Moments of automorphic L-functions at special points Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Alexander Lu Beckwith, B.A., M.S. Graduate Program in Mathematics The Ohio State University 2020 Dissertation committee Dr. Wenzhi Luo, Advisor Dr. James Cogdell Dr. Roman Holowinsky ⃝c Copyright by Alexander Lu Beckwith 2020 Abstract We study the behavior of families of L-functions at exhibiting conductor-dropping behavior. We derive asymptotic expansions of the short interval first and second moments of GL(2) × GL(2) L-functions at special points with power-saving error terms. As a consequence, we obtain an essentially optimal lower bound on the number of cusp forms for the Hecke congruence surfaces Γ0(p) n H of prime level are destroyed under quasiconformal deformation of these surfaces. Additionally, we show that a large number of cusp forms for the Hecke congruence surfaces Γ0(p) n H of prime level are simultaneously destroyed in two directions of the associated Teichmüller space. We also establish upper bounds for the second moment of GL(2) × GL(3) L-functions and the sixth moment of GL(2) L-functions at special points as the spectral parameter varies in a short interval. An auxiliary twisted spectral large sieve inequality for short intervals is derived for the latter two results. ii Dedicated to my grandparents iii Acknowledgments During my time as a student I have been very fortunate to have had the guidance and support of a number of wise and thoughtful people. There is a very long list of individuals whom I count myself lucky to have met and to whom I wish to extend my thanks in the following—I regret that I have surely neglected some from what appears below. First and foremost, to my advisor, Wenzhi Luo, for his thoughtful guidance, constant and enduring support, for his many kind words of encouragement, and for his boundless patience: Thank you, a thousand times, thank you. To Roman Holowinsky: for all his work helping graduate students navigate the nonacademic post- academic world, from which I have personally benefited so much. To Jim Cogdell: for organizing the automorphic forms seminar that I and many other students have benefited from greatly over the years. To Ghaith Hiary: for serving on my candidacy committee, and for teaching a topics course on computa- tional number theory that I found to be a formative experience. To my undergraduate professors, Judy Holdener, Bob Milnikel, and Marie Snipes: for encouraging me to study math a decade ago now. To my mentor and friend Jessen Book: for his wisdom and guidance when I needed a push in the right direction many years ago. And most of all, to my family, for all their love and support they have given me, no matter how much I stumbled. Thank you. iv Vita 1992 . Born in Midland, Michigan 2014 . B.A. in Mathematics, Concentration in Humane Studies, Kenyon College Gambier, Ohio 2017 . M.S. in Mathematics, The Ohio State University Columbus, Ohio 2014-2015 . University Fellow The Ohio State University Columbus, Ohio 2015-Present . Graduate Teaching Associate, Graduate Research Associate The Ohio State University Columbus, Ohio Fields of Study Major field: Mathematics Specialization: Analytic number theory, automorphic forms v Contents Abstract ................................................... ii Acknowledgments .............................................. iv Vita ..................................................... v Notation and conventions ......................................... ix 1 Introduction 1 I Preliminaries 7 2 Preliminaries on automorphic forms 8 2.1 Automorphic forms for GL(2; R) .................................. 9 2.2 Automorphic forms for GL(3; R) .................................. 33 3 Preliminaries on automorphic L-functions 42 3.1 Automorphic L-functions ...................................... 42 3.2 On conductor-dropping behavior of L-functions in the spectral aspect ............. 48 4 Preliminaries on the spectral theory of hyperbolic Riemann surfaces and Teichmüller theory 51 4.1 Riemann surfaces ........................................... 51 4.2 Some remarks on Teichmüller theory and hyperbolic 2-surfaces ................. 52 4.3 The Phillips-Sarnak deformation theory of discrete groups .................... 55 II Upper bounds for moments of L-functions at special points 58 5 A short interval large sieve inequality with spectral twists 59 5.1 Background and survey of existing literature ........................... 60 5.2 The short interval twisted spectral large sieve inequality ..................... 67 5.3 Proof of Theorem 5.2.1 ........................................ 73 5.4 Proof of Lemmas 5.2.4 and 5.2.5 .................................. 75 6 The second moment of GL(2) × GL(3) L-functions at special points over a shortened spectral interval 83 6.1 Background and survey of existing literature ........................... 84 6.2 Preliminaries on GL(2) × GL(3) Rankin-Selberg L-functions ................... 86 6.3 The shortened spectral interval second moment of GL(2) × GL(3) L-functions at special points 88 6.4 The shortened spectral interval sixth moment of GL(2) L-functions at the special point ... 93 vi Contents III Asymptotic expansions of moments of L-functions at special points 97 7 The nonvanishing of GL(2) × GL(2) L-functions at special points over shortened spectral intervals 100 7.1 Properties of the Rankin-Selberg L-function ............................ 102 7.2 An asymptotic expansion of the first moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals .................................. 103 8 The simultaneous nonvanishing of GL(2) × GL(2) L-functions at special points 117 8.1 Proof of Theorem 8.0.1: an asymptotic expansion of the second moment of GL(2) × GL(2) L-functions at special points over shortened spectral intervals .................. 119 IV Future directions 138 9 Future directions 139 9.1 Direct analogues ........................................... 140 9.2 An asymptotic expansion of the fourth moment of GL(2) L-functions at special points .... 140 9.3 The first moment of GL(2) × GL(3) L-functions at special points ................ 141 9.4 The prime geodesic theorem ..................................... 142 9.5 Special point subconvexity problems ................................ 144 9.6 Higher-order Fermi golden rules ................................... 147 9.7 The newform Weyl law ........................................ 148 References 148 V Appendices 157 A Compendium of special functions and their properties 158 A.1 Bessel functions ............................................ 158 B Asymptotic behavoir of oscillatory integral 164 C Miscellaneous bounds for character sums 172 C.1 Compendium of properties of Kloosterman sums and Ramanujan sums and special character sums .................................................. 172 vii List of Figures Chapter 2 2.1 Fundamental domain F0;1 for the modular group SL(2; Z) .................... 13 2.2 Some examples of fundamental domains for the Hecke congruence subgroups Γ0(N) ..... 14 2.3 Some examples of quotient surfaces for the Hecke congruence subgroups Γ0(N) ........ 15 2.4 Plot of cuspidal and residual spectrum of ∆ and poles of Eisenstein series for SL(2; Z) .... 22 viii List of Tables Chapter 3 3.1 Some effects of conductor-dropping for L-functions in the spectral aspect ........... 49 Chapter 6 6.1 Status of upper bounds for families of L-functions at special points ............... 85 ix Notation and conventions We standardize the common notation and conventions used throughout this document. • N: the set of natural numbers 1; 2; 3;::: • H: the upper half of the complex plane, H = fx + iy : x; y 2 R; y > 0g • SL(2; R), SL(2; Z): the special linear group over R, Z, respectively • GL(n; R): the general linear group of degree n over R • Γ0(N): the Hecke congruence subgroup of level N • Γa: the stability group of a cusp a of Γ • νf (n): the normalized Fourier coefficients at 1 of an automorphic form over GL(2; R) • λf (n): the Hecke eigenvalues of an automorphic form f • λf : the Laplace eigenvalue of an automorphic form f • ΦΓ(s);'Γ(s): the scattering matrix for Γ and the determinant of ΦΓ(s) • S(n; m; c): the Kloosterman sums for SL(2; Z), X∗ an + an S(n; m; c) = S11(n; m; c) = e : c a(c) Sab(m; n; c) denotes the Kloosterman sums of a discrete subgroup Γ ⊆ SL(2; R) attached to cusps a and b. As we will deal primarily with Kloosterman sums for the full modular group, we will not use this notation much. • δ(condition): the delta symbol • δ: the reflection operator, unless it appears in an exponent, in which case it is a positive constant. • h·; ·i: the Petersson inner product • k · k: the Petersson norm, unless looking at a sequence, in which case k · k denotes the (possibly truncated) `2-norm x List of Tables • 'e: the Mellin transform of a function ' (satisfying the appropriate conditions) defined by Z 1 dx 'e(s) = '(x)xs 0 x • 'b: the Fourier transforms of a function ' (satisfying the appropriate conditions) defined by Z 'b(ξ) = '(x)e(−ξx)dx R • τ2; τ3: the binary and ternary divisor functions defined by X X X τ2(n) = 1 and τ3(n) = 1 = τ2(n1): n1n2=n n1n2n3=n n1n2=n Note that these are distinct from the divisor functions τit given by X d it τ (n) = : it k dk=n When the order of the divisor function is a natural number (i.e., τk with k 2 N), we mean the k-ary th divisor function. We reserve σk for the k -power-of-divisors function. − t−T 2 • `a˚u¯sfi¯sfi˚i`a‹nffl(t): the Gaussian function centered at T of width M, `a˚u¯sfi¯sfi˚i`a‹nffl(t) = e ( M ) GT;M GT;M • e(x): the additive character e(x) = e2πix • hT;M (t): usually the (even) twisted Gaussian centered at T of width M it −it `a˚u¯sfi¯sfi˚i`a‹nffl m `a˚u¯sfi¯sfi˚i`a‹nffl m hT;M (t; m; n) = (t) + (−t) GT;M n GT;M n • ρ, denote the weight function and phase function, respectively, in an oscillatory integral, except when possibly denotes an SL(3; Z) Maass cusp form.