Hecke operator

Let be the vector space of (entire) modular forms of weight , see Modular form or [a1]. Then the Hecke operator is defined for by

(a1)

where , the upper half-plane. One (easily) proves that if .

If , , is the Fourier expansion of , then

with

Note that

so that, in particular, the commute.

The discriminant form

where is the Ramanujan function, is a simultaneous eigenfunction of all .

Formula (a1) can be regarded as coming from an operation on lattices in the complex plane, , where the sum is over all sublattices of of index . This geometric definition, [a4], makes (a1) easier to understand.

There are Hecke operators in much more general settings, e.g. for suitable subgroups of the modular group . A quite abstract group setting follows, [a6].

Let be a group and a subgroup. Another subgroup is commensurable with if is of finite index in both and . Let . This is a subgroup of that contains .

Now, let be the -module of all formal sums , i.e. the free Abelian group on the double cosets of in . There is an associative multiplication on , defined as follows. Let , . Then the product is clearly a (disjoint) union of double cosets. It gives a product , provided multiplicities are taken into account. More precisely, let , . Then

Now, let . Then . (The restriction of the to is needed to keep things, e.g. the sets , , finite.)

Let be a subset of containing and multiplicatively closed. Then one defines as the submodule of spanned by the for . This gives a subring of . Finally, one defines , the Hecke algebra of as .

In many situations the double cosets act on forms, functions, etc., which gives Hecke operators. See [a5] for an example in the case of double cosets with respect to the principal congruence subgroup

which gives rise to the (usual) Hecke operators for modular forms.

In [a6] this setting is used to define Hecke operators for the case of adelic groups.

Modular forms turn up all over mathematics and physics and, hence, so do the Hecke operators. See the references for a variety of uses of them.

References

[a1]	T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976) pp. 120ff
[a2]	N. Hurt, "Quantum chaos and mesoscopic systems" , Kluwer Acad. Publ. (1997) pp. 101; 163ff
[a3]	M.I. Knopp, "Modular functions in analytic number theory" , Markham Publ. (1970)
[a4]	A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) pp. Chap. II
[a5]	R.A. Rankin, "Modular forms and functions" , Cambridge Univ. Press (1977) pp. Chap. 9
[a6]	G. Shimura, "Euler products and Eisenstein series" , Amer. Math. Soc. (1997) pp. Sect. 11
[a7]	A.B. Venkov, "Spectral theory of automorphic functions" , Kluwer Acad. Publ. (1990) pp. 34; 59
[a8]	D. Bump, "Automorphic forms and representations" , Cambridge Univ. Press (1997)
[a9]	N.E. Hurt, "Exponential sums and coding theory. A review" Acta Applic. Math. , 46 (1997) pp. 49–91