Thesis advisor: Benjamin Howard / I prove the Chai-Faltings version of the Eichler-Shimura congruence relation for simple GSpin Shimura varieties with hyperspecial level structures at a prime p. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1997. / Includes bibliographical references (p. 70-71).

In this thesis we study Kudla’s special cycles of codimension 𝑟 on a unitary Shimura variety Sh(U(𝑚 − 1,1)) together with an embedding of a Shimura subvariety Sh(U(𝑚 − 1,1)). We prove that when 𝑟 = 𝑛 − 𝑚, for certain cuspidal automorphic representations 𝜋 of the quasi-split unitary group U(𝑟,𝑟) and certain cusp forms ⨍ ∈ 𝜋, the geometric volume of the pullbackof the arithmetic theta lift of ⨍ equals the special value of the standard 𝐿-function of 𝜋 at 𝑠 = (𝑚 − 𝑟 + 1)/2. As ingredients of the proof, we also give an exposition of Kudla’s geometric Siegel-Weil formula and Yuan-Zhang-Zhang’s pullback formula in the setting of unitary Shimura varieties, as well as Qin’s integral representation result for 𝐿-functions of quasi-split unitary groups.

Thesis advisor: Ben Howard / The results in this dissertation are on the intersection behavior of certain special cycles on GSpin(n, 2) Shimura varieties for n > 1. In particular, we will determine when the intersection of the special cycles defined by a collection of special endomorphisms consists of isolated points in terms of the fundamental matrix of this collection. These generalize the corresponding results in the lower dimensional cases proved by Kudla and Rapoport. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In this dissertation, we study etale correspondence of hyperbolic curves with unbounded dynamics. Mochizuki proved that over a field of characteristic 0, such curves are always Shimura curves. We explore variants of this question in positive characteristic, using graph theory, l-adic local systems, and Barsotti-Tate groups. Given a correspondence with unbounded dynamics, we construct an infinite graph with a large group of ”algebraic” automorphisms and roughly measures the ”generic dynamics” of the correspondence. We construct a specialization map to a graph representing the actual dynamics. Along the way, we formulate conjectures that etale correspondences with unbounded dynamics behave similarly to Hecke correspondences of Shimura curves. Using graph theory, we show that type (3,3) etale correspondences verify various parts of this philosophy. Key in the second half of this dissertation is a recent p-adic Langlands correspondence, due to Abe, which answers affirmatively the petites camarades conjecture of Deligne in the case of curves. This allows us the build a correspondence between rank 2 l-adic local systems with trivial determinant and Frobenius traces in Q and certain height 2, dimension 1 Barsotti-Tate groups. We formulate a conjecture on the fields of definitions of certain compatible systems of l-adic representations. Relatedly, we conjecture that the Barsotti-Tate groups over complete curves in positive characteristic may be ”algebraized” to abelian schemes.

In this thesis, we study the compatibility between local and global Langlands correspondences for \(GL_n\). This generalizes the compatibility between local and global class field theory and is related to deep conjectures in algebraic geometry and harmonic analysis, such as the Ramanujan-Petersson conjecture and the weight monodromy conjecture. Let L be a CM field. We consider the case when \(\Pi\) is a cuspidal automorphic representation of \(GL_n(\mathbb{A}_L^\infty)\), which is conjugate self-dual and regular algebraic. Under these assumptions, there is an l-adic Galois representation \(R_l(\Pi)\) associated to \(\Pi\), which is known to be compatible with the local Langlands correspondence in most cases (for example, when n is odd) and up to semisimplification in general. In this thesis, we complete the proof of the compatibility when \(l \neq p\) by identifying the monodromy operator N on both the local and the global sides. On the local side, the identification amounts to proving the Ramanujan-Petersson conjecture for \(\Pi\) as above. On the global side it amounts to proving the weight-monodromy conjecture for part of the cohomology of a certain Shimura variety. / Mathematics

This licentiate thesis consists of two papers. In paper I the tautological ring of a Hilbert modular variety at an unramified prime is computed. The method of van der Geer in the case of A_{g} is extended to deal with the case of the Hilbert modular variety, which is more complicated. An example involving the unitary group is given which shows that this method cannot be used to compute the tautological rings of all Shimura varieties of Hodge type. In paper II we compute the pushforward map from a sub flag variety defined by a Levi subgroup to the Siegel flag variety. Specifically, this is the Levi factor of the parabolic associated with the maximal rational boundary component of the Siegel Shimura datum. The method involves an explicit understanding of the pullback map and an application of the self intersection formula.

Nous étudions la stratification de Newton des variétés de Shimura de type PEL aux places de bonne réduction. Nous considérons la strate basique de certaines variétés de Shimura simples de type PEL modulo une place de bonne réduction. Sous des hypothèses simplificatrices nous prouvons une relation entre la cohomologie l-adique de ce strate basique et la cohomologie de la variété de Shimura complexe. En particulier, nous obtenons des formules explicites pour le nombre de points dans la strate basique sur des corps finis, en termes de représentations automorphes. Nous obtenons les résultats à l'aide de la formule des traces et de la troncature de la formule de Kottwitz pour le nombre de points sur une variété de Shimura sur un corps fini. Nous montrons, en utilisant la formule des traces, que n'importe quelle strate de Newton d'une variété de Shimura de type PEL de type (A) est non vide en une place de bonne réduction. Ce résultat a déjà été établi par Viehmann-Wedhorn; nous donnons une nouvelle preuve de ce théorème. Considérons la strate basique des variétés de Shimura associées à certains groupes unitaires dans les cas où cette strate est une variété finie. Alors, nous démontrons un résultat d' équidistribution pour les opérateurs de Hecke agissant sur cette strate. Nous relions le taux de convergence avec celui de la conjecture de Ramanujan. Dans nos formules ne figurent que des représentations automorphes cuspidales sur Gl_n pour lesquelles cette conjecture est connue, et nous obtenons donc des estimations très bonnes sur la vitesse de convergence. En collaboration avec Erez Lapid nous calculons le module de Jacquet d'une représentation en échelle pour tout sous-groupe parabolique standard du groupe général linéaire sur un corps local non-archimédien. / We study the Newton stratification of Shimura varieties of PEL type, at the places of good reduction. We consider the basic stratum of certain simple Shimura varieties of PEL type at a place of good reduction. Under simplifying hypotheses we prove a relation between the l-adic cohomology of this basic stratum and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields, in terms of automorphic representations. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite field. We prove, using the trace formula that any Newton stratum of a Shimura variety of PEL-type of type (A) is non-empty at a prime of good reduction. This result is already established by Viehmann-Wedhorn; we give a new proof of this theorem. We consider the basic stratum of Shimura varieties associated to certain unitary groups in cases where this stratum is a finite variety. Then, we prove an equidistribution result for Hecke operators acting on the basic stratum. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain particular cuspidal automorphic representations on Gl_n. The Ramanujan conjecture turns out to be known for these automorphic representations, and therefore we obtain very sharp estimates on the rate of convergence. We prove that any connected reductive group G over a non-Archimedean local field has a cuspidal representation. Together with Erez Lapid we compute the Jacquet module of a Ladder representation at any standard parabolic subgroup of the general linear group over a non-Archimedean local field.

One of the central themes of modern Number Theory is to study properties of Galois and automorphic representations and connections between them. In our dissertation, we describe two different projects that study properties of these objects. In our first project, which is analytic in nature, we consider Artin representations of Q of dimension 3 that are self-dual. We show that these occur with density 0 when counted using the conductor. This provides evidence that self-dual representations should be rare in all dimensions. Our second project, which is more algebraic in nature, is related to automorphic representations. We show the existence of canonical models for certain unitary Shimura varieties. This should help us in computing certain cohomology groups of these varieties, in which regular algebraic automorphic representations having useful properties should be found.