STNB 2015 (29th year)
Computing equations of elliptic curves over number fields via $p$-adic methods
Presenters
Xavier Guitart and Xevi Guitart
Abstract
It is conjectured that to every modular form over a number field which is a Hecke eigenform with rational coefficients is attached an elliptic curve with the same $L$-function. This is known for modular forms over the rationals; moreover, the so-called Eichler-Shimura construction gives an explicit method for computing the curve from the modular form, which has been used to produce extensive tables of curves over $\mathbb{Q}$ (e.g. Cremona's tables). The Eichler-Shimura construction is known to generalize (at least partially) to forms over totally real number fields. However, over fields that have complex places no explicit construction seems to be available.
The aim of the talk is to describe a conjectural construction which applies under certain additional conditions and is a natural generalization of the $p$-adic uniformizations arising in the theory of Stark-Heegner/Darmon points. I will also report on some numerical computations supporting the conjecture in the case when the number field has exactly one complex place.
This is joint work with Marc Masdeu and Haluk Sengun.
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