Related papers: Vector-valued modular forms and the Mock Theta Con…
Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find the four tenth-order mock theta…
We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta…
We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions.…
Recently, Andrews, Dixit and Yee defined two partition functions $p_{\omega}(n)$ and $p_{\nu}(n)$ that are related with Ramanujan's mock theta functions $\omega(q)$ and $\nu(q)$, respectively. In this paper, we present two variable…
We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group…
The multiple zeta values (MZV) are a set of real numbers with a beautiful structure as an algebra over the rational numbers. They are related to maybe the most important conjecture on mathematics today, the Riemann hypothesis. In this paper…
The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP)…
We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of…
Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these…
We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for…
We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…
We prove the rank-4 case of the conjecture of Ha-Hai-Nghia for the invariant subspace of the truncated polynomial ring $\mathcal{Q}_m(n)=\mathbb{F}_q[x_1,\dots,x_n]/(x_1^{q^m},\dots,x_n^{q^m}),$ under a new, explicit technical hypothesis.…
In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…
In 1947, Lehmer conjectured that the Ramanujan \tau-function \tau(m) is non-vanishing for all positive integers m, where \tau(m) are the Fourier coefficients of the cusp form \Delta of weight 12. It is known that Lehmer's conjecture can be…
We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the…
Let $f(q)=1+\sum_{n=1}^{\infty} \alpha(n)q^n$ be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form $$\alpha(n)= \sum_{c\leq\sqrt{n}}…
We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an…
Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…
Recently, Keith investigated reciprocals of false theta functions and proved some interesting results such as congruences, asymptotic bounds, and combinatorial identities. At the end of his paper, Keith posed a conjecture on congruences…
We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first…