mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Hecke algebra is the

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

generated by

Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , 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 rep ...

s, which are named after

Erich Hecke Erich Hecke (; 20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He ...

Properties

The algebra is a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

. 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 In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...

with respect to the Petersson inner product. Therefore, the

spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...

implies that there is a basis of modular forms that are

eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

s for these Hecke operators. Each of these basic forms possesses an

Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ...

. More precisely, its

Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used ...

is the

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...

that has

s 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. These play a central role in the Langlands correspondence. The derived Hecke algebra is a further generalization of Hecke algebras to derived functors. It was introduced by Peter Schneider in 2015 who, together with Rachel Ollivier, used them to study the ''p''-adic Langlands correspondence. It is the subject of several conjectures on the cohomology of arithmetic groups by Akshay Venkatesh and his collaborators.

Notes

References

* * Abstract algebra Number theory Modular forms {{algebra-stub

Properties

Generalizations

See also

Notes

References