Talk:Hecke character

In the section "Definition using ideles" a definition is given and then it is pointed out that the definition is ambiguous. The ambiguity is not resolved so it is unclear which of the two potential definitions of Hecke character are intended. Dodgejoel (talk) 01:14, 21 August 2010 (UTC)


 * In fact, the definition of Hecke character depends on the author. E.g. Neukirch's book defines it as a unitary character (see Definition VII.6.11), but Schappacher's book Periods of Hecke characters follows SGA4½ (p. 208) and considers "quasicharacters". Perhaps this should be made clearer. RobHar (talk) 03:06, 21 August 2010 (UTC)

The definition using ideals is not clear to me. What does this long unbroken sentence mean: "A Hecke character with modulus m is a group homomorphism from Im into the nonzero complex numbers such that on ideals (a) in Pm its value is equal to the value at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all archimedean completions of K where each local component of the homomorphism has the same real part (in the exponent)"? I'm not sure "product of the multiplicative groups of all archimedean completions of K" is the right space. Also, I am confused about the "same real part (in the exponent)". Which exponent? Furthermore, the definition given does not appear to be supported by any of the references. Lang and Narkiewicz give only the idelic version and Neukirch restricts to unitary characters and doesn't mention real places in an infinite part of the modulus. Would it make more sense to first discuss an infinity type, the space R which Neukirch defines, give an explanation of what a continuous character on R must look like. Also, would it make sense to give one of the original definitions of Hecke with an example in a quadratic case, explaining how it is a special case of Hecke's original definition. Goodwill125 (talk) 16:58, 5 December 2010 (UTC)