Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions

TL;DR

This paper develops new methods to interpret non-critical L-function values of modular forms as mock period functions and sesquiharmonic Maass forms, establishing a novel connection in number theory.

Contribution

It introduces a new completion technique for 1-cohomology, linking non-critical L-values to mock period functions and sesquiharmonic Maass forms, and proves an Eichler-Shimura-type isomorphism.

Findings

Non-critical L-values can be represented as mock period functions.

Non-critical values are encoded into sesquiharmonic Maass forms.

An Eichler-Shimura-type isomorphism for mock period functions is established.

Abstract

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2,Q) cusp forms. We prove that a generating series of non-critical values can be interpreted as a mock period function we define in analogy with period polynomials. Further, we prove that non-critical values can be encoded into a sesquiharmonic Maass form. Finally, we formulate and prove an Eichler-Shimura-type isomorphism for the space of mock period functions.