Remarks on modular symbols for Maass wave forms

Manin, Yuri I.

doi:10.2140/ant.2010.4.1091

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Remarks on modular symbols for Maass wave forms

Yuri I. Manin

Vol. 4 (2010), No. 8, 1091–1114

DOI: 10.2140/ant.2010.4.1091

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 nonpositive rational ends, with values in analytic functions (pseudomeasures in the sense of Manin and Marcolli). We explain the basic issues and draw an analogy with the $p$ -adic case. We then construct the new modular symbols, followed by the related Lévy–Mellin transforms. This work builds on the fundamental study of Lewis and Zagier (2001).

To Professor F. Hirzebruch, with friendship and admiration, for his anniversary.

Keywords

Maass modular forms, modular symbols

Mathematical Subject Classification 2000

Primary: 11F37

Milestones

Received: 12 January 2010

Accepted: 1 October 2010

Published: 24 February 2011

Authors

Yuri I. Manin
Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn Germany	Department of Mathematics Northwestern University Evanston, IL 60208-2730 United States

Vol. 4, No. 8, 2010