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 $X_{0}(p)$ be the moduli space of pairs $(E,E')$ where $E$ and $E'$ are $p$-isogenous elliptic curves. This has maps $\pi_{1}$ and $\pi_{2}$ to the moduli space of elliptic curves $X$, sending $(E,E')$ to $E$ and $E'$ respectively:
We can think of the Hecke correspondence as the multivalued function $\pi_{2 *}\circ\pi_{1}^{*}$. Although this is a multivalued function, upon linearizing (by, say, passing to modular forms i.e. global sections of the sheaf $\omega^{k}$) we get a legitimate function
where $T_{p}$ is the $p$-th Hecke operator.
Alternatively, we can embed $X_{0}(p)$ inside $X\times X$ 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 $X$ we have $Bun_{G}$ and instead of $X_{0}$ we have the Hecke stack $Hck$, which parametrizes pairs of bundles $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ together with a modification which is an isomorphism between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ 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.