Modular form

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a modular form is a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> on the <a href="/facts/Upper_half-plane/7fPqrowr">complex upper half-plane</a>, 
 
 
 
 
 
 H
 
 
 
 
 {\displaystyle {\mathcal {H}}}
 
, that roughly satisfies a <a href="/facts/Functional_equation/4ovU4CAt">functional equation</a> with respect to the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> of the <a href="/facts/Modular_group/wL6l14Hk">modular group</a> and a growth condition. The theory of modular forms has origins in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, with important connections with <a href="/facts/Number_theory/zhoa7zrc">number theory</a>. Modular forms also appear in other areas, such as <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, <a href="/facts/Sphere_packing/NY2FKyMf">sphere packing</a>, and <a href="/facts/String_theory/HUJFIPYH">string theory</a>.
Modular form theory is a special case of the more general theory of <a href="/facts/Automorphic_form/H0Pn7MyP">automorphic forms</a>, which are functions defined on <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a> that transform nicely with respect to the action of certain <a href="/facts/Discrete_subgroup/HIjkpJhS">discrete subgroups</a>, generalizing the example of the modular group 
 
 
 
 
 
 S
 L
 
 
 2
 
 
 (
 
 Z
 
 )
 ⊂
 
 
 S
 L
 
 
 2
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}
 
. Every modular form is attached to a <a href="/facts/Galois_representation/fYDgqs8d">Galois representation</a>.
The term "modular form", as a systematic description, is usually attributed to <a href="/facts/Erich_Hecke/VUPBbT10">Erich Hecke</a>. The importance of modular forms across multiple field of mathematics has been humorously represented in a possibly apocryphal quote attributed to <a href="/facts/Martin_Eichler/hsR857YN">Martin Eichler</a> describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.

Modular form open-in-new

Modular form