Probability and Measure

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

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 $AdS_3/CFT_2$ 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 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.

Quantum Black Holes, Wall Crossing, and Mock Modular Forms

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M24. The reason is yet a mystery.

Notes on the K3 Surface and the Mathieu Group M 24

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M_24. The reason is yet a mystery.

Notes on the K3 Surface and the Mathieu group M_24

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N
 e
 (n) (resp. N
 o
 (n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincare series whose “holomorphic part” is q
 -1
 f(q
 24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N
 e
 (n) and N
 o
 (n).

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The f(q) mock theta function conjecture and partition ranks

which were defined by Ramanujan and Watson decades ago. In his last letter to Hardy dated January 1920 (see pages 127-131 of [27]), Ramanujan lists 17 such functions, and he gives 2 more in his “Lost Notebook” [27]. In his paper “The final problem: An account of the mock theta functions” [32], Watson defines 3 further functions. Surprisingly, much remains unknown about these enigmatic series. Ramanujan’s claims about their analytic properties remain open, and there is even debate concerning the rigorous definition of such a function. Despite these seemingly problematic issues, Ramanujan’s mock theta functions indeed possess many striking properties, and they have been the subject of an astonishing number of important works (for example, see [5, 6, 7, 8, 12, 13, 14, 18, 19, 20, 23, 27, 28, 32, 33, 35, 36] to name a few). Watson predicted this high level of activity in his 1936 Presidential Address to the London Mathematical Society with his prophetic words (see page 80 of [32]):

/pdf/dyson-s-ranks-and-maass-forms-3pf9ic5y0c.pdf

Dyson’s ranks and Maass forms

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for p(n). In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct a new class of functions that we call quasi-weak Maass forms because they have quasi-modular forms as components. As an application, we prove two conjectures of Andrews [1, Conjectures 11, 13]. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves and also their derivatives. As a side product, we introduce a new method that enables us to prove transformation laws for generating functions over incomplete lattices

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On the explicit construction of higher deformations of partition statistics

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salie sums, and a well known reformulation of Salie sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincare series in Kohnen’s Γ0(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.

/pdf/arithmetic-properties-of-coefficients-of-half-integral-2pkt5v0l90.pdf

Kathrin Bringmann

Papers

The f(q) mock theta function conjecture and partition ranks

Dyson’s ranks and Maass forms

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

On the explicit construction of higher deformations of partition statistics

Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series