DOCSLIB.ORG

Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms A. Knightly and C. Li December 12, 2012 Abstract We give an adelic treatment of the Kuznetsov trace formula as a rel- ative trace formula on GL(2) over Q. The result is a variant which in- corporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. We include a proof of a Weil bound for the general- ized twisted Kloosterman sums which arise on the geometric side. As an application, we show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidis- tributed relative to the Sato-Tate measure in the limit as the level goes to infinity. Contents 1 Introduction 3 1.1 Somehistory ............................. 3 1.2 Overviewofthecontents. 6 1.3 Acknowledgements .......................... 9 2 Preliminaries 10 2.1 NotationandHaarmeasure . 10 2.2 Characters and Dirichlet L-functions ................ 12 3 Bi-K -invariant functions on GL2(R) 15 3.1 Severalguises∞ ............................. 15 3.2 TheHarish-Chandratransform . 18 3.3 TheMellintransform. 19 3.4 TheSelbergtransform . 20 3.5 The principal series of G(R)..................... 22 4 Maass cusp forms 26 4.1 Cuspformsofweight0........................ 26 4.2 Heckeoperators............................ 28 4.3 AdelicMaassforms.......................... 31 1 5 Eisenstein series 36 5.1 Induced representations of G(A) .................. 36 5.2 DefinitionofEisensteinseries . 39 5.3 The finite part of φ .......................... 40 K K1(N) 5.4 An orthogonal basis for H(χ1,χ2) ∞× ............ 43 5.5 Evaluationofthebasiselements . 46 5.6 FourierexpansionofEisensteinseries. 48 5.7 Meromorphiccontinuation. 51 5.8 Charactersums............................ 53 6 The kernel of R(f) 55 6.1 Thespectraldecomposition . 55 6.2 Kernelfunctions ........................... 59 6.3 A spectral lower bound for Kh h∗ (x,x)............... 62 6.4 The spectral form of the kernel∗ of R(f) .............. 65 7 A Fourier trace formula for GL(2) 70 7.1 Convergenceofthespectralside . 70 7.2 Cuspidalcontribution . 70 7.3 Residualcontribution . 74 7.4 Continuouscontribution . 74 7.5 Geometricside ............................ 77 7.5.1 Firstcellterm ........................ 78 7.5.2 Secondcellterms. 80 7.6 Finalformulas ............................ 84 7.7 Classicalderivation. 87 8 Validity of the KTF for a broader class of h 90 8.1 Preliminaries ............................. 91 8.2 Smoothtruncation . 100 8.3 Comparing the KTF for h and hT .................104 8.4 R0(f) for f not smooth or compactly supported. 106 8.5 ProofofProposition8.29 . 112 9 Kloosterman sums 119 9.1 AboundfortwistedKloostermansums . 120 9.2 Factorization ............................. 128 9.3 ProofofTheorem9.2. 130 10 Equidistribution of Hecke eigenvalues 132 Notation index 142 Subject index 145 2 1 Introduction 1.1 Some history A Fourier trace formula for GL(2) is an identity between a product of two Fourier coefficients, averaged over a family of automorphic forms on GL(2), and a series involving Kloosterman sums and the Bessel J-function. The first example, arising from Petersson’s computation of the Fourier coefficients of Poincar´eseries in 1932 [P1] and his introduction of the inner product in 1939 [P2], has the form Γ(k 1) am(f)an(f) k S(m, n; c) 4π√mn − k 1 2 = δm,n+2πi Jk 1( ), (4π√mn) − f c − c f (N) c NZ+ ∈FXk k k ∈X where (N) is an orthogonal basis for the space of cusp forms S (Γ (N)), and Fk k 0 S(m, n; c)= e2πi(mx+nx)/c xx 1 mod c ≡X is a Kloosterman sum. Because of the existence of the Weil bound S(m, n; c) τ(c)(m,n,c)1/2c1/2 (1.1) | |≤ where τ is the divisor function, and the bound k 1 1/2 Jk 1(x) min(x − ,x− ) − ≪ for the Bessel function, the Petersson formula is useful for approximating ex- pressions involving Fourier coefficients of cusp forms. For example, Selberg used it in 1964 ([Sel3]) to obtain the nontrivial bound (k 1)/2+1/4+ε an(f)= O(n − ) (1.2) (k 1)/2+ε in the direction of the Ramanujan-Petersson conjecture an(f)= O(n − ) subsequently proven by Deligne. In his paper, Selberg mentioned the problem of extending his method to the case of Maass forms. This was begun in the late 1970’s independently by Bruggeman and Kuznetsov ([Brug], [Ku]). The left-hand side of the above Petersson formula is now replaced by a sum of the form am(uj)an(uj ) h(tj ) 2 , (1.3) uj cosh(πtj ) u Xj ∈F k k where m,n > 0, is an (orthogonal) basis of Maass cusp forms of weight k = 0 F 1 2 and level N = 1, tj is the spectral parameter defined by ∆uj = ( 4 + tj )uj for the Laplacian ∆, and h(t) is an even holomorphic function with sufficient decay. 3 There is a companion term coming from the weight 0 part of the continuous spectrum, describable in terms of the Eisenstein series 1 y1/2+s E(s,z)= (Re(s) > 1 ,y > 0,z = x + iy). 2 cz + d 1+2s 2 c,d Z (c,dX∈)=1 | | More accurately, it involves the analytic continuation to s on the imaginary line. This analytic continuation is provided by the Fourier expansion 1/2+s 1/2 s √π Γ(s)ζ(2s) E(s,z)= y + y − (1.4) Γ(1/2+ s)ζ(1+2s) 2y1/2π1/2+s + σ (m) m sK (2π m y)e2πimx. Γ(1/2+ s)ζ(1+2s) 2s | | s | | m=0 X6 2s Here σ2s(m)= 0<d m d is the divisor sum, and Ks is the K-Bessel function. The continuous contribution| to the Kuznetsov/Bruggeman formula is the fol- P lowing integral of the product of two Fourier coefficients of E(it,z) against the function h(t): it 1 ∞ (m/n) σ (m)σ (n) 2it 2it h(t)dt. (1.5) π ζ(1+2it) 2 Z−∞ | | The Fourier trace formula is then the equality between the sum of (1.3) and (1.5) on the so-called spectral side, with the geometric side given by δ ∞ 2i S(m, n; c) ∞ 4π√mn h(t) t m,n h(t) tanh(πt) tdt + J ( ) dt. π2 π c 2it c cosh(πt) Z−∞ c Z+ Z−∞ X∈ (1.6) Using this together with the Weil bound (9.2), Kuznetsov proved a mean-square estimate for the Fourier coefficients an(uj ) ([Ku], Theorem 6), which immedi- ately implies the bound a (u ) n1/4+ε n j ≪j,ε ε in the direction of the (still open) Ramanujan conjecture an(uj )= O(n ). (See also [Brug], 4.) This extended Selberg’s result (1.2) to the case of Maass forms. Kuznetsov§ also “inverted” the formula to give a variant in which a gen- eral test function appears on the geometric side in place of the Bessel integral. (Motohashi has given an interesting conceptual explanation of this, showing that the procedure is reversible, [Mo2].) This allows for important applications to bounding sums of Kloosterman sums. Namely, Kuznetsov proved that the estimate S(m, n; c) Xθ+ε (1.7) c ≪m,n,ε c X X≤ 1 holds with θ = 6 ([Ku], Theorem 3). The Weil bound alone yields only θ = 1 2 , showing that Kuznetsov’s method detects considerable cancellation among the Kloosterman sums due to the oscillations in their arguments. Linnik had 4 conjectured in 1962 that (1.7) holds with θ = 0, and Selberg remarked that this would imply the Ramanujan-Petersson conjecture for holomorphic cusp forms of level 1, ([Sel3]; see also 4 of [Mu]). By studying the Dirichlet series § S(m, n; c) Z(s,m,n)= , c2s c X Selberg also codified a relationship between sums of Kloosterman sums and the 1 smallest eigenvalue λ1 of the Laplacian, leading him to conjecture that λ1 4 3 ≥ for congruence subgroups. He obtained the inequality λ1 16 using the Weil bound (9.2). This inequality is also a consequence of the gen≥eralized Kuznetsov formula given in 1982 by Deshouillers and Iwaniec ([DI]). Fourier trace formulas have since become a staple tool in analytic number theory. We mention here a sampling of notable results in which they have played a role. Deshouillers and Iwaniec used the Kuznetsov formula to deduce bounds for very general weighted averages of Kloostermans sums, showing in particular that Linnik’s conjecture holds on average ([DI], 1.4). They list some interesting consequences in 1.5 of their paper. For example,§ there are infinitely many primes p for which p+1§ has a prime factor greater than p21/32. They also give applications to the Brun-Titchmarsh theorem and to mean-value theorems for primes in arithmetic progressions (see also [Iw1], 12-13). Suppose f(x) Z[x] is a quadratic polynomial with§ negative discriminant. ∈ ν If p is prime and ν is a root of f in Z/pZ, then the fractional part p [0, 1) is independent of the choice of representative for ν in Z. Duke, Friedlander,{ }∈ and Iwaniec proved that for (p,ν) ranging over all such pairs, the set of these fractional parts is uniformly distributed in [0, 1], i.e. for any 0 α<β 1, ≤ ≤ # (p,ν) p x,f(ν) 0 mod p,α ν <β { | ≤ ≡ ≤ { p } } (β α) # p x p prime ∼ − { ≤ | } as x ([DFI]). Their proof uses the Kuznetsov formula to bound a certain related→ Poincar´eseries∞ via its spectral expansion. See also Chapter 21 of [IK]. Applications of Fourier trace formulas to the theory of L-functions abound. Using the results of [DI], Conrey showed in 1989 that more than 40% of the zeros of the Riemann zeta function are on the critical line ([Con]).1 Motohashi’s book [Mo1] discusses other applications to ζ(s), including the asymptotic formula for its fourth moment.
Recommended publications
  • Identities Between Hecke Eigenforms Have Been Studied by Many Authors

    Identities Between Hecke Eigenforms Have Been Studied by Many Authors

    Identities between Hecke Eigenforms D. Bao February 20, 2018 Abstract In this paper, we study solutions to h = af 2 + bfg + g2, where f,g,h are Hecke newforms with respect to Γ1(N) of weight k > 2 and a, b = 0. We show that the number of solutions is finite for 6 all N. Assuming Maeda’s conjecture, we prove that the Petersson inner product f 2, g is nonzero, where f and g are any nonzero cusp h i eigenforms for SL2(Z) of weight k and 2k, respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp 2 n eigenforms for SL2(Z) of the form X + i=1 αiYi = 0 all are forced by dimension considerations. We also give a proof using polynomial P identities between eigenforms that the j-function is algebraic on zeros of Eisenstein series of weight 12k. 1 Introduction arXiv:1701.03189v1 [math.NT] 11 Jan 2017 Fourier coefficients of Hecke eigenforms often encode important arithmetic information. Given an identity between eigenforms, one obtains nontrivial relations between their Fourier coefficients and may in further obtain solu- tions to certain related problems in number theory. For instance, let τ(n) be the nth Fourier coefficient of the weight 12 cusp form ∆ for SL2(Z) given by ∞ ∞ ∆= q (1 qn)24 = τ(n)qn − n=1 n=1 Y X 1 11 and define the weight 11 divisor sum function σ11(n) = d n d . Then the Ramanujan congruence | P τ(n) σ (n) mod691 ≡ 11 can be deduced easily from the identity 1008 756 E E2 = × ∆, 12 − 6 691 where E6 (respectively E12) is the Eisenstein series of weight 6 (respectively 12) for SL2(Z).
  • On Moments of Twisted L-Functions

    On Moments of Twisted L-Functions

    ARIZONA WINTER SCHOOL 2016: ANALYTIC THEORY OF TRACE FUNCTIONS PHILIPPE MICHEL, EPF LAUSANNE 1. Functions on congruence rings As was emphasized by C.F. Gauss in the Disquisitiones Arithmeticae (and probably much earlier), the concept of congruences (a.k.a arithmetic progressions of some modulus q, a.k.a the ring Z=qZ) is central in number theory. It is therefore fundamental to understand spaces of functions over the ring Z=qZ. One of the first classes of such functions to have been studied systematically (and one of the most interesting even today) is the group of Dirichlet's characters of modulus q, (Z\=qZ)×, the group homomorphisms χ :(Z=qZ)× ! C×: Indeed these functions enter crucially in Dirichlet's proof that there are infinitely many primes in any admissible arithmetic progression of modulus q [7]. The studying of integral solutions of diophantine equations is another important source of func- tions on congruence rings: these arise as a manifestation of some general local-global principles or more directly when using the circle method. For instance, while investigating the set of representa- tions of an integer d by some integral, positive, definite, quadratic form in four variables q(x; y; z; t), ie. the integral solutions to the equation (1.1) d = q(x; y; z; t); x; y; z; t 2 Z; Kloosterman came up with the very important function modulo q, the so-called "Kloosterman sums", 1 X m (mod q) 7! Kl (m; q) := p e (mx + y); 2 q q xy=1 (mod q) where eq(·) denote the additive character of (Z=qZ; +) 2πix e (x) := exp( ): q q By evaluating the fourth moment of this function over Z=qZ, Kloosterman established the following non-trivial bound 2=3−1=2+o(1) (1.2) Kl2(m; q) q and deduced from it an asymptotic formula for the number of solutions to the equations (1.1).
  • Modular Forms, the Ramanujan Conjecture and the Jacquet-Langlands Correspondence

    Modular Forms, the Ramanujan Conjecture and the Jacquet-Langlands Correspondence

    Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence Jonathan D. Rogawski1) The theory developed in Chapter 7 relies on a fundamental result (Theorem 7 .1.1) asserting that the space L2(f\50(3) x PGLz(Op)) decomposes as a direct sum of tempered, irreducible representations (see definition below). Here 50(3) is the compact Lie group of 3 x 3 orthogonal matrices of determinant one, and r is a discrete group defined by a definite quaternion algebra D over 0 which is split at p. The embedding of r in 50(3) X PGLz(Op) is defined by identifying 50(3) and PGLz(Op) with the groups of real and p-adic points of the projective group D*/0*. Although this temperedness result can be viewed as a combinatorial state­ ment about the action of the Heckeoperators on the Bruhat-Tits tree associated to PGLz(Op). it is not possible at present to prove it directly. Instead, it is deduced as a corollary of two other results. The first is the Ramanujan-Petersson conjec­ ture for holomorphic modular forms, proved by P. Deligne [D]. The second is the Jacquet-Langlands correspondence for cuspidal representations of GL(2) and multiplicative groups of quaternion algebras [JL]. The proofs of these two results involve essentially disjoint sets of techniques. Deligne's theorem is proved using the Riemann hypothesis for varieties over finite fields (also proved by Deligne) and thus relies on characteristic p algebraic geometry. By contrast, the Jacquet­ Langlands Theorem is analytic in nature. The main tool in its proof is the Seiberg trace formula.
  • Scientific Report for 2013

    Scientific Report for 2013

    Scientific Report for 2013 Impressum: Eigent¨umer,Verleger, Herausgeber: The Erwin Schr¨odingerInternational Institute for Mathematical Physics - U of Vienna (DVR 0065528), Boltzmanngasse 9, A-1090 Vienna. Redaktion: Goulnara Arzhantseva, Joachim Schwermer. Supported by the Austrian Federal Min- istry of Science and Research (BMWF) through the U of Vienna. Contents Preface 3 The Institute and its Mission in 2013 . 3 Scientific activities in 2013 . 4 The ESI in 2013 . 6 Scientific Reports 7 Main Research Programmes . 7 Teichm¨ullerTheory . 7 The Geometry of Topological D-branes, Categories, and Applications . 11 Jets and Quantum Fields for LHC and Future Colliders . 18 GEOQUANT 2013 . 25 Forcing, Large Cardinals and Descriptive Set Theory . 28 Heights in Diophantine Geometry, Group Theory and Additive Combinatorics . 31 Workshops organized independently of the Main Programmes . 36 ESI Anniversary - Two Decades at the Interface of Mathematics and Physics The [Un]reasonable Effectiveness of Mathematics in the Natural Sciences . 36 Word maps and stability of representations . 38 Complexity and dimension theory of skew products systems . 42 Advances in the theory of automorphic forms and their L-functions . 44 Research in Teams . 51 Marcella Hanzer and Goran Muic: Eisenstein Series . 51 Vladimir N. Remeslennikov et al: On the first-order theories of free pro-p groups, group extensions and free product groups . 53 Raimar Wulkenhaar et al: Exactly solvable quantum field theory in four dimensions . 57 Jan Spakula et al: Nuclear dimension and coarse geometry . 60 Alan Carey et al: Non-commutative geometry and spectral invariants . 62 Senior Research Fellows Programme . 64 Vladimir Korepin: The Algebraic Bethe Ansatz . 64 Simon Scott: Logarithmic TQFT, torsion, and trace invariants .
  • On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017

    On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017

    On Some Variants of the Gauss Circle Problem by David Lowry-Duda B.S. in Applied Mathematics, Georgia Institute of Technology, Atlanta, GA, 2011 B.S. in International Affairs and Modern Languages, Georgia Institute of Technology, Atlanta, GA, 2011 M.Sc. in Mathematics, Brown University, Providence, RI, 2015 A dissertation submitted in partial fulfillment of the arXiv:1704.02376v2 [math.NT] 2 May 2017 requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2017 c Copyright 2017 by David Lowry-Duda Abstract of “On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017 The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly accurate. This seemingly simple problem has prompted new ideas in many areas of number theory and mathematics, and it is now recognized as one instance of a general phenomenon. In this work, we describe two variants of the Gauss Circle problem that exhibit similar characteristics. The first variant concerns sums of Fourier coefficients of GL(2) cusp forms. These sums behave very similarly to the error term in the Gauss Circle problem. Normalized correctly, it is conjectured that the two satisfy essentially the same asymptotics. We introduce new Dirichlet series with coefficients that are squares of partial sums of Fourier coefficients of cusp forms. We study the meromorphic properties of these Dirichlet series and use these series to give new perspectives on the mean square of the size of sums of these Fourier coefficients.
  • Algebra & Number Theory Vol. 4 (2010), No. 6

    Algebra & Number Theory Vol. 4 (2010), No. 6

    Algebra & Number Theory Volume 4 2010 No. 6 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale
  • The Role of the Ramanujan Conjecture in Analytic Number Theory

    The Role of the Ramanujan Conjecture in Analytic Number Theory

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 267–320 S 0273-0979(2013)01404-6 Article electronically published on January 14, 2013 THE ROLE OF THE RAMANUJAN CONJECTURE IN ANALYTIC NUMBER THEORY VALENTIN BLOMER AND FARRELL BRUMLEY Dedicated to the 125th birthday of Srinivasa Ramanujan Abstract. We discuss progress towards the Ramanujan conjecture for the group GLn and its relation to various other topics in analytic number theory. Contents 1. Introduction 267 2. Background on Maaß forms 270 3. The Ramanujan conjecture for Maaß forms 276 4. The Ramanujan conjecture for GLn 283 5. Numerical improvements towards the Ramanujan conjecture and applications 290 6. L-functions 294 7. Techniques over Q 298 8. Techniques over number fields 302 9. Perspectives 305 J.-P. Serre’s 1981 letter to J.-M. Deshouillers 307 Acknowledgments 313 About the authors 313 References 313 1. Introduction In a remarkable article [111], published in 1916, Ramanujan considered the func- tion ∞ ∞ Δ(z)=(2π)12e2πiz (1 − e2πinz)24 =(2π)12 τ(n)e2πinz, n=1 n=1 where z ∈ H = {z ∈ C |z>0} is in the upper half-plane. The right hand side is understood as a definition for the arithmetic function τ(n) that nowadays bears Received by the editors June 8, 2012. 2010 Mathematics Subject Classification. Primary 11F70. Key words and phrases. Ramanujan conjecture, L-functions, number fields, non-vanishing, functoriality. The first author was supported by the Volkswagen Foundation and a Starting Grant of the European Research Council. The second author is partially supported by the ANR grant ArShiFo ANR-BLANC-114-2010 and by the Advanced Research Grant 228304 from the European Research Council.
  • Kuznetsov Trace Formula and the Distribution of Fourier Coefficients

    Kuznetsov Trace Formula and the Distribution of Fourier Coefficients

    A GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass Forms Jo~ao Leit~aoGuerreiro Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2016 c 2016 Jo~aoLeit~aoGuerreiro All Rights Reserved ABSTRACT A GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass Forms Jo~ao Leit~aoGuerreiro We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl's law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications. Table of Contents 1 Introduction 1 2 Preliminaries 6 2.1 Maass Forms for SL(2; Z) ................................. 6 2.2 Maass Forms for SL(3; Z) ................................. 8 2.3 Fourier-Whittaker Expansion . 11 2.4 Hecke-Maass Forms and L-functions . 15 2.5 Eisenstein Series and Spectral Decomposition . 18 2.6 Kloosterman Sums .
  • Beyond Endoscopy for the Relative Trace Formula II: Global Theory

    Beyond Endoscopy for the Relative Trace Formula II: Global Theory

    BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY YIANNIS SAKELLARIDIS Abstract. For the group G = PGL2 we perform a comparison between two relative trace formulas: on one hand, the relative trace formula of Jacquet for the quotient T \G/T , where T is a non-trivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to non-standard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ “Beyond Endoscopy” program. Contents 1. Introduction. 1 Part 1. Poisson summation 18 2. Generalities and the baby case. 18 3. Direct proof in the baby case 31 4. Poisson summation for the relative trace formula 41 Part 2. Spectral analysis. 54 5. Main theorems of spectral decomposition 54 6. Completion of proofs 67 7. The formula of Waldspurger 85 Appendix A. Families of locally multiplicative functions 96 Appendix B. F-representations 104 arXiv:1402.3524v3 [math.NT] 9 Mar 2017 References 107 1. Introduction. 1.1. The result of Waldspurger. The celebrated result of Waldspurger [Wal85], relating periods of cusp forms on GL2 over a nonsplit torus (against a character of the torus, but here we will restrict ourselves to the trivial character) with the central special value of the corresponding quadratic base 2010 Mathematics Subject Classification. 11F70. Key words and phrases. relative trace formula, beyond endoscopy, periods, L-functions, Waldspurger’s formula. 1 2 YIANNIS SAKELLARIDIS change L-function, was reproven by Jacquet [Jac86] using the relative trace formula.
  • Modular Forms and L-Functions

    Modular Forms and L-Functions

    MODULAR FORMS AND L-FUNCTIONS A Thesis by TEKIN KARADAĞ Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE Chair of Committee, Mathhew P Young Committee Members, Mohamad Riad Masri Daren B. H. Cline Head of Department, Emil Straube August 2016 Major Subject: Mathematics Copyright 2016 Tekin Karadağ ABSTRACT In this thesis, our main aims are expressing some strong relations between modular forms, Hecke operators, and L-functions. We start with background information for mod- ular forms and give some information about the linear space of modular forms. Next, we introduce Hecke operators and their properties. Also we find a basis of Hecke eigenforms. Then, we explain how to construct L-functions from modular forms. Finally, we give a nice functional equation for completed L-function. ii Dedicated to my nieces Ahsen Yıldırım, Hüma Yıldırım, Meryem Özkale and my nephew Ömer Özkale. iii ACKNOWLEDGMENTS I would like to express my sincere gratitude to my supervisor Matthew P. Young for his excellent guidance, valuable suggestions, encouragement and patience. I would also like to thank Mohamad Riad Masri and Daren B. H. Cline for a careful reading of this thesis. I would like to thank to our Associate Head for Graduate Programs Peter Howard who always finds solutions to graduate students’ problems. I am so grateful to my mother, my father, my sisters for increasing my motivation whenever I need. I would like to thank to my friends Ahmed Sehy Youssef, Alisettar Huseynli, and Veysel Kaşdaş who are always with me whenever I experience difficulties.
  • 18.785 Notes

    18.785 Notes

    Contents 1 Introduction 4 1.1 What is an automorphic form? . 4 1.2 A rough definition of automorphic forms on Lie groups . 5 1.3 Specializing to G = SL(2; R)....................... 5 1.4 Goals for the course . 7 1.5 Recommended Reading . 7 2 Automorphic forms from elliptic functions 8 2.1 Elliptic Functions . 8 2.2 Constructing elliptic functions . 9 2.3 Examples of Automorphic Forms: Eisenstein Series . 14 2.4 The Fourier expansion of G2k ...................... 17 2.5 The j-function and elliptic curves . 19 3 The geometry of the upper half plane 19 3.1 The topological space ΓnH ........................ 20 3.2 Discrete subgroups of SL(2; R) ..................... 22 3.3 Arithmetic subgroups of SL(2; Q).................... 23 3.4 Linear fractional transformations . 24 3.5 Example: the structure of SL(2; Z)................... 27 3.6 Fundamental domains . 28 3.7 ΓnH∗ as a topological space . 31 3.8 ΓnH∗ as a Riemann surface . 34 3.9 A few basics about compact Riemann surfaces . 35 3.10 The genus of X(Γ) . 37 4 Automorphic Forms for Fuchsian Groups 40 4.1 A general definition of classical automorphic forms . 40 4.2 Dimensions of spaces of modular forms . 42 4.3 The Riemann-Roch theorem . 43 4.4 Proof of dimension formulas . 44 4.5 Modular forms as sections of line bundles . 46 4.6 Poincar´eSeries . 48 4.7 Fourier coefficients of Poincar´eseries . 50 4.8 The Hilbert space of cusp forms . 54 4.9 Basic estimates for Kloosterman sums . 56 4.10 The size of Fourier coefficients for general cusp forms .
  • Modular Forms and Hecke Operators

    Modular Forms and Hecke Operators

    MODULAR FORMS AND HECKE OPERATORS SAMANDA (YUTING) HU Abstract. This paper focuses on discussing Hecke operators in the theory of modular forms and its relation to Hecke rings which occur in representation theory. We will first introduce the basic definitions and properties of modular forms and Hecke operators. In particular, we will show that the space of cusp forms of an arbitrary weight with respect to a specific congruence subgroup is an inner product space and discuss that the space has an orthogonal basis of simultaneous eigenvectors for the Hecke operators. We will then shift the focus to Hecke rings and do relevant computations regarding the Hecke ring of GL2(Qp). Contents 1. Introduction 1 2. Modular Form and Congruence Subgroup 2 + 3. Action of GL2 (Q) on H 3 4. Hecke Operators 4 4.1. Double Coset Operator 4 4.2. hdi and Tp operator 6 4.3. Petersson Inner Product and Adjoints of Hecke Operators 11 5. Another Interpretation of Hecke Algebra 13 5.1. Hecke Ring 14 5.2. Satake Isomorphism 20 Acknowledgement 23 References 23 1. Introduction Modular forms are functions on the complex upper half plane that satisfy cer- tain transformation conditions and holomorphy conditions. The theory of modular forms is a significant branch of modern number theory { for instance, the Modu- larity Theorem suggests that all rational elliptic curves arise from modular forms with some restrictions. This theorem was essential to the proof of Fermat's Last Theorem by Richard Taylor and Andrew Wiles. In this paper, we aim to discuss Hecke operators in the theory of modular forms.