Remarks on modular symbols for Maass wave forms

TL;DR

This paper introduces modular symbols for Maass wave cusp forms, exploring their properties as finitely additive functions with values in analytic functions, and relates them to Lévý–Mellin transforms, extending foundational work by Lewis and Zagier.

Contribution

It constructs modular symbols for Maass wave forms and connects them to Lévý–Mellin transforms, providing new insights into their structure and properties.

Findings

Modular symbols for Maass wave cusp forms are finitely additive functions.

Construction of modular symbols and Lévý–Mellin transforms for Maass forms.

Extension of Lewis–Zagier's foundational work on modular symbols.

Abstract

In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions (pseudo--measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related L\'evy--Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis--Zagier fundamental study [LZ2].