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.

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

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.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20150号 / 理博第4235号 / 新制||理||1609(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 市野 篤史, 教授 雪江 明彦, 教授 池田 保 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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.

Cette these concerne la geometrie de la correspondance de Langlands p-adique. On donne la formalisation des methodes de Emerton, qui permettrait d'etablir la conjecture de Fontaine-Mazur dans le cas general des groupes unitaires. Puis, on verifie que ce formalism est satisfait dans la cas de U(3) ou on utilise la construction de Breuil-Herzig pour la correspondence p-adique. De point de vue local, on commence l'etude de cohomologie modulo p et p-adiques de tour de Lubin-Tate pour GL_2(Q_p). En particulier, on demontre que on peut retrouver la correspondence de Langlands p-adique dans la cohomologie completee de tour de Lubin-Tate. / This thesis concerns the geometry behind the p-adic local Langlands correspondence. We give a formalism of methods of Emerton, which would permit to establish the Fontaine-Mazur conjecture in the general case for unitary groups. Then, we verify that our formalism works well in the case of U(3) where we use the construction of Breuil-Herzig as the input for the p-adic correspondence.From the local viewpoint, we start a study of the modulo p and p-adic cohomology of the Lubin-Tate tower for GL_2(Q_p). In particular, we show that we can find the local p-adic Langlands correspondence in the completed cohomology of the Lubin-Tate tower.