Quantum modular forms and Hecke operators

Abstract

It is known that there are one-to-one correspondences among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.

A note on additive twists, reciprocity laws and quantum modular forms

“…For k > 2 this is however not an interesting example in our context since this is the restriction of a continuous function1 (because of the absolute convergence of the Dirichlet series (3.1) at the special value s = k − 1) as was also noted in [2,Section 1.4.1]. Using [9,Theorem 1.4] one can see that the the central value s = k/2 of the additive twists of the L-function of an integral weight cusp form f considered as a function of the twisting 1Quantum modularity of Eichler integrals of integral weight cusp forms were studied by Lee in [8]. In this work the notion of regularity used in the definition of quantum modular forms is smooth instead of continuous.…”