number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, the Eichler–Shimura congruence relation expresses the local ''L''-function of a

modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) 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 grou ...

at a

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'' in terms of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s of

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 repr ...

s. It was introduced by and generalized by . Roughly speaking, it says that the correspondence on the modular curve inducing the

''T''_''p'' is congruent mod ''p'' to the sum of the

Frobenius map In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...

''Frob'' and its transpose ''Ver''. In other words, :''T''_''p'' = ''Frob'' + ''Ver'' as endomorphisms of the Jacobian ''J''₀(''N'')_{F_''p''} of the modular curve ''X''₀''N'' over the finite field F_''p''. The Eichler–Shimura congruence relation and its generalizations to

Shimura varieties In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are no ...

play a pivotal role in the

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

, by identifying a part of the

Hasse–Weil zeta function In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reduc ...

of a modular curve or a more general modular variety, with the product of

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. This integral transform is closely connected to the theory of Dirichlet series, and is often used i ...

s of weight 2

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...

s or a product of analogous automorphic ''L''-functions.

References

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Goro Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...

, ''Introduction to the arithmetic theory of automorphic functions'', Publ. of Math. Soc. of Japan, 11, 1971 {{DEFAULTSORT:Eichler-Shimura congruence relation Modular forms Zeta and L-functions Theorems in number theory