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Title: Random matrix theory and L-functions in function fields
Author: Bueno de Andrade, Júlio César
ISNI:       0000 0004 2739 1830
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2012
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It is an important problem in analytic number theory to estimate mean values of the Riemann zeta-function and other L-functions. The study of moments of L-functions has some important applications, such as to give information about the maximal order of the Riemann zeta-function on the critical line, the Lindelof Hypothesis for L-functions and non-vanishing results. Furthermore, according to the Katz-Sarnak philosophy [Katz-Sar99a, Katz-Sar99b] it is believed that the understanding of mean values of different families of L-functions may reveal the symmetry of such families. The analogy between characteristic polynomials of random matrices and L- functions was first studied by Keating and Snaith [Kea-SnaOOa, Kea-SnaOOb]. For example, they were able to conjecture asymptotic formulae for the moments of L- functions in different families. The purpose of this thesis is to study moments of L-functions over function fields, since in this case the L-functions satisfy a Riemann Hypothesis and one may give a spectral interpretation for such L-functions as the characteristic polynomial of a unitary matrix. Thus, we expect that the analogy between characteristic polynomials and L-functions can be further understood in this scenario. In this thesis, we study power moments of a family of L-functions associated with hyperelliptic curves of genus 9 over a fixed finite field lFq in the limit as g-->∞, which is the opposite limit considered by the programme of Katz and Sarnak. Specifically, we compute some average value theorems of L-functions of curves and we extend to the function field setting the heuristic for integral moments and ratios of L-functions previously developed by Conrey et. al [CFKRS05, Conr-Far-Zir] for the number field case.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available