In mathematics, a modular form is a (complex) analytic function on the upper half-plane,






H





{\displaystyle \,{\mathcal {H}}\,}

, that satisfies:

a kind of functional equation with respect to the group action of the modular group,
and a growth condition.
The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, 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} )}

.
The term "modular form", as a systematic description, is usually attributed to Hecke.
Each modular form is attached to a Galois representation.

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