Let K be a number field with ring of integers OK . We compute the local factors of the normal zeta functions of the Heisenberg groups H(OK) at rational primes which are unramified in K. These factors… (More)

Let F be a totally real field and p ≥ 3 a prime. If ρ : Gal(F/F ) → GL2(Fp) is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture… (More)

Let F be a totally real field and ρ : Gal(F/F )→ GL2(Fp) a Galois representation whose restriction to a decomposition group at some place dividing p is irreducible. Suppose that ρ is modular of some… (More)

In this mostly expository article we give a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. We also discuss recent… (More)

Let F be a totally ramified extension of Qp. We consider supersingular representations of GL2(F ) whose socles as GL2(OF )-modules are of a certain form that is expected to appear in the mod p local… (More)

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal… (More)

Let O be the ring of integers of a p-adic field and p its maximal ideal. We compute the Jordan-Hölder decomposition of the reduction modulo p of the cuspidal representations of GL2(O/p) for e ≥ 1. We… (More)

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of… (More)

These notes are based on lectures given by the author at the winter school on Galois theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre’s… (More)

Given a quadratic number field k = Q( √ d) with narrow class number h+d and discriminant ∆k, let Od be the orthogonal Z-group of the associated norm form qd. In this paper we describe the structure… (More)