We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham e-Theses
You are in:

Arithmetic of metaplectic modular forms

MERCURI, SALVATORE,MARCO (2019) Arithmetic of metaplectic modular forms. Doctoral thesis, Durham University.

PDF - Accepted Version


Modular forms came to the attention of number theorists through the wealth of their arithmetic behaviour, the development and applications of which continue to surprise. Arithmetic data of associated $L$-functions have conjectured links to fundamental questions, for example the generalised Riemann hypothesis and the BSD conjecture; special values of $L$-functions and their $p$-adic analogues have had a key role in progress towards BSD. Modular forms of half-integral weight have a number-theoretic history spanning as far back as that of their integral-weight counterparts, but their arithmetic theory has long been latent. Being fundamental variants of integral-weight modular forms, a fully fledged theory of half-integral weight modular forms has high potential for impact in areas of number theory.

In this thesis, we develop four key areas in the arithmeticity of Siegel modular forms of half-integral weight, focusing on the behaviour of their Fourier coefficients and associated $L$-functions as follows: an analogue of Garrett's conjecture on the precise algebraicity of Klingen Eisenstein series and of the decomposition $\mathcal{M}_k = \mathcal{S}_k\oplus\mathcal{E}_k$; the precise algebraicity of special $L$-values; the existence of $p$-adic $L$-functions; and, for vector-valued modular forms, an explicit Rankin-Selberg integral expression. Some of the results, such as special values of $L$-functions, are further refinements of existing theorems; others, such as the construction of $p$-adic $L$-functions, are entirely new.

The multifaceted nature of modular forms is a considerable characteristic of theirs. Classically developed as analytic objects, integral-weight modular forms have been reinterpreted algebraically in terms of automorphic representations and associations to motives. Since the algebraic viewpoint remains insufficient for our purposes we focus on the analytic theory and methods of proof for half-integral weight modular forms, using Shimura's theory of Hecke operators and his Rankin-Selberg expression as a basis, and modifying the established methods of Harris, Sturm, and Panchishkin to prove our results.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Siegel modular forms; modular forms of half-integral weight; arithmetic of automorphic forms; Rankin-Selberg method; arithmetic of Eisenstein series; special values; p-adic L-function; vector-valued modular forms
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2019
Copyright:Copyright of this thesis is held by the author
Deposited On:04 Jun 2019 13:33

Social bookmarking: del.icio.usConnoteaBibSonomyCiteULikeFacebookTwitter