Modular form: Difference between revisions

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===The Riemann surface ''G''\H^∗===

Let {{mvar|G}} be a subgroup of {{math|SL(2, '''Z''')}} that is of finite [[Index of a subgroup|index]]. Such a group {{mvar|G}} [[Group action (mathematics)|acts]] on '''H''' in the same way as {{math|SL(2, '''Z''')}}. The [[quotient topological space]] ''G''\/'''H''' can be shown to be a [[Hausdorff space]]. Typically it is not compact, but can be [[compactification (mathematics)|compactified]] by adding a finite number of points called ''cusps''. These are points at the boundary of '''H''', i.e. in '''[[Rational numbers|Q]]'''∪{∞},<ref group="note">Here, a matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> sends ∞ to ''a''/''c''.</ref> such that there is a parabolic element of {{mvar|G}} (a matrix with [[trace of a matrix|trace]] ±2) fixing the point. This yields a compact topological space ''G''\'''H'''^∗. What is more, it can be endowed with the structure of a [[Riemann surface]], which allows one to speak of holo- and meromorphic functions.

Important examples are, for any positive integer ''N'', either one of the [[congruence subgroup]]s