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 advisor: Benjamin Howard / We study two moduli problems involving false elliptic curves with complex multiplication (CM), generalizing theorems about the arithmetic degree of certain moduli spaces of CM elliptic curves. The first moduli problem generalizes a space considered by Howard and Yang, and the formula for its arithmetic degree can be seen as a calculation of the intersection multiplicity of two CM divisors on a Shimura curve. This formula is an extension of the Gross-Zagier theorem on singular moduli to certain Shimura curves. The second moduli problem we consider deals with special endomorphisms of false elliptic curves. The formula for its arithmetic degree generalizes a theorem of Kudla, Rapoport, and Yang. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Proefschrift Universiteit van Amsterdam. / Met lit. opg. - Met samenvatting in het Nederlands.

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

Dans cette thèse nous étudions deux problèmes. Le premier est la non-existence de pointsrationnels non spéciaux sur des quotients d’Atkin-Lehner de courbes de Shimura. Le se-cond est l’absence de surfaces abéliennes rationnelles à multiplication potentiellementquaternioniques munies d’une structure de niveau. Ces deux problèmes sont liés car unesurface abélienne rationnelle simple à multiplication potentiellement quaternionique cor-respond à un point rationnel non spécial sur un certain quotient d’Atkin-Lehner de courbede Shimura.Dans une première partie nous expliquons comment vérifier un critère de Parent etYafaev en grande généralité pour prouver que dans les conditions du cas non ramifié deOgg, et si p est assez grand par rapport à q, alors le quotient X^pq/w_q n’a pas de pointrationnel non spécial.Dans une seconde partie nous déterminons une borne effective pour les structures deniveaux possibles pour une surface abélienne rationnelle acquérant sur un corps quadra-tique imaginaire fixé multiplication par un ordre fixé dans une algèbre de quaternions. / In this thesis we study two problems. The first one is the non-existence of rational non-special points on Atkin-Lehner quotients of Shimura curves. The second one is the absence of rational abelian surfaces with potential quaternionique multiplication endowed with a level structure. These two problems are linked because a simple rational abelian surface with potential quaternionique multiplication is associated to a rational non-special point on an Atkin-Lehner quotients of Shimura curve. In a first part of our work we explain how to verify in wide generality a criterium of Parent and Yafaev in order to prove that in the conditions of Ogg's non ramified case, and if $p$ is big enough compared two $q$, then the quotient $X^{pq}/w_q$ has no non-special rational point. In a second part we determine an effective born for possible level structures on rational abelian surfaces having, over a fixed quadratic field, multiplication by a fixed order in a quaternion algebra

On étudie dans un cadre abstrait des critères de semi-simplicité pour des représentations l-adiques de groupes profinis. On applique les résultats obtenus pour montrer que les relations d'Eichler-Shimura généralisées entraînent la semi-simplicitéde certaines représentations galoisiennes non triviales qui apparaissent dans la cohomologie des variétés de Shimura unitaires. Les résultats les plus intéressants sont obtenus pour les variétés de Shimura unitaires de signature $(n,0)^a \times (n-1,1)^b \times (1,n-1)^c \times (0,n)^d$. / We prove several abstract criteria for semi-simplicity of l-adic representations for profinite groups. As an application, we show that generalised Eichler-Shimura relations imply the semi-simplicity of a non-trivial subspace of middle cohomology of unitary Shimura varieties. The most complete results are obtained for unitary Shimura varieties of signature $(n,0)^a \times (n-1,1)^b \times (1,n-1)^c \times (0,n)^d$.

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.