Editing Modular form (section)

==Modular functions==
When the weight ''k'' is zero, it can be shown using [[Liouville's theorem (complex analysis)|Liouville's theorem]] that the only modular forms are constant functions. However, relaxing the requirement that ''f'' be holomorphic leads to the notion of ''modular functions''. A function ''f'' : '''H''' → '''C''' is called modular if it satisfies the following properties:

* ''f'' is [[meromorphic function|meromorphic]] in the open [[upper half-plane]] ''H''
* For every integer [[matrix (mathematics)|matrix]] <math>\begin{pmatrix}a & b \\ c & d \end{pmatrix}</math> in the [[modular group|modular group {{math|Γ}}]], <math> f\left(\frac{az+b}{cz+d}\right) = f(z)</math>.
* The second condition implies that ''f'' is periodic, and therefore has a [[Fourier series]]. The third condition is that this series is of the form
::<math>f(z) = \sum_{n=-m}^\infty a_n e^{2i\pi nz}.</math>   
It is often written in terms of <math>q=\exp(2\pi i z)</math> (the square of the [[nome (mathematics)|nome]]), as:
::<math>f(z)=\sum_{n=-m}^\infty a_n q^n.</math>
This is also referred to as the ''q''-expansion of ''f'' ([[q-expansion principle]]). The coefficients <math>a_n</math> are known as the Fourier coefficients of ''f'', and the number ''m'' is called the order of the pole of ''f'' at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-''n'' coefficients are non-zero, so the ''q''-expansion is bounded below, guaranteeing that it is meromorphic at ''q''&nbsp;=&nbsp;0.&nbsp;<ref group="note">A [[meromorphic]] function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a [[Pole (complex analysis)|pole]] at ''q''&nbsp;=&nbsp;0, not an [[essential singularity]] as exp(1/''q'') has.</ref>

Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that ''f'' be meromorphic in the open upper half-plane and that ''f'' be invariant with respect to a sub-group of the modular group of finite index.<ref>{{Cite book |last1=Chandrasekharan |first1=K. |title=Elliptic functions |publisher=Springer-Verlag |year=1985 |isbn=3-540-15295-4}} p. 15</ref> This is not adhered to in this article.

Another way to phrase the definition of modular functions is to use [[elliptic curve]]s: every lattice Λ determines an [[elliptic curve]] '''C'''/Λ over '''C'''; two lattices determine [[isomorphic]] elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number {{mvar|α}}. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the [[j-invariant]] ''j''(''z'') of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the [[moduli problem|moduli space]] of isomorphism classes of complex elliptic curves.

A modular form ''f'' that vanishes at {{math|''q'' {{=}} 0}} (equivalently, {{math|''a''<sub>0</sub> {{=}} 0}}, also paraphrased as {{math|''z'' {{=}} ''i''∞}}) is called a ''[[cusp form]]'' (''Spitzenform'' in [[German language|German]]). The smallest ''n'' such that {{math|''a<sub>n</sub>'' ≠ 0}} is the order of the zero of ''f'' at {{math|''i''∞}}.

A ''[[modular unit]]'' is a modular function whose poles and zeroes are confined to the cusps.<ref>{{Citation| last1=Kubert | first1=Daniel S. | author1-link=Daniel Kubert | last2=Lang | first2=Serge | author2-link=Serge Lang | title=Modular units | url=https://books.google.com/books?id=BwwzmZjjVdgC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] | isbn=978-0-387-90517-4 |mr=648603 | year=1981 | volume=244 | zbl=0492.12002 | page=24 }}</ref>