In number theory and complex analysis, a modular form is a type of function of a complex number variable that possesses a high degree of symmetry, of a certain kind. Similarly to a periodic function of a real variable, a modular form repeats or transforms in a certain way when its argument is subjected to a particular transformation. Unlike an ordinary periodic function, its symmetries include transformations such as replacing a complex number z by −1/z, and the transformation law is not an exact symmetry of the function, but more like the transformation law of a quasiperiodic function: the function picks up an additional factor, depending on the transformation. Modular forms serve as an important bridge between complex analysis, number theory, and geometry. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. More precisely, a modular form is a holomorphic function on the complex upper half-plane that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. A modular form is a special case of an automorphic form, 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 Erich Hecke. The importance of modular forms across multiple fields of mathematics has been humorously represented in a possibly apocryphal quote attributed to Martin Eichler describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.
Definition In general, given a subgroup Γ < SL 2 ( Z ) {\displaystyle \Gamma <{\text{SL}}_{2}(\mathbb {Z} )} of finite index (called an arithmetic group), a modular form of level Γ {\displaystyle \Gamma } and weight k {\displaystyle k} is a holomorphic function f : H → C {\displaystyle f:{\mathcal {H}}\to \mathbb {C} } from the upper half-plane H = { z ∈ C | ℑ z > 0 } {\displaystyle {\mathcal {H}}=\{z\in \mathbb {C} |\Im z>0\}} satisfying the following two conditions:
Automorphy condition: for any γ ∈ Γ {\displaystyle \gamma \in \Gamma } , we have f ( γ ( z ) ) = ( c z + d ) k f ( z ) {\displaystyle f(\gamma (z))=(cz+d)^{k}f(z)} , and Growth condition: for any γ ∈ SL 2 ( Z ) {\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )} , the function ( c z + d ) − k f ( γ ( z ) ) {\displaystyle (cz+d)^{-k}f(\gamma (z))} is bounded for im ( z ) → ∞ {\displaystyle {\text{im}}(z)\to \infty } . In addition, a modular form is called a cusp form if it satisfies the following growth condition:
Cuspidal condition: For any γ ∈ SL 2 ( Z ) {\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )} , we have ( c z + d ) − k f ( γ ( z ) ) → 0 {\displaystyle (cz+d)^{-k}f(\gamma (z))\to 0} as im ( z ) → ∞ {\displaystyle {\text{im}}(z)\to \infty } . Note that γ {\displaystyle \gamma } is a matrix
γ = ( a b c d ) ∈ SL 2 ( Z ) , {\textstyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}_{2}(\mathbb {Z} ),}
identified with the function γ ( z ) = ( a z + b ) / ( c z + d ) {\textstyle \gamma (z)=(az+b)/(cz+d)} . The identification of functions with matrices makes function composition equivalent to matrix multiplication.
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