Notes on Harmonic Maass 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 + .. -
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. -
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. -
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. -
Modular Forms in String Theory and Moonshine
(Mock) Modular Forms in String Theory and Moonshine Miranda C. N. Cheng Korteweg-de-Vries Institute of Mathematics and Institute of Physics, University of Amsterdam, Amsterdam, the Netherlands Abstract Lecture notes for the Asian Winter School at OIST, Jan 2016. Contents 1 Lecture 1: 2d CFT and Modular Objects 1 1.1 Partition Function of a 2d CFT . .1 1.2 Modular Forms . .4 1.3 Example: The Free Boson . .8 1.4 Example: Ising Model . 10 1.5 N = 2 SCA, Elliptic Genus, and Jacobi Forms . 11 1.6 Symmetries and Twined Functions . 16 1.7 Orbifolding . 18 2 Lecture 2: Moonshine and Physics 22 2.1 Monstrous Moonshine . 22 2.2 M24 Moonshine . 24 2.3 Mock Modular Forms . 27 2.4 Umbral Moonshine . 31 2.5 Moonshine and String Theory . 33 2.6 Other Moonshine . 35 References 35 1 1 Lecture 1: 2d CFT and Modular Objects We assume basic knowledge of 2d CFTs. 1.1 Partition Function of a 2d CFT For the convenience of discussion we focus on theories with a Lagrangian description and in particular have a description as sigma models. This in- cludes, for instance, non-linear sigma models on Calabi{Yau manifolds and WZW models. The general lessons we draw are however applicable to generic 2d CFTs. An important restriction though is that the CFT has a discrete spectrum. What are we quantising? Hence, the Hilbert space V is obtained by quantising LM = the free loop space of maps S1 ! M. Recall that in the usual radial quantisation of 2d CFTs we consider a plane with 2 special points: the point of origin and that of infinity. -
Weierstrass Mock Modular Forms and Elliptic Curves
WEIERSTRASS MOCK MODULAR FORMS AND ELLIPTIC CURVES CLAUDIA ALFES, MICHAEL GRIFFIN, KEN ONO, AND LARRY ROLEN Abstract. Mock modular forms, which give the theoretical framework for Ramanujan’s enig- matic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E=Q. We show that mock modular forms which arise from Weierstrass ζ-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently stud- ied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form. 1. Introduction and Statement of Results The theory of mock modular forms, which provides the underlying theoretical framework for Ramanujan’s enigmatic mock theta functions [10, 11, 63, 64], has recently played important roles in combinatorics, number theory, mathematical physics, and representation theory (see [50, 51, 63]). Here we consider mock modular forms and the arithmetic of elliptic curves. We first recall the notion of a harmonic weak Maass form which was introduced by Bruinier and Funke [15]. Here we let z := x + iy 2 H, where x; y 2 R, and we let q := e2πiz. -
Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5
Preprint typeset in JHEP style - HYPER VERSION BONN-TH-2010-13 LMU-ASC 102/10 Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes Murad Alim1, Babak Haghighat2,3, Michael Hecht4, Albrecht Klemm3, Marco Rauch3 and Thomas Wotschke3 1Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA 2Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD Utrecht, The Netherlands 3Bethe Center for Theoretical Physics, University of Bonn, Nussallee 12, D-53115 Bonn, Germany 4Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, D-80333 Munich, Germany Abstract: Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrap- ping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of = 4 topological Yang-Mills the- 2 N ory on È and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The arXiv:1012.1608v1 [hep-th] 7 Dec 2010 non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge. Contents 1. Introduction 2 2. -
Cycle Integrals of the J-Function and Mock Modular Forms
CYCLE INTEGRALS OF THE J-FUNCTION AND MOCK MODULAR FORMS W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´ Abstract. In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are given in terms of cycle integrals of the modular j-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coefficient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coefficients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product. 1. Introduction Mock modular forms, especially those of weight 1/2, have attracted much attention recently. This is mostly due to the discovery of Zwegers [51, 52] that Ramanujan's mock theta functions can be completed to become modular by the addition of a certain non-holomorphic function on the upper half plane H. This complement is associated to a modular form of weight 3/2, the shadow of the mock theta function. Consider, for example, the q-series 2 X qn f(τ) = q−1=24 q = e(τ) = e2πiτ ; τ 2 H: (1 + q)2 ··· (1 + qn)2 n≥0 Up to the factor q−1=24 this is one of Ramanujan's original mock theta functions. -
Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms
HARMONIC MAASS FORMS, MOCK MODULAR FORMS, AND QUANTUM MODULAR FORMS KEN ONO Abstract. This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few. Here we discuss the essential facts in the theory, and consider some applications in number theory. This mathematics has an unlikely beginning: the mystery of Ramanujan’s enigmatic “last letter” to Hardy written three months before his untimely death. Section 15 gives examples of projects which arise naturally from the mathematics described here. Modular forms are key objects in modern mathematics. Indeed, modular forms play cru- cial roles in algebraic number theory, algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics. The recent history of the subject includes (to name a few) great successes on the Birch and Swinnerton-Dyer Con- jecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat’s Last Theorem. These celebrated works are dramatic examples of the evolution of mathematics. Here we tell a different story, the mathematics of harmonic Maass forms, mock modular forms, and quantum modular forms. This story has an unlikely beginning, the mysterious last letter of the Ramanujan. 1. Ramanujan The story begins with the legend of Ramanujan, the amateur Indian genius who discovered formulas and identities without any rigorous training1 in mathematics. He recorded his findings in mysterious notebooks without providing any indication of proofs. -
Mathematisches Forschungsinstitut Oberwolfach Modular Forms
Mathematisches Forschungsinstitut Oberwolfach Report No. 22/2014 DOI: 10.4171/OWR/2014/22 Modular Forms Organised by Jan Hendrik Bruinier, Darmstadt Atsushi Ichino, Kyoto Tamotsu Ikeda, Kyoto Ozlem¨ Imamoglu, Z¨urich 27 April – 3 May 2014 Abstract. The theory of Modular Forms has been central in mathematics with a rich history and connections to many other areas of mathematics. The workshop explored recent developments and future directions with a particular focus on connections to the theory of periods. Mathematics Subject Classification (2010): 11xx. Introduction by the Organisers The workshop Modular Forms, organized by Jan Hendrik Bruinier (Darmstadt), Atsushi Ichino (Kyoto), Tamotsu Ikeda (Kyoto) and Ozlem¨ Imamoglu (Z¨urich) consisted of 19 one-hour long lectures and covered various recent developments in the theory of modular and automorphic forms and related fields. A particular focus was on the connection of modular forms to periods, since there have been important developments in that direction in recent years. In this context, the topics that the workshop addressed include the global Gross- Prasad conjecture and its analogs, which predict a relationship between periods of automorphic forms and central values of L-functions, the theory of liftings and their applications to period relations, as well as explicit aspects of these formulas and relations with a view towards the arithmetic properties of periods. There are two fundamental ways in which automorphic forms are related to periods. First, according to the conjectures of Deligne, Beilinson and Scholl, spe- cial values of motivic automorphic L-functions at integral arguments should be given by periods and encode important arithmetic information, such as ranks of 1222 Oberwolfach Report 22/2014 Chow groups and Selmer groups. -
Quantum Black Holes, Wall Crossing, and Mock Modular Forms Arxiv
Preprint typeset in JHEP style - HYPER VERSION Quantum Black Holes, Wall Crossing, and Mock Modular Forms Atish Dabholkar1;2, Sameer Murthy3, and Don Zagier4;5 1Theory Division, CERN, PH-TH Case C01600, CH-1211, 23 Geneva, Switzerland 2Sorbonne Universit´es,UPMC Univ Paris 06 UMR 7589, LPTHE, F-75005, Paris, France CNRS, UMR 7589, LPTHE, F-75005, Paris, France 3NIKHEF theory group, Science Park 105, 1098 XG Amsterdam, The Netherlands 4Max-Planck-Institut f¨ur Mathematik,Vivatsgasse 7, 53111 Bonn, Germany 5Coll`egede France, 3 Rue d'Ulm, 75005 Paris, France Emails: [email protected], [email protected], [email protected] Abstract: We show that the meromorphic Jacobi form that counts the quarter-BPS states in N = 4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes are Fourier coefficients of this mock Jacobi form, while the Appell-Lerch sum captures the degeneracies of multi-centered black holes which decay upon wall-crossing. The completion of the mock Jacobi form restores the modular symmetries expected from AdS3=CF T2 holography but has a holomorphic anomaly reflecting the non-compactness of the microscopic CFT. For every positive integral value m of the magnetic charge invariant of the black hole, our analysis leads arXiv:1208.4074v2 [hep-th] 3 Apr 2014 to a special mock Jacobi form of weight two and index m, which we characterize uniquely up to a Jacobi cusp form. This family of special forms and another closely related family of weight-one forms contain almost all the known mock modular forms including the mock theta functions of Ramanujan, the generating function of Hurwitz-Kronecker class numbers, the mock modular forms appearing in the Mathieu and Umbral moonshine, as well as an infinite number of new examples. -
Theta Lifts for Lorentzian Lattices and Coefficients of Mock Theta Functions
THETA LIFTS FOR LORENTZIAN LATTICES AND COEFFICIENTS OF MOCK THETA FUNCTIONS JAN HENDRIK BRUINIER AND MARKUS SCHWAGENSCHEIDT Abstract. We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews' spt-function, and Ramanujan's mock theta functions. 1. Introduction Special values of modular forms at complex multiplication points play a prominent role in number theory. For instance, they are crucial in explicit class field theory and in the Chowla-Selberg formula. Moreover, the evaluation of automorphic Green functions at CM points determines archimedean height pairings appearing in the Gross-Zagier formula and some of its generalizations [23]. These can be computed using suitable see-saw dual pairs for regularized theta lifts of harmonic Maass forms to orthogonal groups of signature (2; n), see [34] or [15], for example. In the present paper, we study similar special values of regularized theta lifts for Lorentzian lattices, that is, for even lattices of signature (1; n). It was shown in [16] that these values ap- pear naturally as vanishing orders of Borcherds products along boundary divisors of toroidal compactifications of orthogonal Shimura varieties. Here we compute the special values in sev- eral different ways. As an application, we derive general recursive formulas for the coefficients of mock theta functions. These include the Hurwitz-Kronecker class number relations and some generalizations as special cases. We give a more detailed outline of our results now.