Theta Functions, Gauss Sums and Modular Forms

Abstract

We present some results related to the areas of theta functions, modular forms, Gauss sums and reciprocity. After a review of background material, we recount the elementary theory of modular forms on congruence subgroups and provide a proof of the transformation law for Jacobi's theta function using special values of zeta functions. We present a new proof, obtained during work with Michael Eastwood, of Jacobi's theorem that every integer is a sum of four squares. Our proof is based on theta functions but emphasises the geometry of the thrice-punctured sphere. Next, we detail some investigations into quadratic Gauss sums. We include a new proof of the Landsberg-Schaar relation by elementary methods, together with a second based on evaluations of Gauss sums. We give elementary proofs of generalised and twisted Landsberg-Schaar relations, and use these results to answer a research problem posed by Berndt, Evans and Williams. We conclude by proving some sextic and octic local analogues of the Landsberg-Schaar relation. Finally, we give yet another proof of the Landsberg-Schaar relation based on the relationship between Mellin transforms and asymptotic expansions. This proof makes clear the relationship between the Landsberg-Schaar relation and the existence of a metaplectic Eisenstein series with certain properties. We note that one may promote this correspondence to the setting of number fields, and furthermore, that the higher theta functions constructed by Banks, Bump and Lieman are ideal candidates for future investigations of such correspondences.