Let l be a prime and let q be a prime power not divisible by l. Put G=GI[subscript n](F[subscript q])and fix a representation pi of G over a sufficiently large finite field, k, of characteristic l, so that pi is cuspidal but not supercuspidal. We compute the W(k)[G]-endomorphism ring of the projective envelope of pi under the assumption that l>n. / text

Dans cette thèse, on construit le morphisme terme constant pour les groupes de cohomologie l-adique à supports compacts des champs classifiants des G-Chtoucas. Ensuite on définit la partie cuspidale de ces groupes de cohomologie et on montre qu'elle est de dimension finie. De plus, on montre que la partie cuspidale coïncide avec la partie Hecke-finie au sens rationnel. / In this thesis, we construct the constant term morphism for the l-adic cohomology groups with compact supports of the classifying stacks of the G-Shtukas. Then we define the cuspidal part of these cohomology groups and we prove that it is of finite dimension. Moreover, we show that the cuspidal part coincides with the Hecke-finite part in the rational sense.

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[pt] Neste trabalho, estudamos superfícies com curvatura média afim zero. Elas são chamadas de superfícies mínimas afins e para superfícies convexas, também são chamadas de superfícies máximas afins. Provamos que uma superfície mínima euclidiana também é uma superfície mínima afim se, e somente se, as linhas de curvatura da superfície mínima euclidiana conjugada são planas. Para uma superfície máxima afim, descrevemos como recuperá-la do campo de vetor conormal ao longo de uma determinada curva. Para algumas escolhas do vector conormal, a superfície máxima é singular e descrevemos as condições sob as quais as singularidades são arestas cuspidais ou swallowtails. / [en] In this work we study surfaces with zero affine mean curvature. They are called affine minimal surfaces and for convex surfaces, they are also called affine maximal surfaces. We prove that an euclidean minimal surface is also an affine minimal surface if and only if the curvature lines of the conjugate euclidean minimal surface are planar. For an affine maximal surface, we describe how to recover it from the conormal vector field along a given curve. For some choices of the conormal vector, the maximal surface is singular and we describe conditions under which the singularities are cuspidal edges or swallowtails.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-19T18:27:28Z No. of bitstreams: 1 santiagomilerquispemamani.pdf: 952447 bytes, checksum: 6008ae3816024f866eea3c17d560372d (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-07T12:39:21Z (GMT) No. of bitstreams: 1 santiagomilerquispemamani.pdf: 952447 bytes, checksum: 6008ae3816024f866eea3c17d560372d (MD5) / Made available in DSpace on 2017-02-07T12:39:21Z (GMT). No. of bitstreams: 1 santiagomilerquispemamani.pdf: 952447 bytes, checksum: 6008ae3816024f866eea3c17d560372d (MD5) Previous issue date: 2016-04-25 / O objetivo deste trabalho é apresentar o resultado dado por Bass em [4], que determina uma condição no domínio de integridade R para que todo módulo finitamente gerado e livre de torção seja escrito como soma direta de módulos de posto 1. Mostramos que uma condição necessária é que todo ideal em R seja gerado por dois elementos, ou seja, que esses domínios sejam quase domínios de Dedekind. Em seguida, aplicamos o resultado na descrição de módulos livres de torção e de posto finito sobre os anéis de coordenadas de curvas singulares, cujas singularidades são nós ou cúspides. / The aim of this paper is to present the result given by Bass in [4], which determines a condition on the integral domain R so that every finitely generated torsion free module is written as a direct sum of modules of rank 1. We show that a necessary condition is that all ideal in R is generated by two elements, in other words, that these domains are almost Dedekind domains. Then, we apply the result in the description of torsion free modules of finite rank over the coordinate rings of singular curves, whose singularities are nodal or cuspidal.

Soit X1 une courbe projective lisse et géométriquement connexe sur un corps fini Fq avec q = pn éléments où p est un nombre premier. Soit X le changement de base de X1 à une clôture algébrique de Fq. Nous donnons une formule pour le nombre des systèmes locaux ℓ-adiques (ℓ ≠ p) irréductibles de rang donné sur X fixé par l’endomorphisme de Frobenius. Nous montrons que ce nombre est semblable à une formule de point fixe de Lefschetz pour une variété sur Fq, ce qui généralise un résultat de Drinfeld en rang 2 et prouve une conjecture de Deligne. Pour ce faire, nous passerons du côté automorphe, utiliserons la formule des traces d’Arthur non-invariante, et relierons le nombre cherché avec le nombre Fq-points de l’espace des modules des fibrés de Higgs stables. / Let X1 be a projective, smooth and geometrically connected curve over Fq with q = pn elements where p is a prime number, and let X be its base change to an algebraic closure of Fq.We give a formula for the number of irreducible ℓ-adic local systems (ℓ ≠ p) with a fixed rank over X fixed by the Frobenius endomorphism.We prove that this number behaves like a Lefschetz fixed point formula for a variety over Fq, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. To do this, we pass to the automorphic side by Langlands correspondence, then use Arthur’s non-invariant trace formula and link this number to the number of Fq-points of the moduli space of stable Higgs bundles.