Vector-valued modular forms and the Mock Theta Conjectures

The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann-Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews-Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modular forms which transform according to the Weil representation. We then show that the difference of these vectors lies in a zero-dimensional vector space.Comment: 11 page

Hecke grids and congruences for weakly holomorphic modular forms

Let $U(p)$ denote the Atkin operator of prime index $p$. Honda and Kaneko proved infinite families of congruences of the form $f|U(p) \equiv 0 \pmod{p}$ for weakly holomorphic modular forms of low weight and level and primes $p$ in certain residue classes, and conjectured the existence of similar congruences modulo higher powers of $p$. Partial results on some of these conjectures were proved recently by Guerzhoy. We construct infinite families of weakly holomorphic modular forms on the Fricke groups $\Gamma^*(N)$ for $N=1,2,3,4$ and describe explicitly the action of the Hecke algebra on these forms. As a corollary, we obtain strengthened versions of all of the congruences conjectured by Honda and Kaneko

Shifted polyharmonic Maass forms for PSL(2,Z)

We study the vector space V_k^m(\lambda) of shifted polyharmonic Maass forms of weight k \in 2Z, depth m \geq 0, and shift \lambda \in C. This space is composed of real-analytic modular forms of weight k for PSL(2,Z) with moderate growth at the cusp which are annihilated by (\Delta_k - \lambda)^m, where \Delta_k is the weight k hyperbolic Laplacian. We treat the case \lambda \neq 0, complementing work of the second and third authors on polyharmonic Maass forms (with no shift). We show that V_k^m(\lambda) is finite-dimensional and bound its dimension. We explain the role of the real-analytic Eisenstein series E_k(z,s) with \lambda=s(s+k-1) and of the differential operator d/ds in this theory.Comment: 34 page