Eichler–Shimura congruence relation

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 Eichler (1954) and generalized by Shimura (1958). 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
Not to be confused with Eichler–Shimura isomorphism.

as endomorphisms of the Jacobian J0(N)Fp of the modular curve X0N 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.

References
  • Eichler, Martin (1954), "Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion", Archiv für mathematische Logik und Grundlagenforschung, 5: 355–366, doi:10.1007/BF01898377, ISSN 0003-9268, MR 0063406
  • Piatetski-Shapiro, Ilya (1972). "Zeta functions of modular curves". Modular functions of one variable II. Lecture Notes in Mathematics. 349. Antwerp. pp. 317–360.
  • Shimura, Goro (1958), "Correspondances modulaires et les fonctions ζ de courbes algébriques", Journal of the Mathematical Society of Japan, 10: 1–28, doi:10.4099/jmath.10.1, ISSN 0025-5645, MR 0095173
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. of Math. Soc. of Japan, 11, 1971
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