Eichler–Shimura congruence relation should not be confused with Eichler–Shimura isomorphism.
In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was introduced by and generalized by . Roughly speaking, it says that the correspondence on the modular curve inducing the Hecke operator Tp is congruent mod p to the sum of the Frobenius map Frob and its transpose Ver. In other words,
Tp = Frob + Ver as endomorphisms of the Jacobian J0(N)Fp of the modular curve X0(N) over the finite field Fp.
The Eichler–Shimura congruence relation and its generalizations to Shimura varieties play a pivotal role in the Langlands program, by identifying a part of the Hasse–Weil zeta function of a modular curve or a more general modular variety, with the product of Mellin transforms of weight 2 modular forms or a product of analogous automorphic L-functions.
. Ilya Piatetski-Shapiro . Zeta functions of modular curves . Modular functions of one variable II . 1972 . Antwerp . Lecture Notes in Mathematics . 349 . 317–360.