Editing Modular form (section)

===Consequences===
The theory of Riemann surfaces can be applied to ''G''\'''H'''<sup>∗</sup> to obtain further information about modular forms and functions. For example, the spaces {{math|''M<sub>k</sub>''(''G'')}} and {{math|''S<sub>k</sub>''(''G'')}} are finite-dimensional, and their dimensions can be computed thanks to the [[Riemann–Roch theorem]] in terms of the geometry of the {{mvar|G}}-action on '''H'''.<ref>{{Citation | last1=Shimura | first1=Goro | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11}}, Theorem 2.33, Proposition 2.26</ref> For example,

:<math>\dim_\mathbf{C} M_k\left(\text{SL}(2, \mathbf{Z})\right) = \begin{cases}
  \left\lfloor k/12 \right\rfloor     & k \equiv 2 \pmod{12} \\
  \left\lfloor k/12 \right\rfloor + 1 & \text{otherwise}
\end{cases}</math>

where <math>\lfloor \cdot \rfloor</math> denotes the [[floor function]] and <math>k</math> is even.

The modular functions constitute the [[function field of an algebraic variety|field of functions]] of the Riemann surface, and hence form a field of [[transcendence degree]] one (over '''C'''). If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of [[pole (complex analysis)|pole]]s of ''f'' in the [[closure (mathematics)|closure]] of the [[fundamental region]] ''R''<sub>Γ</sub>.It can be shown that the field of modular function of level ''N'' (''N'' ≥ 1) is generated by the functions ''j''(''z'') and ''j''(''Nz'').<ref>{{Citation |last=Milne |first=James |title=Modular Functions and Modular Forms |url=https://www.jmilne.org/math/CourseNotes/MF.pdf#page=88 |year=2010 |page=88 }}, Theorem 6.1.</ref>