Modular forms on the upper half-plane

  • Gérard Lion
  • Michèle Vergne
Part of the Progress in Mathematics book series (PM, volume 6)


Let us consider the action of SL(2,ℝ) on the upper half-plane. Let Γ be a discrete subgroup of SL(2,ℝ) and X a character of Γ. We wish to construct holomorphic functions f on P+ such that:
$$ f\left( {\frac{{az + b}}{{cz + d}}} \right) = x(y){(cz + d)^{k}}f(z) $$
for every \( y = \left( {\begin{array}{*{20}{c}}a b \\c d \\\end{array}} \right) \) in Γ. (If f satisfies also the additional condition to be holomorphic at the cusps, f is called a modular form of type (k,x) for Γ.)


Holomorphic Function Modular Form Fundamental Domain Cusp Form Discrete Subgroup 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Gérard Lion
    • 1
  • Michèle Vergne
    • 2
  1. 1.X U.E.R. de Sciences EconomiquesUniversité de ParisNanterreFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations