Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology and string theory.

Related Topics:
Mathematics - Analytic function - Upper half plane - Functional equation - Complex analysis - Number theory - Algebraic topology - String theory

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Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups. It was developed, historically speaking, in three or four periods of development: in connection with the theory of elliptic functions, in the first part of the nineteenth century; by Felix Klein and others towards the end of the nineteenth century, as the automorphic form concept was understood (for one variable); by Erich Hecke from about 1925; and in the 1960s, as the needs of number theory and the formulation of the Taniyama-Shimura conjecture in particular made it clear that modular forms are deeply implicated.

Related Topics:
Automorphic form - Discrete group - Elliptic function - Nineteenth century - Felix Klein - Erich Hecke - Taniyama-Shimura conjecture

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The term modular form, as a systematic description, is usually attributed to Hecke. Curiously, G. H. Hardy is said to have banned it, in his circle of students; the deep studies made on the particular cusp form highlighted by Srinivasa Ramanujan often do not use the modern term. A modular function is in practical terms a modular form of weight 0; but to be strictly accurate modular functions are meromorphic functions rather than analytic.

Related Topics:
G. H. Hardy - Cusp form - Srinivasa Ramanujan - Modular function - Meromorphic function

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