Theta Functions, Gauss Sums and Modular Forms Benjamin Moore Thesis submitted for the degree of Master of Philosophy in Pure Mathematics at The University of Adelaide Faculty of Engineering, Computer and Mathematical Sciences School of Mathematical Sciences May 2020 An artist's conception of the action of Γ0(4) on the unit disk. Drawing by Katarzyna Nurowska; used with permission. Contents Introduction 1 1 Overview of differential geometry, Fourier analysis and number theory 5 1.1 Manifolds and differential geometry . .6 1.1.1 Group actions on topological spaces . .6 1.1.2 Vector bundles and connections . 11 1.1.3 Projective and conformal structures . 14 1.2 Fourier analysis . 15 1.2.1 Fourier analysis on finite abelian groups . 15 1.2.2 Poisson summation . 16 1.2.3 Functional equations for theta functions . 18 1.3 Algebraic number fields and zeta functions . 19 1.3.1 The arithmetic of algebraic number fields . 20 1.3.2 The Hilbert symbol and reciprocity laws . 23 1.3.3 Dirichlet characters and Gauss sums . 25 1.3.4 Dedekind zeta functions and Dirichlet L-functions . 27 1.3.5 The Mordell{Tornheim zeta function . 28 2 Modular forms 31 2.1 Elementary aspects of modular forms . 32 2.1.1 Congruence subgroups of SL(2; Z)................................. 32 2.1.2 Eisenstein series and cusp forms . 35 2.1.3 Modular forms as sections of line bundles . 38 2.1.4 Dimension formulae and the Riemann{Roch Theorem . 40 2.2 Jacobi's theta function . 45 2.2.1 Jacobi's transformation law . 45 2.2.2 Jacobi's transformation law via special values of zeta functions . 45 2.2.3 Modularity of theta functions . 50 2.3 Modular forms and Jacobi's theorem on sums of four squares . 52 2.3.1 The dimension of M2(Γ0(4)) . 53 2.3.2 Eisenstein series for M2(Γ0(4)) . 54 2.3.3 Quasimodularity via L-functions . 57 3 Projective geometry and Jacobi's four square theorem 61 3.1 The geometry of punctured spheres . 62 3.1.1 The twice-punctured sphere . 62 3.1.2 The thrice-punctured sphere . 63 3.1.3 Puncture repair for compact Riemann surfaces . 65 3.2 A digression into function theory . 67 3.2.1 Jacobi's theta function . 67 3.2.2 A quasi-modular Eisenstein series . 68 3.3 A projectively invariant differential operator . 70 3.3.1 Projective invariance and an SL(2; Z) representation . 71 3.3.2 Proof of quasi-modularity . 72 3.3.3 Proof of Jacobi's theorem . 73 CONTENTS 4 Gauss sums over Q 77 4.1 The Landsberg{Schaar relation over Q ................................... 78 4.1.1 The asymptotic expansion of Jacobi's theta function . 79 4.1.2 The Landsberg{Schaar relation and quadratic reciprocity . 80 4.2 An elementary proof of the Landsberg{Schaar relation by induction . 83 4.2.1 Introduction . 85 4.2.2 Counting solutions to x2 = a mod pk ................................ 87 4.2.3 Evaluation of Φ(pk; 2l)........................................ 89 4.2.4 Induction . 92 4.3 Evaluating Gauss sums over Q ....................................... 93 4.3.1 The Jacobi symbol . 93 4.3.2 No linear term . 93 4.3.3 Linear terms . 96 4.4 Another elementary proof of the Landsberg{Schaar relation . 99 4.5 A generalised Landsberg{Schaar relation . 101 4.5.1 Analytic proof . 101 4.5.2 Elementary proof . 103 4.6 A generalised twisted Landsberg{Schaar relation . 104 4.6.1 Analytic proof . 105 4.6.2 Elementary proof . 107 4.6.3 An elementary proof of Berndt's reciprocity relation . 107 4.7 Gauss sums of higher degree . 108 4.7.1 Reduction of exponent for odd prime moduli coprime to the degree . 109 4.7.2 Reduction to sums of lower degree . 112 4.8 Twisted Gauss sums . 112 4.8.1 Evaluating twisted Gauss sums . 113 4.8.2 From higher-degree Gauss sums to twisted quadratic Gauss sums . 119 4.8.3 A generalised quartic Landsberg{Schaar relation . 121 4.8.4 A generalised sextic Landsberg{Schaar relation . 123 4.8.5 A generalised octic Landsberg{Schaar relation . 125 5 Gauss sums over algebraic number fields 131 5.1 Theta functions associated to number fields . 131 5.1.1 Theta functions over totally real number fields . 132 5.1.2 Asymptotic expansions of theta functions . 133 5.2 Generalised Hecke reciprocity . 135 5.2.1 Theta functions . 135 5.2.2 Another asymptotic expansion . 136 5.3 Twisted Hecke reciprocity . 138 5.3.1 Analytic proof . 138 5.3.2 Towards local quartic Hecke reciprocity . 140 6 Asymptotic expansions of metaplectic Eisenstein series 143 6.1 The Landsberg{Schaar relation and the Riemann zeta function . 144 6.1.1 Asymptotic expansions and Mellin transforms . 144 6.1.2 Dirichlet L-functions and yet another proof of the Landsberg{Schaar relation . 145 6.2 The metaplectic group and the Landsberg{Schaar relation over Q .................... 149 6.2.1 Construction of the Eisenstein series . 150 6.2.2 The Fourier expansion of E0(z; s).................................. 151 6.2.3 The asymptotic expansion of E0(z; s)............................... 158 6.3 Metaplectic covers of GL(n) and theta functions of higher degree . 161 6.3.1 Automorphic forms on GL(n).................................... 162 6.3.2 Metaplectic theta functions of degree two and three . 166 6.3.3 Metaplectic theta functions on n-fold covers of GL(n)...................... 170 Citation listing for included publications [Moo20] B. Moore. \A proof of the Landsberg{Schaar relation by finite methods". In: Ramanujan J. (2020). doi: 10.1007/s11139-019-00195-4. i 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. ii Statement I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any.