Hecke algebra

For other uses, see Hecke algebra (disambiguation).

In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke.

Properties

The algebra is a commutative ring.

In the classical elliptic modular form theory, the Hecke operators T_n with n coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the reciprocal of the Hecke polynomial, a quadratic polynomial in p^−s. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of τ(n).

Generalizations

The classical Hecke algebra has been generalized to other settings, such as the Hecke algebra of a locally compact group and spherical Hecke algebra that arise when modular forms and other automorphic forms are viewed using adelic groups.

See also

• Abstract algebra
• Wiles's proof of Fermat's Last Theorem

References

• Bump, Daniel (1997). Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics. Vol. 55. Cambridge University Press.
• Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. New York: Springer Science+Business Media.