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Thesis advisor: Mark Reeder / This thesis contributes to the proof of the conjectural local Langlands correspondence in the case of small residue characteristic. Let G be an absolutely simple split reductive group over a finite extension k of ℚ_p. To each point in the Bruhat-Tits building of G(K), Moy and Prasad have attached a filtration of G(K) by bounded subgroups. In the first main result of this thesis we give necessary and sufficient conditions for the first Moy-Prasad filtration quotient to have stable functionals for the action of the reductive quotient (this result is joint with Jessica Fintzen). Our work extends earlier results by Reeder and Yu, who gave a classification in the case when p is sufficiently large. By passing to a finite unramified extension of k if necessary, we obtain new supercuspidal representations of G(k) when p is small. Next we consider G = G₂. For this case we explicitly describe the locus of stable functionals on the first Moy-Prasad filtration quotient for every point in the Bruhat-Tits building. Our description is in terms of the invariant theory of SL₂ x SL₂. This allows us to construct a previously unknown representation π of G₂(ℚ₂) using the construction of Reeder-Yu. We then prove that there exists a unique Langlands parameter that satisfies the local degree conjecture of Hiraga, Ichino, and Ikeda with respect to π. We give an explicit construction of this parameter. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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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

Cette thèse traite de deux problèmes dans le cadre du programme de Langlands. Pour le premier problème, dans la situation de $\GL_2 $ et un cocaractère non minuscule, nous fournissons un contre-exemple (sous certaines hypothèses naturelles) à la conjecture de Rapoport-Zink, communiquée par Laurent Fargues. Le deuxième problème concerne un résultat dans le programme de Langlands $p$-adique. Soit $A$ une algèbre $\qp$-affinoïde, au sens de Tate. Nous développons une théorie d'un espace localement convexe en $A$-modules parallèle au traitement dans le cas d'un corps par Schneider et Teitelbaum. Nous montrons qu'il existe une application d'intégration liant une catégorie de représentations localement analytiques en $A$ -modules et des modules de distribution séparés relatif. Il existe une théorie de cohomologie localement analytique pour ces objets et une version du Lemme de Shapiro. Dans le cas d'un corps, ceci a été substantiellement développé par Kohlhaase. Comme une application, nous proposons une correspondance de Langlands $p$-adique en families: Pour un $(\varphi, \Gamma)$-module trianguline et régulière de dimension 2 sur l'anneau de Robba relatif $\Robba_A$ nous construisons une $\GL_2(\qp)$-représentation localement analytique en $A$-modules. Il s'agit d'un travail en commun avec Joaquin Rodrigues. / This thesis deals with two problems within the Langlands program. For the first problem, in the situation of $\GL_2$ and a non-minuscule cocharacter, we provide a counter-example (under some natural assumptions) to the Rapoport-Zink conjecture, communicated to us by Laurent Fargues.The second problem deals with a result in the $p$-adic Langlands program. Let $A$ be a $\qp$-affinoid algebra, in the sense of Tate. We develop a theory of locally convex $A$-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in $A$-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma. In the case of a field this has been substantially developed by Kohlhaase. As an application we propose a $p$-adic Langlands correspondence in families: For a regular trianguline $(\varphi,\Gamma)$-module of dimension 2 over the relative Robba ring $\Robba_A$ we construct a locally analytic $\GL_2(\qp)$-representation in $A$-modules. This is joint work with Joaquin Rodrigues.

Let E/F be a quadratic extension of p-adic fields. The local Langlands correspondence establishes a bijection between n-dimensional Frobenius semisimple representations of the Weil-Deligne group of E and smooth, irreducible representations of GL(n, E). We reinterpret this bijection in the setting of the Weil restriction of scalars Res(GL(n), E/F), and show that the Asai L-function and epsilon factor on the analytic side match up with the expected Artin L-function and epsilon factor on the Galois side.

Let D be a division algebra over a p-adic field of characteristic 0. We investigate the mod-p supersingular representations of GL(2, D) by computing a basis for the space of invariants of a certain quotient under the pro-p Iwahori subgroup. This generalizes the previous works of Hendel and Schein.

Hida a étudié l'image de la représentation galoisienne associée à une famille p-adique de Hida de formes automorphes. Il a montré que l'image d'une famille non CM de formes modulaires classiques ordinaires contient un sous-groupe de congruence. Il a aussi lié le niveau optimal du groupe de congruence à l'idéal de congruence entre la famille de Hida non-CM et des familles CM. Cette thèse se divise en deux parties. La première partie est à généraliser ce genre de résultats dans le cas ordinaire pour les familles de Hida sur les groupes réductifs sous les hypothèses techniques. La deuxième partie se consacre à étudier les cas concrets. On montre que les hypothèses techniques sont satisfaites. Donc le même type de résultats est établi automatiquement. / Hida has studied the image of Galois representation associated to a p-adic Hida family of automorphic forms. He has proved that the image of a non-CM family ofordinary classic modular forms contains a congruence subgroup. He also related the optimal level of congruence subgroup to the congruence ideal between the non-CM Hida family and the CM ones. This thesis is divided into two parts. In the first part,we generalize this type of results to ordinary Hida families over reductive groups under some technical hypothesis. In the second part, we consider concrete cases. We prove that the technical hypothesis are satised for these cases. Hence the same type of results is established automatically.

In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.

This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.