Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms
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Traces of Reciprocal Singular Moduli, We Require the Theta Functions
TRACES OF RECIPROCAL SINGULAR MODULI CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT Abstract. We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight 3/2 whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla-Millson theta lift of Bruinier and Funke to meromorphic modular functions. 1. Introduction and statement of results The special values of the modular j-invariant 1 2 3 j(z)= q− + 744 + 196884q + 21493760q + 864299970q + ... at imaginary quadratic points in the upper half-plane are called singular moduli. By the theory of complex multiplication they are algebraic integers in the ray class fields of certain orders in imaginary quadratic fields. In particular, their traces j(zQ) trj(D)= ΓQ Q + /Γ ∈QXD | | are known to be rational integers. Here + denotes the set of positive definite quadratic QD forms of discriminant D < 0, on which Γ = SL2(Z) acts with finitely many orbits. Further, ΓQ is the stabilizer of Q in Γ = PSL2(Z) and zQ H is the CM point associated to Q. In his seminal paper “Traces of singular moduli”∈ (see [Zag02]) Zagier showed that the generating series 1 D 1 3 4 7 8 q− 2 tr (D)q− = q− 2+248q 492q + 4119q 7256q + ... − − J − − − D<X0 arXiv:1905.07944v2 [math.NT] 2 Apr 2020 of traces of CM values of J = j 744 is a weakly holomorphic modular form of weight − 3/2 for Γ0(4). It follows from this result, by adding a multiple of Zagier’s mock modular Eisenstein series of weight 3/2 (see [Zag75]), that the generating series 1 D 1 4 7 8 q− + 60 tr (D)q− = q− + 60 864q + 3375q 8000q + .. -
A Life Inspired by an Unexpected Genius
Quanta Magazine A Life Inspired by an Unexpected Genius The mathematician Ken Ono believes that the story of Srinivasa Ramanujan — mathematical savant and two-time college dropout — holds valuable lessons for how we find and reward hidden genius. By John Pavlus The mathematician Ken Ono in his office at Emory University in Atlanta. For the first 27 years of his life, the mathematician Ken Ono was a screw-up, a disappointment and a failure. At least, that’s how he saw himself. The youngest son of first-generation Japanese immigrants to the United States, Ono grew up under relentless pressure to achieve academically. His parents set an unusually high bar. Ono’s father, an eminent mathematician who accepted an invitation from J. Robert Oppenheimer to join the Institute for Advanced Study in Princeton, N.J., expected his son to follow in his footsteps. Ono’s mother, meanwhile, was a quintessential “tiger parent,” discouraging any interests unrelated to the steady accumulation of scholarly credentials. This intellectual crucible produced the desired results — Ono studied mathematics and launched a promising academic career — but at great emotional cost. As a teenager, Ono became so desperate https://www.quantamagazine.org/the-mathematician-ken-onos-life-inspired-by-ramanujan-20160519/ May 19, 2016 Quanta Magazine to escape his parents’ expectations that he dropped out of high school. He later earned admission to the University of Chicago but had an apathetic attitude toward his studies, preferring to party with his fraternity brothers. He eventually discovered a genuine enthusiasm for mathematics, became a professor, and started a family, but fear of failure still weighed so heavily on Ono that he attempted suicide while attending an academic conference. -
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. -
Arxiv:1906.07410V4 [Math.NT] 29 Sep 2020 Ainlpwrof Power Rational Ftetidodrmc Ht Function Theta Mock Applications
MOCK MODULAR EISENSTEIN SERIES WITH NEBENTYPUS MICHAEL H. MERTENS, KEN ONO, AND LARRY ROLEN In celebration of Bruce Berndt’s 80th birthday Abstract. By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) “small divisors” summatory functions sm σψ (n). More precisely, in terms of the weight 2 quasimodular Eisenstein series E2(τ) and a generic Shimura theta function θψ(τ), we show that there is a constant αψ for which ∞ E + · E2(τ) 1 sm n ψ (τ) := αψ + X σψ (n)q θψ(τ) θψ(τ) n=1 is a half integral weight (polar) mock modular form. These include generating functions for combinato- rial objects such as the Andrews spt-function and the “consecutive parts” partition function. Finally, in analogy with Serre’s result that the weight 2 Eisenstein series is a p-adic modular form, we show that these forms possess canonical congruences with modular forms. 1. Introduction and statement of results In the theory of mock theta functions and its applications to combinatorics as developed by Andrews, Hickerson, Watson, and many others, various formulas for q-series representations have played an important role. For instance, the generating function R(ζ; q) for partitions organized by their ranks is given by: n n n2 n (3 +1) m n q 1 ζ ( 1) q 2 R(ζ; q) := N(m,n)ζ q = −1 = − − n , (ζq; q)n(ζ q; q)n (q; q)∞ 1 ζq n≥0 n≥0 n∈Z − mX∈Z X X where N(m,n) is the number of partitions of n of rank m and (a; q) := n−1(1 aqj) is the usual n j=0 − q-Pochhammer symbol. -
Congruences Between Modular Forms
CONGRUENCES BETWEEN MODULAR FORMS FRANK CALEGARI Contents 1. Basics 1 1.1. Introduction 1 1.2. What is a modular form? 4 1.3. The q-expansion priniciple 14 1.4. Hecke operators 14 1.5. The Frobenius morphism 18 1.6. The Hasse invariant 18 1.7. The Cartier operator on curves 19 1.8. Lifting the Hasse invariant 20 2. p-adic modular forms 20 2.1. p-adic modular forms: The Serre approach 20 2.2. The ordinary projection 24 2.3. Why p-adic modular forms are not good enough 25 3. The canonical subgroup 26 3.1. Canonical subgroups for general p 28 3.2. The curves Xrig[r] 29 3.3. The reason everything works 31 3.4. Overconvergent p-adic modular forms 33 3.5. Compact operators and spectral expansions 33 3.6. Classical Forms 35 3.7. The characteristic power series 36 3.8. The Spectral conjecture 36 3.9. The invariant pairing 38 3.10. A special case of the spectral conjecture 39 3.11. Some heuristics 40 4. Examples 41 4.1. An example: N = 1 and p = 2; the Watson approach 41 4.2. An example: N = 1 and p = 2; the Coleman approach 42 4.3. An example: the coefficients of c(n) modulo powers of p 43 4.4. An example: convergence slower than O(pn) 44 4.5. Forms of half integral weight 45 4.6. An example: congruences for p(n) modulo powers of p 45 4.7. An example: congruences for the partition function modulo powers of 5 47 4.8. -
UBC Math Distinguished Lecture Series Ken Ono (Emory University)
PIMS- UBC Math Distinguished Lecture Series Ken Ono (Emory University) Nov 9, 2017 University of British Columbia Room: ESB 2012 Vancouver, BC 3:30pm Polya’s Program for the Riemann Hypothesis and Related Problems In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. We prove the hyperbolicity of 100% of the Jensen poly- nomials of every degree. We obtain a general theorem which models such polyno- mials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier. KEN ONO is the Asa Griggs Candler Professor of Mathematics at Emory University. He is considered to be an expert in the theory of integer partitions and modular forms. His contributions include several monographs and over 150 research and popular articles in number theory, combinatorics and algebra. He received his Ph.D. from UCLA and has received many awards for his research in number theory, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Fellowship. He was awarded a Presidential Early Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and he was named the National Science Foundation’s Distinguished Teaching Scholar in 2005. He serves as Editor-in-Chief for several journals and is an editor of The Ramanujan Journal. Visit his web page at http://www.mathcs.emory.edu/~ono/ Light refreshments will be served in ESB 4133 from 3:00pm For more details please visit the PIMS Webpage at: https://www.pims.math.ca/scientific-event/171109-pumdcko www.pims.math.ca @pimsmath facebook.com/pimsmath . -
25 Modular Forms and L-Series
18.783 Elliptic Curves Spring 2015 Lecture #25 05/12/2015 25 Modular forms and L-series As we will show in the next lecture, Fermat's Last Theorem is a direct consequence of the following theorem [11, 12]. Theorem 25.1 (Taylor-Wiles). Every semistable elliptic curve E=Q is modular. In fact, as a result of subsequent work [3], we now have the stronger result, proving what was previously known as the modularity conjecture (or Taniyama-Shimura-Weil conjecture). Theorem 25.2 (Breuil-Conrad-Diamond-Taylor). Every elliptic curve E=Q is modular. Our goal in this lecture is to explain what it means for an elliptic curve over Q to be modular (we will also define the term semistable). This requires us to delve briefly into the theory of modular forms. Our goal in doing so is simply to understand the definitions and the terminology; we will omit all but the most straight-forward proofs. 25.1 Modular forms Definition 25.3. A holomorphic function f : H ! C is a weak modular form of weight k for a congruence subgroup Γ if f(γτ) = (cτ + d)kf(τ) a b for all γ = c d 2 Γ. The j-function j(τ) is a weak modular form of weight 0 for SL2(Z), and j(Nτ) is a weak modular form of weight 0 for Γ0(N). For an example of a weak modular form of positive weight, recall the Eisenstein series X0 1 X0 1 G (τ) := G ([1; τ]) := = ; k k !k (m + nτ)k !2[1,τ] m;n2Z 1 which, for k ≥ 3, is a weak modular form of weight k for SL2(Z). -
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. -
P-Adic Analysis and Mock Modular Forms
P -ADIC ANALYSIS AND MOCK MODULAR FORMS A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI`I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS AUGUST 2010 By Zachary A. Kent Dissertation Committee: Pavel Guerzhoy, Chairperson Christopher Allday Ron Brown Karl Heinz Dovermann Theresa Greaney Abstract A mock modular form f + is the holomorphic part of a harmonic Maass form f. The non-holomorphic part of f is a period integral of a cusp form g, which we call the shadow of f +. The study of mock modular forms and mock theta functions is one of the most active areas in number theory with important works by Bringmann, Ono, Zagier, Zwegers, among many others. The theory has many wide-ranging applica- tions: additive number theory, elliptic curves, mathematical physics, representation theory, and many others. We consider arithmetic properties of mock modular forms in three different set- tings: zeros of a certain family of modular forms, coupling the Fourier coefficients of mock modular forms and their shadows, and critical values of modular L-functions. For a prime p > 3, we consider j-zeros of a certain family of modular forms called Eisenstein series. When the weight of the Eisenstein series is p − 1, the j-zeros are j-invariants of elliptic curves with supersingular reduction modulo p. We lift these j-zeros to a p-adic field, and show that when the weights of two Eisenstein series are p-adically close, then there are j-zeros of both series that are p-adically close. -
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 . -
CM Values and Fourier Coefficients of Harmonic Maass Forms
CM values and Fourier coefficients of harmonic Maass forms Vom Fachbereich Mathematik der Technischen Universit¨at Darmstadt zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Math. Claudia Alfes aus Dorsten Referent: Prof. Dr. J. H. Bruinier 1. Korreferent: Prof. Ken Ono, PhD 2. Korreferent: Prof. Dr. Nils Scheithauer Tag der Einreichung: 11. Dezember 2014 Tag der mundlichen¨ Prufung:¨ 5. Februar 2015 Darmstadt 2015 D 17 ii Danksagung Hiermit m¨ochte ich allen danken, die mich w¨ahrend meines Studiums und meiner Promotion, in- und außerhalb der Universit¨at, unterstutzt¨ und begleitet haben. Ein besonders großer Dank geht an meinen Doktorvater Professor Dr. Jan Hendrik Bruinier, der mir die Anregung fur¨ das Thema der Arbeit gegeben und mich stets gefordert, gef¨ordert und motiviert hat. Außerdem danke ich Professor Ken Ono, der mich ebenfalls auf viele interessante Fragestellungen aufmerksam gemacht hat und mir stets mit Rat und Tat zur Seite stand. Vielen Dank auch an Professor Nils Scheithauer fur¨ interessante Hinweise und fur¨ die Bereitschaft, die Arbeit zu begutachten. Ein weiteres großes Dankesch¨on geht an Dr. Stephan Ehlen, von ihm habe ich viel gelernt und insbesondere große Hilfe bei den in Sage angefertigten Rechnungen erhalten. Vielen Dank auch an Anna von Pippich fur¨ viele hilfreiche Hinweise. Weiterer Dank geht an die fleißigen Korrekturleser Yingkun Li, Sebastian Opitz, Stefan Schmid und Markus Schwagenscheidt die viel Zeit investiert haben, viele nutzliche¨ Anmerkungen hatten und zahlreiche Tippfehler gefunden haben. Daruber¨ hinaus danke ich der Deutschen Forschungsgemeinschaft, aus deren Mitteln meine Stelle an der Technischen Universit¨at Darmstadt zu Teilen im Rahmen des Projektes Schwache Maaß-Formen" und der Forschergruppe Symmetrie, Geometrie und " " Arithmetik" finanziert wurde. -
CONGRUENCES of SIEGEL EISENSTEIN SERIES of DEGREE TWO 11 Cubic Lifts to Siegel Cusp Forms of Scalar Weight Other Than Three (Cf
CONGRUENCES OF SIEGEL EISENSTEIN SERIES OF DEGREE TWO TAKUYA YAMAUCHI Abstract. In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp4(Z). 1. Introduction The congruences for automorphic forms have been studied throughly in several settings. The well-known prototype seems to be one related to Ramanujan delta function and the Eisenstein series of weight 12 for SL2(Z). Apart from its own interests, various important applications, for instance Iwasawa theory, have been made since many individual examples found before. In [24] Ribet tactfully used Galois representations for elliptic cusp forms of level one to obtain the converse of the Herbrand’s theorem in the class numbers for cyclotomic fields. Herbrand-Ribet’s theorem is now understood in the context of Hida theory and Iwasawa main conjecture [1]. The notion of congruence modules is developed by many authors (cf. [13],[22] among others). In this paper we study the congruence primes between Siegel Eisenstein series of degree 2 and Siegel cusp forms with respect to Sp4(Z). To explain the main claims we need to fix the notation. Let H = Z M (C) tZ = Z > 0 be the Siegel upper half space of degree 2 and Γ := 2 { ∈ 2 | } t 02 E2 02 E2 Sp4(Z) = γ GL4(Z) γ γ = where E2 is the identity matrix ( ∈ E 0 ! E 0 ! ) − 2 2 − 2 2 −1 A B arXiv:1912.03847v2 [math.NT] 13 Dec 2019 of size 2. The group Γ acts on H2 by γZ := (AZ + B)(CZ + D) for γ = Γ C D! ∈ and Z H .