The Second Moment of Rankin-Selberg L-functions, Hybrid Subconvexity Bounds, and Related Topics Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Zhilin Ye, B.S. Graduate Program in Department of Mathematics The Ohio State University 2014 Dissertation Committee: Roman Holowinsky, Advisor James Cogdell Wenzhi Luo Mark Berliner c Copyright by Zhilin Ye 2014 Abstract In this thesis, we study three problems related to subconvexity bounds of Rankin- Selberg L-functions. Let M; N be two coprime square-free integers. Let f be either a holomorphic or a Maaß cusp form of level N. Using a large sieve inequality, we establish P ( j) 2 a bound for an unamplified 2nd moment average, such as g L (1=2 + it; f ⊗ g) t; j, M + M2=3−βN4=3 (MN) where β ≈ 1=500 and g ranges over an orthonormal basis of holo- morphic cusp forms of level M and a fixed weight for any j. As a consequence, we obtain a subconvexity bound for L( j) (1=2 + it; f ⊗ g) for any N < M satisfying the conditions above. Moreover, by symmetry, we establish a level aspect hybrid subconvexity bound for any coprime square-free M and N when both forms are holomorphic. We also establish a k=2 1=2 −1=6+ sup-norm bound for holomorphic forms such that ky f (z)k1 k N k f k2. Further- more, we establish an equidistribution property of mod c reciprocals which is natural to the subconvexity problem and second moments of Rankin-Selberg L-functions. ii This is dedicated to the one I love. iii Acknowledgments I would like to express my special appreciation and thanks to my advisor Professor Roman Holowinsky for his tremendous mentoring, encouraging, and priceless advice on my research. I would also like to thank my committee members, Professor James Cogdell, Professor Wenzhi Luo, and Professor Mark Berliner for serving as my committee members even at hardship. I also want to thank my family, my friends and all of people who have been there to support me during my life as a Ph.D. candidate. iv Vita 2006-2010 . .B.S., Mathematics, Zhejiang University, Hangzhou, Zhejiang, P.R. China. 2010-2014 . .Ph.D., Mathematics, The Ohio State University, Columbus, OH, USA. Fields of Study Major Field: Mathematics Studies in Analytic Number Theory: Roman Holowinsky v Table of Contents Page Abstract......................................... ii Dedication........................................ iii Acknowledgments.................................... iv Vita...........................................v 1. Introduction....................................1 1.1 L-functions, their properties, and the subconvexity problem.......1 1.2 Known subconvexity bounds and their applications............ 12 1.2.1 The Burgess’s bound and subconvexity bound for GL1 ...... 12 1.2.2 The subconvexity bound for GL2 case............... 18 1.2.3 The QUE conjecture and the subconvexity problem of higher rank 20 2. Preliminaries and Past Works........................... 27 2.1 Automorphic forms and the normalization................. 27 2.2 Rankin-Selberg convolution and L-functions............... 31 2.3 Properties of Bessel functions....................... 34 2.4 The Voronoi formula, trace formulae and large sieve inequalities..... 40 2.5 Jutila’s version of circle method...................... 47 2.6 Past works on hybrid subconvexity bounds for Rankin-Selberg L-functions 48 2.7 Past works on Sup-norm bounds for modular forms............ 51 2.8 Past works on the distribution of inverses mod c .............. 54 3. Main Results................................... 58 3.1 Hybrid subconvexity bounds for Rankin-Selberg L-functions....... 58 3.2 Sup-norm bounds for holomorphic modular forms............ 60 3.3 The distribution of inverses mod c ..................... 61 vi 4. The Proof of Hybrid Subconvexity Bounds for Rankin-Selberg L-functions.. 63 4.1 The deduction of Theorem 3.1.1 and Corollary 3.1.2 from Proposition 3.1.1.................................... 63 4.2 A sketch proof of Propositon 3.1.1..................... 66 4.3 The proof of Proposition 3.1.1....................... 70 4.3.1 Step 1. Reducing to sums of Kloosterman sums via trace for- mula................................ 71 4.3.2 Step 2. Removing large and small values of D.......... 71 4.3.3 Step 3. Applying the Voronoi formula to convert Kloosterman Sums to Ramanujan sums..................... 73 4.3.4 Step 4. Treating the zero shift................... 75 4.3.5 The sum of shifted sums...................... 75 4.3.6 Step 5. Applying the circle method................ 76 4.3.7 Step 6. Applying Vornoi formula twice to regenerate Klooster- mann sums............................. 78 4.3.8 Step 7. Applying the large sieve type inequality to the sum of Kloostermann sums........................ 80 4.3.9 Conclusion and the final bound.................. 82 4.4 The estimation of weight functions.................... 84 4.5 The proof of Theorem 3.1.3........................ 91 5. The Proof of Sup-norm Bounds for Holomorphic Modular Forms........ 96 5.1 Preliminary................................. 96 5.1.1 The Sup-norm via Fourier expansion............... 96 5.1.2 Pretrace formula for holomorphic cusp forms........... 96 5.1.3 Amplification method....................... 97 5.1.4 Counting lattice points....................... 100 5.1.5 The estimation of parabolic matrices............... 104 5.2 The proof of Theorem 3.2.1........................ 107 6. The Proof of the Distribution of Inverses mod c ................. 110 6.1 Wilton’s bound for Eisenstein series.................... 110 6.2 The proof of Theorem 3.3.1........................ 119 Bibliography...................................... 122 vii Chapter 1: Introduction 1.1 L-functions, their properties, and the subconvexity problem The study of L-functions plays a key role in number theory. The most famous L- function is the Riemman Zeta function, which is defined as X1 1 ζ(s):= ns n=1 when Re (s) > 1. It has a meromorphic continuation to any complex number s. Fur- thermore, by the unique factorization of the integers, we have the Euler product when Re (s) > 1 Y ζ(s) = 1 − p−s−1 ; p primes which relates ζ(s) to prime numbers. The locations of poles and zeros of the Riemann Zeta function carry highly nontrivial information about the distribution of prime numbers. For example, we can prove that there are infinitely many prime numbers by showing that ζ(s) has a pole at s = 1. A deeper result is the Prime Number Theorem Theorem 1.1.1 (Prime Number Theorem). Let π(x) denote the number of primes p 6 x. As x ! 1, we have x π(x) ∼ : log x 1 This theorem is equivalent to the fact that ζ(s) , 0 when Re (s) = 1. For a complete proof, see [IK04] Chapter 2. These examples illustrate why the study of the locations of zeros of ζ(s) has attracted number theorists for over 100 years. In 1859, B. Riemann made a conjecture about the lo- cations of these zeros. His conjecture, which is now called the Riemann Hypothesis, claims that all the solutions of the equation ζ(s) = 0 such that 0 < Re (s) < 1 lie on the vertical Re 1 line (s) = 2 . This conjecture is still open. In order to study prime numbers in arithmetic progressions, L. Dirichlet (1837) invented multiplicative characters χ(mod q) and associated a Dirichlet L-functions to each of them. The explicit definition is as follows. Denote by Z the set of all integers. Denote by C× the set of all nonzero complex numbers. Let q > 1 be a positive integer. Denote by (Z=qZ)× the modulo q multiplicative × × group. Let χq : (Z=qZ) ! C be a character. A Dirichlet L-function is defined as 1 X χq(n) L(s; χ ) = ; q ns n=1 when Re (s) > 0. It also has a meromorphic continuation to any complex number s and can be rewritten as the Euler product − Y −s 1 L(s; χq) = 1 − χq(p)p : p primes Just like the the Riemann Zeta function, the zeros of Dirichlet L-functions carry infor- mation about prime numbers. For example, we can prove that there are infinitely many primes p such that p ≡ 1(mod 3) by showing that L(s; χ3) , 0 when Re (s) = 1. For a complete proof, see [IK04] Chapter 2. 2 The Generalized Riemann Hypothesis suggests that all the zeros of L(s; χq) in the strip Re Re 1 0 < (s) < 1 lie on the vertical line (s) = 2 for any Dirichilet character χq modulo q. There are other arithmetic objects with L-functions associated to them. For example, 2 let SL2(R) be the group of 2-by-2 matrices with determinant 1. Let H be the upper-half ! a b plane of C. Any γ = 2 SL (R) acts on H2 via Mobius¨ action c d 2 az + b γ:z = : cz + d Let M > 0 be an integer and κ > 0 be an even integer. Let ( ! ) a b Γ (M):= 2 SL (Z): c ≡ 0(mod M) 0 c d 2 be a congruence subgroup of SL2(Z). The holomorphic cusp forms with weight κ, level M, and trivial nebentypus are holo- morphic functions on the upper half-plane F : H2 ! C satisfying F(γ:z) = (cz + d)κF(z); when ! a b γ = 2 Γ (M); c d 0 and vanishing at every cusp, e.g. F(a:z) ! 0 as Im (z) ! 1 for any a 2 SL2(Z). κ Let f (z) = y 2 F(z) where F is a holomorphic cusp form with weight κ, level M and trivial nebentypus. We call such f (z) a Hecke newform, if it is not generated by cusp 3 forms with lower levels (see [IK04] Chapter 14, [Cas73] for more details) and it is the eigenfunction of Hecke operators Tm for all m > 0, where ! 1 X X az + b T f (z):= p f : m d m ad=m 06b<d (a;M)=1 Denote by λ f (m) the corresponding eigenvalues.