Twisted Fermat curves over totally real fields

Twisted Fermat curves over totally real fields

Darmon points on elliptic curves over totally real fields

Let F be a number field, let K be a quadratic extension of F , and let E be an elliptic curve over F of conductor an ideal N of F . We assume that there is a newform f of weight 2 on Γ0(N ) over F (for the notion of modular forms over number fields other than Q, see, e.g., [Byg99]) whose L-function coincides with that of E. For example, this is known to be the case if F = Q by [BCDT01] and if F...

Darmon points on elliptic curves over totally real fields

Let F be a number field and let K be a quadratic extension of F . The goal of the theory of Stark-Heegner points is to generalize the construction of Heegner points on elliptic curves over quadratic imaginary fields. If F is totally real and K is totally complex (i.e., a CM field), then under fairly general hypotheses, there is a notion of Heegner points on E defined over suitable ring class fi...

Representations over Totally Real Fields

In this paper, we study the level lowering problem for mod 2 representations of the absolute Galois group of a totally real field F. In the case F = Q, this was done by Buzzard; here, we generalise some of Buzzard’s results to higher weight and arbitrary totally real fields, using Rajaei’s generalisation of Ribet’s theorem and previous work of Fujiwara and the author. 2000 Mathematics Subject C...

Perfect Forms over Totally Real Number Fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronöı and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect f...