Extending a catalog of mock and quantum modular forms to an infinite class

Abstract

Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms Vmn having weight 3/2 eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the V m1 , and show that the functions in this infinite class are both mock modular and quantum modular forms.where τ is in the upper half-plane H = {τ ∈ C | Im(τ )}, and q = e 2πiτ is one of the most well-known half-integral weight modular forms. Quotients of the form c j=1 η(a j τ ) bj , for b j ∈ Z are called eta-quotients, and have been of interest in many areas of mathematics, including combinatorics, number theory, and representation theory (see [16], [1], [7], for example). Classifications of eta-quotients have also been of interest. Dummit, Kisilevsky, and McKay [8] classified all multiplicative eta-products (restricting to positive b j ), and later Martin [14] classified multiplicative integer weight eta-quotients. Furthering these classifications, Lemke Oliver [15] recently classified all eta-quotients which are also theta functions, which are of the form θ χ (τ ) := n χ(n)n ν q n 2 ,where χ is an even (resp. odd) Dirichlet character, ν is 0 (resp. 1), and the sum above is over n ∈ Z when χ is trivial, or over n ∈ N when χ is not trivial. These functions are known to be ordinary modular forms of weight 1/2 + ν.We refer to a function which is both an eta-quotient and a theta function as an eta-theta function. In Lemke Oliver's classification there are six odd (character) eta-theta functions E m , and eighteen even (character) eta-theta functions e n (some of which are twists by certain principal characters).Modular theta functions also arise in the theory of harmonic Maass forms and mock modular forms via the shadow operator. A harmonic Maass form f , as originally defined by Bruinier and Funke [5], naturally decompose into two parts as f = f + + f − , where f + is called the holomorphic part of f , and f − is called the non-holomorphic part of f . All of Ramanujan's original mock theta functions turn out to be examples of holomorphic parts of harmonic Maass forms [20], and we now define after Zagier [18] a mock modular form to be any such holomorphic part of a harmonic Maass form (for which the non-holomorphic part is not