Mathematics

This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain p -adic L -functions. First seen in Serre's realization of p -adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {\em Kummer congruences}) to certain other L -functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.

We relate the study of Landau-Siegel zeros to the ranks of Jacobians J 0 (q) of modular curves for large primes q . By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level q have analytic rank ?? . We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank ?? . In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes q we show the rank of J 0 (q) is asymptotically equal to the rank predicted by the Brumer-Murty conjecture.

In this paper we study the exceptional zero phenomenon for Hilbert modular forms in the anticyclotomic setting. We prove a formula expressing the leading term of the p-adic L-functions via arithmetic L-invariants.

We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system Φ . We introduce the generating function Z(s) of a packing, an exponential series in four variables with an Apollonian symmetry group, which relates to Weyl-Kac characters of Φ . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ , with automorphic Weyl denominators, we express Z(s) in terms of Jacobi theta functions and the Siegel modular form ? 5 . We also show that the domain of convergence of Z(s) is the Tits cone of Φ , and discover that this domain inherits the intricate geometric structure of Apollonian packings.

The goal of this PhD thesis is to study a diophantine approximation problem stated by Schmidt in 1967. The problem aim to study the approximation of a subspace of R n by rational subspaces, not necessarily of the same dimension, by determining the exponents of approximation associated to the various angles between the subspaces.

Khnichin's theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss--Kuzmin theorem to several thousand high precision (up to more than 53000 significant digits) initial Stieltjes constants γ n , n=0,1,...,5000 in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We study also the normality of these important constants.

In this paper we connect a celebrated theorem of Nyman and Beurling on the equivalence between the Riemann hypothesis and the density of some functional space in L 2 (0,1) to a trigonometric series considered first by Hardy and Littlewood. We highlight some of its curious analytical and arithmetical properties.

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than Z . We then study the growth of the p ??-Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers k?�K?�F in which [F:K] is not a p -power extension.

Let d?? . We study a subspace of the space of automorphic forms of GL d over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place ??of F , and we consider the subspace A St consisting of automorphic forms such that the local component at ??of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of A St as automorphic representation and gives a multiplicity one type statement. For the other theorem (Theorem 12), we construct, using the geometry of the Bruhat-Tits building, an analogue of modular symbols in A St integrally (that is, in the space of Z -valued automorphic forms). We show that the quotient is finite and give a bound on the exponent of this quotient.

We consider a family of abelian surfaces over Q arising as Prym varieties of double covers of genus- 1 curves by genus- 3 curves. These abelian surfaces carry a polarization of type (1,2) and we show that the average size of the Selmer group of this polarization equals 3 . Moreover we show that the average size of the 2 -Selmer group of the abelian surfaces in the same family is bounded above by 5 . This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding F 4 ??E 6 , invariant theory, a classical geometric construction due to Pantazis, a study of Néron component groups of Prym surfaces and Bhargava's orbit-counting techniques.