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Type of character in number theory
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map.
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in C {\displaystyle \mathbb {C} } ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character χ {\displaystyle \chi } is the largest ideal m {\displaystyle {\mathfrak {m}}} such that χ {\displaystyle \chi } is a Hecke character mod m {\displaystyle {\mathfrak {m}}} . Here we say that χ {\displaystyle \chi } is a Hecke character mod m {\displaystyle {\mathfrak {m}}} if χ {\displaystyle \chi } (considered as a character on the idele group) is trivial on the group of finite ideles whose every ν {\displaystyle \nu } -adic component lies in 1 + m O ν {\displaystyle 1+{\mathfrak {m}}O_{\nu }} .
A Größencharakter (often written Grössencharakter, Grossencharacter, etc.), origin of a Hecke character, going back to Hecke, is defined in terms of a character on the group of fractional ideals. For a number field K {\displaystyle K} , let m = m f m ∞ {\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{f}{\mathfrak {m}}_{\infty }} be a K {\displaystyle K} -modulus, with m f {\displaystyle {\mathfrak {m}}_{f}} , the "finite part", being an integral ideal of K {\displaystyle K} and m ∞ {\displaystyle {\mathfrak {m}}_{\infty }} , the "infinite part", being a (formal) product of real places of K {\displaystyle K} . Let I m {\displaystyle I_{\mathfrak {m}}} denote the group of fractional ideals of K {\displaystyle K} relatively prime to m f {\displaystyle {\mathfrak {m}}_{f}} and let P m {\displaystyle P_{\mathfrak {m}}} denote the subgroup of principal fractional ideals ( a ) {\displaystyle (a)} where a {\displaystyle a} is near 1 {\displaystyle 1} at each place of m {\displaystyle {\mathfrak {m}}} in accordance with the multiplicities of its factors. That is, for each finite place ν {\displaystyle \nu } in m f {\displaystyle {\mathfrak {m}}_{f}} , the order o r d ν ( a − 1 ) {\displaystyle ord_{\nu }(a-1)} is at least as large as the exponent for ν {\displaystyle \nu } in m f {\displaystyle {\mathfrak {m}}_{f}} , and a {\displaystyle a} is positive under each real embedding in m ∞ {\displaystyle {\mathfrak {m}}_{\infty }} . A Größencharakter with modulus m {\displaystyle {\mathfrak {m}}} is a group homomorphism from I m {\displaystyle I_{\mathfrak {m}}} into the nonzero complex numbers such that on ideals ( a ) {\displaystyle (a)} in P m {\displaystyle P_{\mathfrak {m}}} its value is equal to the value at a {\displaystyle a} of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of K {\displaystyle K} where each local component of the homomorphism has the same real part (in the exponent). (Here we embed a {\displaystyle a} into the product of Archimedean completions of K {\displaystyle K} using embeddings corresponding to the various Archimedean places on K {\displaystyle K} .) Thus a Größencharakter may be defined on the ray class group modulo m {\displaystyle {\mathfrak {m}}} , which is the quotient I m / P m {\displaystyle I_{\mathfrak {m}}/P_{\mathfrak {m}}} .
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared. The role of the infinite part m∞ is now subsumed under the notion of an infinity-type.
Relationship between Größencharakter and Hecke character[edit]A Hecke character and a Größencharakter are essentially the same notion with a one-to-one correspondence[how?]. The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct L-functions (sometimes referred to as Hecke L-functions)[1] that extend the notion of a Dirichlet L-function from the rationals to other number fields. For a Größencharakter χ, its L-function is defined to be the Dirichlet series
carried out over integral ideals relatively prime to the modulus m {\displaystyle {\mathfrak {m}}} of the Größencharakter. Here N ( I ) {\displaystyle N(I)} denotes the ideal norm. The common real part condition governing the behavior of Größencharakter on the subgroups P m {\displaystyle P_{\mathfrak {m}}} implies these Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these L-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at ' s = 1 {\displaystyle s=1} when the character is trivial. For primitive Größencharakter (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate character.
Consider a character ψ {\displaystyle \psi } of the idele class group, taken to be a map into the unit circle which is 1 on principal ideles and on an exceptional finite set S {\displaystyle S} containing all infinite places. Then ψ {\displaystyle \psi } generates a character χ {\displaystyle \chi } of the ideal group I S {\displaystyle I^{S}} , which is the free abelian group on the prime ideals not in S {\displaystyle S} .[2] Take a uniformising element π {\displaystyle \pi } for each prime p {\displaystyle {\mathfrak {p}}} not in S {\displaystyle S} and define a map Π {\displaystyle \Pi } from I S {\displaystyle I^{S}} to idele classes by mapping each p {\displaystyle {\mathfrak {p}}} to the class of the idele which is π {\displaystyle \pi } in the p {\displaystyle {\mathfrak {p}}} coordinate and 1 {\displaystyle 1} everywhere else. Let χ {\displaystyle \chi } be the composite of Π {\displaystyle \Pi } and ψ {\displaystyle \psi } . Then χ {\displaystyle \chi } is well-defined as a character on the ideal group.[3]
In the opposite direction, given an admissible character χ {\displaystyle \chi } of I S {\displaystyle I^{S}} there corresponds a unique idele class character ψ {\displaystyle \psi } .[4] Here admissible refers to the existence of a modulus m {\displaystyle {\mathfrak {m}}} based on the set S {\displaystyle S} such that the character χ {\displaystyle \chi } evaluates to 1 {\displaystyle 1} on the ideals which are 1 mod m {\displaystyle {\mathfrak {m}}} .[5]
The characters are 'big' in the sense that the infinity-type when present non-trivially means these characters are not of finite order. The finite-order Hecke characters are all, in a sense, accounted for by class field theory: their L-functions are Artin L-functions, as Artin reciprocity shows. But even a field as simple as the Gaussian field has Hecke characters that go beyond finite order in a serious way (see the example below). Later developments in complex multiplication theory indicated that the proper place of the 'big' characters was to provide the Hasse–Weil L-functions for an important class of algebraic varieties (or even motives).
Hecke's original proof of the functional equation for L(s,χ) used an explicit theta-function. John Tate's 1950 Princeton doctoral dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions. A similar theory was independently developed by Kenkichi Iwasawa which was the subject of his 1950 ICM talk. A later reformulation in a Bourbaki seminar by Weil 1966 showed that parts of Tate's proof could be expressed by distribution theory: the space of distributions (for Schwartz–Bruhat test functions) on the adele group of K transforming under the action of the ideles by a given χ has dimension 1.
Algebraic Hecke characters[edit]An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0. Such characters occur in class field theory and the theory of complex multiplication.[6]
Indeed let E be an elliptic curve defined over a number field F with complex multiplication by the imaginary quadratic field K, and suppose that K is contained in F. Then there is an algebraic Hecke character χ for F, with exceptional set S the set of primes of bad reduction of E together with the infinite places. This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial of the Frobenius endomorphism. As a consequence, the Hasse–Weil zeta function for E is a product of two Dirichlet series, for χ and its complex conjugate.[7]
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