Uniformization of modular elliptic curves via $p$-adic periods

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 [GMS14], we associated, building on works of Darmon [Dar01] and Greenberg [Gre09], a $p$-adic lattice $\Lambda$ to $f$, under certain hypothesis, and implicitly conjectured that $\Lambda$ 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 $f$, a 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 gives extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields.