Arithmetic of metaplectic modular forms

Abstract

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 -functions have conjectured links to fundamental questions, for example the generalised Riemann hypothesis and the BSD conjecture; special values of -functions and their -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 -functions as follows: an analogue of Garrett's conjecture on the precise algebraicity of Klingen Eisenstein series and of the decomposition ; the precise algebraicity of special -values; the existence of -adic -functions; and, for vector-valued modular forms, an explicit Rankin-Selberg integral expression. Some of the results, such as special values of -functions, are further refinements of existing theorems; others, such as the construction of -adic -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.