Amanda Folsom

The spt-function of Andrews.

Recently, Andrews introduced the function s(n) = spt(n) which counts the number of smallest parts among the integer partitions of n. We show that its generating function satisfies an identity analogous to Ramanujans mock theta identities. As a consequence, we are able to completely determine the parity of s(n). Using another type of identity, one based on Hecke operators, we obtain a complete multiplicative theory for s(n) modulo 3. These congruences confirm unpublished conjectures of Garvan and Sellers. Our methods generalize to all integral moduli.

q -series and weight 3/2 Maass forms

Despite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers{Fine basic hypergeometric series.

A short proof of the Mock Theta Conjectures using Maass forms

A celebrated work of D. Hickerson gives a proof of the Mock Theta Conjectures using Hecke-type identities discovered by G. Andrews. Here, we respond to a remark by K. Bringmann, K. Ono and R. Rhoades and provide a short proof of the Mock Theta Conjectures by realizing each side of the identities as the holomorphic projection of a harmonic weak Maass form.

The probability that the number of points on the Jacobian of a genus 2 curve is prime

In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus g = 2 and study various related issues, such as the probability of cyclicity and the probability of primality of the number of points on the curve itself. Finally, we discuss the asymptotic behavior for g →∞ .

Graded dimensions of principal subspaces and modular Andrews–Gordon-type series

Our results in this paper are threefold: First, we establish the modular properties of the graded dimensions of principal subspaces of level one standard modules for , and of principal subspaces of certain higher level standard modules for . Second, we establish the modular properties of families of q-series that appear in identities due to Warnaar and Zudilin, which generalize Macdonalds identities and the Rogers–Ramanujan identities. Third, we formulate a number of conjectures regarding the modularity of series of this type related to AN-1 root systems.

Conic theta functions and their relations to theta functions

It is natural to ask when the spherical volume defined by the intersection of a sphere at the apex of an integer polyhedral cone is rational. We use number theoretic methods to study a new class of polyhedral functions called conic theta functions, which are closely related to classical theta functions. We show that if K is a Weyl chamber for any finite reflection group, then its conic theta function lies in a graded ring of classical theta functions and in this sense is ‘almost’ modular. It is then natural to ask whether or not the conic theta functions are themselves modular, and we prove that (generally) they are not. In other words, we uncover some connections between the class of integer polyhedral cones that have a rational solid angle, and the class of conic theta functions that are almost modular.

Quantum mock modular forms arising from eta–theta functions

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta–theta functions by constructing mock modular forms from the eta–theta functions with even characters, such that the shadows of these mock modular forms are given by the eta–theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta–theta functions, as well as some curious algebraic identities.

A CHARACTERIZATION OF THE MODULAR UNITS

We provide an exact formula for the complex exponents in the modular product expansion of the modular units in terms of the Kubert–Lang structure theory, and deduce a characterization of the modular units in terms of the growth of these exponents, answering a question posed by Kohnen.