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Dimension Formulas

  • Fred Diamond
  • Jerry Shurman
Part of the Graduate Texts in Mathematics book series (GTM, volume 228)

Abstract

For any congruence subgroup \(\Gamma\) of \(\mathrm{SL_2}(\mathbb{Z})\), the compactified modular curve \(\textit{X}(\Gamma)\) is now a Riemann surface. The genus of \(\textit{X}(\Gamma)\), its number of elliptic points, its number of its cusps, and the meromorphic functions and meromorphic differentials on \(\textit{X}(\Gamma)\) are all used by Riemann surface theory to determine dimension formulas for the vector spaces \(\mathcal{M}_\textit{k}(\Gamma)\) and \(\mathcal{S}_\textit{k}(\Gamma)\).

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Copyright information

© Springer Science+Business Media New York 2005

Authors and Affiliations

  • Fred Diamond
    • 1
  • Jerry Shurman
    • 2
  1. 1.Department of MathematicsKing’s College London StrandLondonUK
  2. 2.Department of MathematicsReed CollegePortlandUSA

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