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 .[ 1] [ 2]
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 .[ 3] 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 .[ 4] [ 5] 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 ).[ 6]
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 .[ 7] These play a central role in the Langlands correspondence .[ 8]
The derived Hecke algebra is a further generalization of Hecke algebras to derived functors .[ 8] [ 9] [ 10] It was introduced by Peter Schneider in 2015 who, together with Rachel Ollivier , used them to study the p -adic Langlands correspondence.[ 8] [ 9] [ 10] [ 11] It is the subject of several conjectures on the cohomology of arithmetic groups by Akshay Venkatesh and his collaborators.[ 8] [ 10] [ 12] [ 13] [ 14]
See also
Notes
^ Serre 1973 , Ch. VII, § 5. Corollary 2.
^ Bump 1997 , Theorem 1.4.2, p. 45.
^ Bump 1997 , Theorem 1.4.3, p. 46.
^ Serre 1973 , Ch. VII, § 5. Corollary 3.
^ Bump 1997 , §1.4, pp. 47–49.
^ Bump 1997 , §1.4, p. 49.
^ Bump 1997 , §2.2, p. 162.
^ a b c d Feng, Tony; Harris, Michael (2024). "Derived structures in the Langlands correspondence". arXiv :2409.03035 [math.NT ].
^ a b Schneider, Peter (2015). "Smooth representations and Hecke modules in characteristic p". Pacific Journal of Mathematics . 279 (1): 447– 464. doi :10.2140/pjm.2015.279.447 . ISSN 0030-8730 .
^ a b c Venkatesh, Akshay (2019). "Derived Hecke algebra and cohomology of arithmetic groups". Forum of Mathematics, Pi . 7 . arXiv :1608.07234 . doi :10.1017/fmp.2019.6 . ISSN 2050-5086 .
^ Rachel, Ollivier; Schneider, Peter (2019). "The modular pro-p Iwahori-Hecke Ext-algebra" (PDF) . In Aizenbud, Avraham; Gourevitch, Dmitry; Kazhdan, David ; Lapid, Erez M. (eds.). Representations of Reductive Groups . Proceedings of Symposia in Pure Mathematics . Vol. 101. American Mathematical Society . arXiv :1807.10232 . doi :10.1090/pspum/101 . ISBN 978-1-4704-4284-2 .
^ Galatius, Søren ; Venkatesh, Akshay (2018). "Derived Galois deformation rings". Advances in Mathematics . 327 : 470– 623. doi :10.1016/j.aim.2017.08.016 . ISSN 0001-8708 .
^ Prasanna, Kartik; Venkatesh, Akshay (2021). "Automorphic cohomology, motivic cohomology, and the adjoint L -function" . Astérisque . 428 . ISBN 978-2-85629-943-2 .
^ Darmon, Henri ; Harris, Michael ; Rotger, Victor; Venkatesh, Akshay (2022). "The Derived Hecke Algebra for Dihedral Weight One Forms". Michigan Mathematical Journal . 72 : 145– 207. arXiv :2207.01304 . doi :10.1307/mmj/20217221 . ISSN 0026-2285 .
References