Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

Abstract

The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of SL 2 (Z)-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms.

Eichler integrals and generalized second order Eisenstein series

“…Generalized second order modular forms of type (1, 1) are the same as second order modular forms, introduced by Goldfeld [12]. It is possible to view generalized second order modular forms as components of vector-valued modular forms of suitable arithmetic type [16].…”

“…Leveraging the cocycle in (2.6), we find that Eichler integrals are generalized second order modular forms. The next proposition can be viewed as a special case of Theorem 3.7 of [16] when using the language of vector-valued modular forms.…”

“…Remark 3.2. The Eisenstein series in (3.2) is a component of a suitable vector-valued Eisenstein series [16].…”

“…Given even integers k and d ≥ 0, and an integer 0 ≤ j ≤ d, the constant functions and the functions (X − τ) j satisfy the condition on f in Definition 3.1 for the arithmetic types σ = 1 and σ = sym d (X ), respectively. By Proposition 5.4 of [16] the associated generalized second order Eisenstein series are defined for k ≥ 5 + d and yield functions of moderate growth, i.e., generalized second order modular forms. Note that one can improve this bound when employing convexity of additively twisted L-functions as in Lemma 3.11 or in [7].…”

“…Diamantis [7] showed that iterated Eichler-Shimura integrals are also closely related to higher order modular forms, first introduced by Goldfeld [12]. In a similar vein Mertens-Raum [16] showed that both higher order modular forms and iterated Eichler-Shimura integrals can be reinterpreted as components of vector-valued modular forms of higher depth arithmetic types. In this paper, we explore the problem of efficiently computing Eichler-Shimura integrals by means of the aforementioned theoretical frameworks, focusing on the special case of Eichler integrals of cusp forms of level one.…”