Eichler–Shimura congruence relation
Encyclopedia
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 operator
s. 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 and its transpose.
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 transform
s of weight 2 modular form
s or a product of analogous automorphic L-functions.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
at a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p in terms of the eigenvalues of Hecke operator
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
s. Roughly speaking, it says that the correspondence on the modular curve inducing the Hecke operator
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
Tp is congruent mod p to the sum of the Frobenius map and its transpose.
The Eichler–Shimura congruence relation and its generalizations to Shimura varieties
Shimura variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of...
play a pivotal role in the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
, 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 transform
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
s of weight 2 modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s or a product of analogous automorphic L-functions.