Pilar Bayer

Quaternion Orders, Quadratic Forms, and Shimura Curves

Quaternion algebras and quaternion orders Introduction to Shimura curves Quaternion algebras and quadratic forms Embeddings and quadratic forms Hyperbolic fundamental domains for Shimura curves Complex multiplication points in Shimura curves The Poincare package Tables Further contributions to the study of Shimura curves Applications of Shimura curves Bibliography Index.

On values of zeta functions and ℓ-adic Euler characteristics

The values of the zeta function (~(s) of a totally real number field K on the negative integers s = n are, by a theorem of Siegel, rational numbers; they are zero iff n is even. S. Lichtenbaum has made the remarkable conjecture that, for every odd prime #, the f-adic absolute values l(~(-n)[~ can be expressed in the following way as 6tale Euler characteristics of the scheme X=Spec((9)-{points above f}, where (9 is the ring of integers of K:

On the Hasse–Witt Invariants of Modular Curves

We briefly discuss the relationship between several characterizations of the Hasse–Witt invariant of curves in characteristic p with the goal of computing its value in concrete instances. We study its asymptotic behaviour when dealing with the geometric fibres of curves of genus ≥ 2 defined over the rationals. Numerical evidence gathered for several modular curves supports certain conjectural distribution laws.

A reduction point algorithm for cocompact Fuchsian groups and applications

In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations. We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group

Jean-Pierre Serre: An Overview of His Work

\mathbf{SL}(2, \mathbb{Z})

On automorphic forms and hodge theory

and whose roots go back to [9]. The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.

Galois representations of octahedral type and 2-coverings of elliptic curves

For more than five decades, the mathematical contributions of Jean-Pierre Serre have played an essential role in the development of several areas of mathematics. The present paper aims to provide an overview of his work. The selected references include his books, most of the papers collected in his Œuvres, as well as some of his more recent publications.