This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Viswanathan, Vinay Title: Zeros of quadratic forms and the delta method General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. ZEROS OF QUADRATIC FORMS AND THE δ-METHOD VINAY KUMARASWAMY VISWANATHAN A Dissertation Submitted to the University of Bristol in accordance with the requirements for award of the degree of Doctor of Philosophy in the Faculty of Science, School of Mathematics May 2018 Abstract This thesis presents solutions to three problems. First, we show that the optimal 4 covering exponent for the 3-sphere is 3 ; and this is joint work with T. D. Browning and R. S. Steiner. Next, we prove a result involving h(−n), the class number of an imaginary quadratic field with fundamental discriminant −n. We give an asymptotic formula for correlations involving h(−n) and h(−n−l) over fundamental discriminants that avoid the congruence class 1 (mod 8). The result is uniform in the shift l, and along the way we also derive an asymptotic formula for correlations between rQ(n), the number of representations of an integer by a positive definite quadratic form Q. Finally, we study sums of normalised Hecke eigenvalues λ(n) of holomorphic cusp forms over thin sequences. Let F (x) be a diagonal quadratic form in 4 variables, we give an upper bound for the problem of counting integer solutions of bounded height to F (x) = 0 weighted by λ(x1), and as a consequence we derive upper bounds for certain generalised cubic divisor sums. All three problems are solved by counting integer zeros of quadratic forms using the δ-method. To Shakthi Acknowledgments I would like to thank my supervisor Tim Browning for his endless encouragement and unflagging patience. I am grateful to him for always being generous with his time, advice and ideas. I would also like to thank my partner and best friend S. Shakthi, without whom none of this would have been possible. Travelling down this long road was more enriching and reflective because we made the journey together. Author's Declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University's Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate's own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. Signed: Date: Vinay Kumaraswamy Viswanathan Contents Notation 1 1 Introduction 2 1.1 Integral Hasse principle . 3 1.2 Statement of results . 8 1.3 Directions for future work . 12 2 The δ-method 15 2.1 The δ-method and quadratic forms . 15 2.1.1 The δ-method of Duke, Friedlander and Iwaniec . 19 2.2 Other versions of the δ-method . 25 3 Some technical results 28 3.1 Bessel functions . 28 3.2 Summation formulae . 29 3.3 Exponential sums . 30 3.4 Exponential integrals . 31 4 Covering exponent for S3 33 4.1 Introduction . 33 4.2 Preliminaries . 35 4.2.1 Overview . 35 4.2.2 Notation . 36 4.2.3 Activation of the circle method . 36 4.3 Gauss sums and Kloosterman sums . 39 4.4 Oscillatory integrals . 40 4.4.1 Easy estimates . 41 4.4.2 Stationary phase . 43 4.4.3 Hard estimates . 44 4.5 Putting everything together . 50 4.5.1 The main term . 50 4.5.2 The error term . 53 5 Sums of class numbers 60 5.1 Introduction . 60 5.1.1 Correlations involving rQ(n)................... 62 5.2 The main proposition . 62 5.2.1 Applying the δ-method . 64 5.2.2 Analysis of the exponential sum . 65 5.2.3 Estimates for exponential integrals I . 67 5.2.4 Estimates for exponential integrals II . 69 5.2.5 Evaluating Iq(0) ......................... 71 5.2.6 Proof of Proposition 5.2.1 . 72 5.3 Proof of the main theorems . 76 5.3.1 Proof of Theorem 5.1.1 . 76 5.3.2 Reduction to a counting problem . 77 5.3.3 Proof of Theorem 5.1.3 . 84 5.3.4 Proof of Theorems 1.2.4 and 5.1.4 . 86 6 Sums of Hecke eigenvalues over thin sequences 88 6.1 Introduction . 88 6.2 Preliminaries . 92 6.2.1 Some facts about L-functions . 93 6.3 Setting up the δ-method . 94 6.3.1 Applying the Poisson summation formula . 95 6.4 Integral estimates . 96 6.4.1 First steps . 96 6.4.2 Estimates for Iq(c)........................ 97 0 6.4.3 Estimates for Iq(c ; s) and Id;q(c) . 100 6.5 Exponential sums . 108 6.5.1 Evaluation of Sq(n) . 110 6.5.2 Exponential sums in the case where F −1(0; c0) = 0 and c0 6= 0 117 6.5.3 Auxillary estimates . 118 6.6 Proof of Theorem 1.2.5 . 124 6.6.1 Contribution from N (0)(λ; X) .................. 124 6.6.2 Contribution from N (1)(λ; X) .................. 128 6.7 Deduction of Theorem 6.1.1 from Theorem 1.2.5 . 131 Bibliography 134 Notation 1. As is standard in analytic number theory, for any complex number α, we set e(α) = exp(2πiα), and eq(α) is used as shorthand for e(α=q). P∗ 2. By a (mod q) we denote restriction to primitive residue classes modulo q, i.e. (a; q) = 1. P∗ 3. S(m; n; q) = a (mod q) eq(am + an) will denote the Kloosterman sum, where a is the mutliplicative inverse of a modulo q. 4. cq(m) = S(m; 0; q) is defined to be Ramanujan's sum. n 1 5. For a smooth real valued function w : R ! R, we denote by kwkN;1 its L Sobolev norm of order N. P 6. d(n) = djn 1 will denote the divisor function. × 7. For a prime p, and u 2 Q , vp(u) will denote the p−adic valuation of u. 8. Unless stated otherwise, j:j will denote the sup norm on Rn. 9. In this thesis we adopt the following convention. All implicit constants that appear in the error terms will be allowed to depend on the underlying quadratic forms. Any further dependence will be indicated by an appropriate subscript. 1 Chapter 1 Introduction An integral quadratic form Q(x) 2 Z[x1; : : : ; xr] is a homogeneous quadratic polyno- mial in r variables with integer coefficients. More explicitly, for 1 ≤ i; j ≤ r, there P exist integers aij such that Q(x) = 1≤i;j≤r aijxixj. Let A be the symmetric matrix associated to Q. The rank of Q is defined to be the matrix rank of A, and if A is of full rank, the discriminant of Q is defined to be the determinant of A. Quadratic forms arise naturally across mathematics, and the arithmetic of quadratic forms occupies an exalted position within number theory. Perhaps the most fundamental problem is to understand the representation of integers by quadratic forms. Observe that x = 0 is always a solution to the equation Q(x) = 0, so our aim is to understand non-zero solutions to quadratic forms. Given a quadratic form Q and an integer n, define the set R(n; Q) = fx 2 Zr : Q(x) = n; x 6= 0g : Our problem then translates to asking if R(n; Q) is non-empty. Beginning with Brah- magupta (598-670 CE), who derived a method (see [30]) to find infinitely many solu- tions to the `Pell equation' x2 − 92y2 = 1, this question has led to the development of a deep and beautiful theory. The central theme of this thesis is counting integer solutions to quadratic forms using a form of the circle method known as the δ-method. We begin with a brief discussion of the Hasse principle and postpone a discussion of the δ-method to Chap- 2 ter 2. 1.1 Integral Hasse principle For R(n; Q) to be non-empty, it is clearly necessary that the equation satisfies local solubility, i.e. the equation Q(x) = n has a solution in Zp for each place p ≤ 1 (for p = 1, we regard Z1 = R).