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Concept
Modular form
Summary
In mathematics, a modular form is a (complex) analytic function on the upper half-plane 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 .
The term "modular form", as a systematic description, is usually attributed to Hecke.
Each modular form is attached to a Galois representation.
In general, given a subgroup of finite index, called an arithmetic group, a modular form of level and weight is a holomorphic function from the upper half-plane such that two conditions are satisfied:
Automorphy condition: For any there is the equality
Growth condition: For any the function is bounded for
where and the function is identified with the matrix The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication. In addition, it is called a cusp form if it satisfies the following growth condition:
Cuspidal condition: For any the function as
Modular forms can also be interpreted as sections of a specific line bundle on modular varieties. For a modular form of level and weight can be defined as an element ofwhere is a canonical line bundle on the modular curveThe dimensions of these spaces of modular forms can be computed using the Riemann–Roch theorem. The classical modular forms for are sections of a line bundle on the moduli stack of elliptic curves.
A modular function is a function that is invariant with respect to the modular group, but without the condition that f (z) be holomorphic in the upper half-plane (among other requirements).
Official source
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