Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 61-62). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--Ohio State University, 2009. / Title from first page of PDF file. Includes vita. Includes bibliographical references (p. 128-131).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

No description available.

No description available.

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

Cette thèse s'intéresse aux fonctions L, à leurs aspects explicites et à leurs applications Dans le premier chapitre, nous donnons une définition précise de ce que nous appelons une fonction L ainsi que leurs principales propriétés, notamment concernant les invariants appelés paramètres locaux. Ensuite, nous traitons le cas des fonctions L d'Artin. Pour celles-ci, nous avons créé un programme dans le logiciel PARI/GP donnant les coefficients et les invariants d'une fonction L d'Artin lorsque le corps de base est Q.Le deuxième chapitre explicite un théorème dû à Henryk Iwaniec et Emmanuel Kowalski permettant de différencier deux fonctions L générales en considérant leurs paramètres locaux pour tous les premiers jusqu'à une certaine borne théorique.Dans la suite, nous constaterons que distinguer la somme des paramètres locaux de fonctions L d'Artin revient à séparer les caractères associés par les automorphismes de Frobenius. Ce sera l'objet du troisième chapitre qui est à relier au théorème de Chebotarev. En appliquant notre résultat à des caractères conjugués du groupe alterné, on obtient une borne sur un nombre premier p donnant l'écriture de la factorisation modulo p d'un polynôme répondant à certains critères. Ce travail est à comparer avec un résultat de Joël Bellaïche (2013). Nous illustrons enfin numériquement nos résultats en étudiant l'évolution de la borne sur des polynômes de la forme X^n+uX+v avec n=5, 7 et 13. / This thesis focuses on L-functions, their explicit aspects and their applications.In the first chapter, we give a precise definition of L-functions and their main properties, especially about the invariants called local parameters. Then, we deal with Artin L-functions. For them, we have created a computer program in PARI/GP which gives the coefficients and the invariants for an Artin L-function above Q.In the second chapter, we make explicit a theorem of Henryk Iwaniec and Emmanuel Kowalski, which distinguishes between two L-functions by considering their local parameters for primes up to a theoretical bound.Actually, distinguishing between sums of local parameters of Artin L-functions is the same as separating the associated characters by the Frobenius automorphism. This is the subject of the third chapter, that can be related to Chebotarev Theorem. By applying the result to conjugate characters of the alternating group, we get a bound for a prime p giving the factorization modulo $p$ of a certain polynomial. This work has to be compared with a result from Joël Bellaïche (2013).Finally, we numerically illustrate our results by studying the evolution of the bound on polynomials X^n+uX+v, for n=5, 7 and 13.

This thesis is comprised of three problems in number theory. The introduction is Chapter 1. The first problem is to partially generalize the main theorem of Gross, Kohnen and Zagier to higher weight modular forms. In Chapter 2, we present two conjectures which do this and some partial results towards their proofs as well as numerical examples. This work provides a new method to compute coefficients of weight k+1/2 modular forms for k>1 and to compute the square roots of central values of L-functions of weight 2k>2 modular forms. Chapter 3 presents four different interpretations of the main construction in Chapter 2. In particular we prove our conjectures are consistent with those of Beilinson and Bloch. The second problem in this thesis is to find an arithmetic formula for the central value of a certain Hecke L-series in the spirit of Waldspurger's results. This is done in Chapter 4 by using a correspondence between special points in Siegel space and maximal orders in quaternion algebras. The third problem is to find a lower bound for the cardinality of the principal genus group of binary quadratic forms of a fixed discriminant. Chapter 5 is joint work with Jeffrey Stopple and gives two such bounds. / text

Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors. We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. We then give an exposition of the Siegel--Walfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions. Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the Bombieri--Vinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.