Book Review: Super-real fields---Totally ordered fields with additional structure

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.

Twisted Fermat curves over totally real fields

Zp-Extensions of Totally Real Fields

We continue our investigations into complex and p-adic variants of H. M. Stark’s conjectures [St] for an abelian extension of number fields K/k. We have formulated versions of these conjectures at s = 1 using so-called ‘twisted zeta-functions’ (attached to additive characters) to replace the more usual L-functions. The complex version of the conjecture was given in [So3]. In [So4] we formulated...