of elliptic curves associated with modular forms over number fields, providing strong evidence for a conjectured p-adic uniformization linking modular forms and elliptic curves.

Contribution

It introduces a method to explicitly construct elliptic curves from modular forms using p-adic periods, supporting the conjecture with extensive computational data.

Findings

Strong computational evidence supporting the conjecture.

Efficient algorithms for constructing elliptic curves from modular forms.

Creation of large databases of elliptic curves over number fields.

Abstract

The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain provide overwhelming amount of support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields.