A Hecke correspondence is a certain correspondence between moduli stacks of bundles. The integral transform induced by a Hecke correspondence is called a Hecke transform. The objects preserves by such a transform up to a tensor multiple are called Hecke eigensheaves.
These are central objects of interest in geometric Langlands duality.
The reference for this is section 1.4 of Calegari13.
Let be the moduli space of pairs where and are -isogenous elliptic curves. This has maps and to the moduli space of elliptic curves , sending to and respectively:
We can think of the Hecke correspondence as the multivalued function . Although this is a multivalued function, upon linearizing (by, say, passing to modular forms i.e. global sections of the sheaf ) we get a legitimate function
where is the -th Hecke operator.
Alternatively, we can embed inside as the graph of the above-mentioned multivalued function. In this way it is more properly seen as a correspondence.
A reference for this is section 3.7 of Frenkel05. A discussion can also be found in Lafforgue18.
Hecke correspondences for moduli stacks of bundles follow roughly the same idea but instead of we have and instead of we have the Hecke stack , which parametrizes pairs of bundles , together with a modification which is an isomorphism between and away from some points:
Edward Frenkel, section 3.7 in Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172)
Frank Calegari, Congruences between modular forms, 2013 pdf
Vincent Lafforgue, Shtukas for reductive groups and Langlands correspondence for function fields, arXiv:1803.03791
Last revised on July 23, 2023 at 10:24:12. See the history of this page for a list of all contributions to it.