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.
Definition 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 }} .
Größencharakter 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}}}
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