Eichler–Shimura congruence relation
{{Short description|Theorem in number theory}}
{{distinguish|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 {{harvs|txt|last=Eichler|year=1954|authorlink=Martin Eichler}} and generalized by {{harvs|txt|last=Shimura|year=1958|authorlink=Goro Shimura}}. 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.
References
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- {{Citation | last1=Eichler | first1=Martin | authorlink=Martin Eichler | title=Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion | doi=10.1007/BF01898377 |mr=0063406 | year=1954 | journal=Archiv für mathematische Logik und Grundlagenforschung | issn=0003-9268 | volume=5 | issue=4–6 | pages=355–366| s2cid=119801181 }}
- {{cite book |first=Ilya |last=Piatetski-Shapiro |authorlink=Ilya Piatetski-Shapiro |chapter=Zeta functions of modular curves |title=Modular functions of one variable II |year=1972 |location=Antwerp |series=Lecture Notes in Mathematics |volume=349 |pages=317–360}}
- {{Citation | last1=Shimura | first1=Goro | authorlink=Goro Shimura |title=Correspondances modulaires et les fonctions ζ de courbes algébriques | doi=10.2969/jmsj/01010001 |s2cid= 119360118 |mr=0095173 | year=1958 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=10 | pages=1–28| doi-access=free }}
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. of Math. Soc. of Japan, 11, 1971
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