Algebra & Number Theory

Remarks on modular symbols for Maass wave forms

Yuri I. Manin

Full-text: Open access


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).

Article information

Algebra Number Theory, Volume 4, Number 8 (2010), 1091-1114.

Received: 12 January 2010
Accepted: 1 October 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms

Maass modular forms modular symbols


Manin, Yuri I. Remarks on modular symbols for Maass wave forms. Algebra Number Theory 4 (2010), no. 8, 1091--1114. doi:10.2140/ant.2010.4.1091.

Export citation


  • Y. Choie and D. Zagier, “Rational period functions for ${\rm PSL}(2,\mathbf Z)$”, pp. 89–108 in A tribute to Emil Grosswald: number theory and related analysis, edited by M. Knopp and M. Sheingorn, Contemp. Math. 143, Amer. Math. Soc., Providence, RI, 1993.
  • N. Diamantis, “Special values of higher derivatives of $L$-functions”, Forum Math. 11:2 (1999), 229–252.
  • D. Goldfeld, “Special values of derivatives of $L$-functions”, pp. 159–173 in Number theory (Halifax, NS, 1994), edited by K. Dilcher, CMS Conf. Proc. 15, Amer. Math. Soc., Providence, RI, 1995.
  • D. Goldfeld, “Zeta functions formed with modular symbols”, pp. 111–121 in Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), edited by R. S. Doran et al., Proc. Sympos. Pure Math. 66, Amer. Math. Soc., Providence, RI, 1999.
  • J. Hilgert, D. Mayer, and H. Movasati, “Transfer operators for $\Gamma\sb 0(n)$ and the Hecke operators for the period functions of ${\rm PSL}(2,{\mathbb Z})$”, Math. Proc. Cambridge Philos. Soc. 139:1 (2005), 81–116.
  • J. Lewis and D. Zagier, “Period functions and the Selberg zeta function for the modular group”, pp. 83–97 in The mathematical beauty of physics (Saclay, 1996), edited by J. M. Drouffe and J. B. Zuber, Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ, 1997.
  • J. Lewis and D. Zagier, “Period functions for Maass wave forms, I”, Ann. of Math. $(2)$ 153:1 (2001), 191–258.
  • H. Maass, “Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen”, Math. Ann. 121 (1949), 141–183.
  • Y. I. Manin, “Parabolic points and zeta functions of modular curves”, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. In Russian; translated in Math. USSR-Izv. 6 (1972), 19–64.
  • Y. I. Manin, “The periods of modular forms and $p$-adic Hecke series”, Mat. Sb. $($N.S.$)$ 92(134) (1973), 378–401. In Russian; translated in Math. USSR-Sb. 21:3 (1973), 371–393.
  • Y. I. Manin, “The values of $p$-adic Hecke series at integer points of the critical strip”, Math. USSR, Sb. 22:4 (1974), 631–637.
  • Y. I. Manin and M. Marcolli, “Modular shadows and the Lévy–Mellin $\infty$-adic transform”, pp. 189–238 in Modular forms on Schiermonnikoog, edited by B. Edixhoven et al., Cambridge Univ. Press, 2008.
  • S. Marmi, P. Moussa, and J.-C. Yoccoz, “Complex Brjuno functions”, J. Amer. Math. Soc. 14:4 (2001), 783–841.
  • S. Marmi, P. Moussa, and J.-C. Yoccoz, “Some properties of real and complex Brjuno functions”, pp. 601–623 in Frontiers in number theory, physics, and geometry, I, edited by P. Cartier et al., Springer, Berlin, 2006.
  • D. H. Mayer, “Continued fractions and related transformations”, pp. 175–222 in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), edited by T. Bedford et al., Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.
  • D. H. Mayer, “The thermodynamic formalism approach to Selberg's zeta function for ${\rm PSL}(2,{\bf Z})$”, Bull. Amer. Math. Soc. $($N.S.$)$ 25:1 (1991), 55–60.
  • B. Mazur and P. Swinnerton-Dyer, “Arithmetic of Weil curves”, Invent. Math. 25 (1974), 1–61.
  • T. Mühlenbruch, “Hecke operators on period functions for the full modular group”, Int. Math. Res. Not. 2004:77 (2004), 4127–4145.
  • V. V. Shokurov, “Shimura integrals of cusp forms”, Izv. Akad. Nauk SSSR Ser. Mat. 44:3 (1980), 670–718, 720.
  • V. V. Shokurov, “A study of the homology of Kuga varieties”, Izv. Akad. Nauk SSSR Ser. Mat. 44:2 (1980), 443–464, 480. In Russian; translated in Math. USSR-Izv. 16:2 (1981), 399–418.
  • M. M. Višik and Y. I. Manin, “$p$-adic Hecke series of imaginary quadratic fields”, Mat. Sb. $($N.S.$)$ 95(137) (1974), 357–383, 471. In Russian; translated in Math. USSR, Sb. 24 (1976), 345–371.