Asymptotic Fermat for signatures (p,p,2)$(p,p,2)$ and (p,p,3)$(p,p,3)$ over totally real fields

Twisted Fermat curves over totally real fields

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...

Companion Forms Over Totally Real Fields, II

We prove a companion forms theorem for mod l Hilbert modular forms. This work generalises results of Gross and Coleman–Voloch for modular forms over Q, and gives a new proof of their results in many cases.

Companion Forms over Totally Real Fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular fo...