Emmanuel Hallouin

Computing Local Integral Closures

Let A be a discrete valuation ring. We give a new approach to the round4 algorithm which permits the computation of the integral closure of A in a finite separable field extension of its field of fractions.

Cancellation in totally definite quaternion algebras

Abstract In [M.-F. Vignéras, Simplification pour les ordres des corps de quaternions totalement définis, J. reine angew. Math. 286/287 (1976), 257–277.], M.-F. Vignéras proves that only a finite number of Eichler orders of a totally definite quaternion algebra over a number field can have the cancellation property. In this article, we draw up the complete list of the genus of these orders.

Global descent obstructions for varieties

We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having

A graph aided strategy to produce good recursive towers over finite fields

QQ

Computation of a cover of Shimura curves using a Hurwitz space

as field of moduli, that admit models over every completion of

Recursive towers of curves over finite fields using graph theory

QQ

Study and computation of a Hurwitz space and totally real PSL2(F8)-extensions of Q

, but have no model over

From Hodge Index Theorem to the number of points of curves over finite fields

QQ

An unified viewpoint for upper bounds for the number of points of curves over finite fields via Euclidean geometry and semi-definite symmetric Toeplitz matrices

.

Courbes très spéciales mais en aucun cas génériques

We propose a systematic method to produce potentially good recursive towers over finite fields. The graph point of view, so as some magma and sage computations are used in this process. We also establish some theoretical functional criterion ensuring the existence of many rational points on a recursive tower. Both points are illustrated by an example, from the production process, to the theoretical study.