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Thesis advisor: Solomon Friedberg / We study a new lifting of automorphic representations using the theta representation ϴ on the 4-fold cover of the symplectic group, $\overline{\Sp}_{2r}(\A)$. This lifting produces the first examples of CAP representations on higher degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to ϴ. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\Sp}_{2r}(\A)$. Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of $\overline{\Sp}_{2r}(\A)$. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over ℚ. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in ^CG(ℚ_l) as opposed to ^LG(ℚ_l). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map ^CG(ℚ_l) → ^LG(ℚ_l) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = U_n(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross' conjecture. We prove that one can transfer a cuspidal automorphic representation π of G to a π' on Sp_2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer π′ to an automorphic representaion of GL_2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL_2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.

Cette thèse est consacrée à l'étude des représentations automorphes algébriques des groupes linéaires découvertes par Chenevier-Renard. On s'intéresse plus particulièrement à leurs paramètres de Satake. Pour cela, nous utilisons la théorie d'Arthur afin de faire apparaître ces représentations par le biais de représentations automorphes discrètes des groupes spéciaux orthogonaux de réseaux bien choisis. Ensuite, on détermine des propriétés d'opérateurs de Hecke agissant sur ces mêmes réseaux, ce qui nous donne de nombreuses informations sur ces paramètres de Satake. On arrive notamment à déterminer la trace dans la représentation standard de nombreux paramètres de Satake des représentations algébriques évoquées, dont les poids peuvent être arbitrairement grands. Ces résultats nous permettent aussi de déterminer de nombreux opérateurs de Hecke, associés aux voisinage de Kneser, vus comme endomorphismes agissant sur les classes d'isomorphisme des réseaux pairs de déterminant 2 en dimension 23 ou 25. / In this these we study the different algebraic automorphic representations discovered by Chenevier-Renard. We focus more particularly on their Satake parameters. To do so, we use Arthur's theory, which enables us to see these representations through discrete automorphic representations for the special orthogonal group of well chosen lattices. Afterwards, we can compute some properties of Hecke operators acting on these lattices. This gives us a lot of information on these Satake parameters. In particular, we can determine the trace in the standard representation for many of these algebraic representations, which weight can be arbitrarily high. These results also enable us to compute many Hecke operators, connected to the notion of neighbourhood developed by Kneser, seen as linear operators acting on the classes of isomorphism of even lattices with determinant 2 in dimension 23 or 25.

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.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23673号 / 理博第4763号 / 新制||理||1683(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