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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© 2005 Springer Science+Business Media New York
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Diamond, F., Shurman, J. (2005). Dimension Formulas. In: A First Course in Modular Forms. Graduate Texts in Mathematics, vol 228. Springer, New York, NY. https://doi.org/10.1007/978-0-387-27226-9_3
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DOI: https://doi.org/10.1007/978-0-387-27226-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2005-8
Online ISBN: 978-0-387-27226-9
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