No abstract

A "mock modular form" is the holomorphic part of a harmonic Maass form f. The nonholomorphic part of f is a period integral of its "shadow," a cusp form g. A direct method for relating the coefficients of shadows and mock modular forms is not known. We solve these problems when the shadow is an integer weight newform. Our solution is p-adic, and it relies on our definition of an algebraic "regularized mock modular form." As an application, we consider the modular solution to the cubic "arithmeticgeometric mean."harmonic Maass form | mock theta function

In this paper, we explore a method for associating L-series to weakly holomorphic modular forms and then proceed to study their L-values. As our main application, we prove a very curious limiting theorem which relates three "periods" of a mock modular form and its shadow to the ratio of their noncritical L-values. Critical L-values are then shown to fit nicely within the framework of period polynomials and an extended Eichler-Shimura theory recently studied by Guerzhoy, Ono, and the first and third authors.

No abstract

Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the 'half-derivatives' of the Andrews-Gordon functions and with Kashaev's invariant on (2m + 1, 2) torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least 50% of primes. Introduction and statement of results