Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of 'averaging' operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations). These operators can be realised in a number of contexts; the simplest meaning is combinatorial, namely as taking for a given integer n some function f(Λ) defined on lattices to

Σ f(Λ′)

with the sum taken over all the Λ′ that are subgroups of Λ of index n. For example, with n=2 and two dimensions, there are three such Λ′. Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to enlargement of a lattice; these conditions are preserved by the summation and so Hecke operators take modular forms to modular forms. Algebras of Hecke operators are called Hecke algebras, and the most significant basic fact of the theory is that these are commutative rings.

The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Erich Hecke. The idea may be considered to go back to earlier work of Hurwitz, who treated correspondences between modular curves which realise some individual Hecke operator. In fact the algebraic theory of correspondences (relations closed for the Zariski topology) is another and natural way to express the formal sum involved.

A third way to express Hecke operators is as double cosets in the modular group. In the contemporary adelic approach this translates to double cosets with respect to some compact subgroups. In any case, the presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.

In the classical elliptic modular form theory it is shown that the Hecke operators are a C-star algebra with respect to the Peterson inner product; and that therefore the spectral theory implies that there is a basis of modular forms that are eigenfunctions for all Hecke operators. These basic forms all have Euler products (more precisely, their Mellin transforms are Dirichlet series that have Euler products with the factor for each prime p being of degree 2). In the case treated by Mordell, there is a one-dimensional space of cusp forms of weight 12; so that the Euler product must apply to Ramanujan's form.

References

• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 8.)

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