Vol. 4, No. 8, 2010

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Remarks on modular symbols for Maass wave forms

Yuri I. Manin

Vol. 4 (2010), No. 8, 1091–1114

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.

Maass modular forms, modular symbols
Mathematical Subject Classification 2000
Primary: 11F37
Received: 12 January 2010
Accepted: 1 October 2010
Published: 24 February 2011
Yuri I. Manin
Max-Planck-Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
United States