Quasi-character

A continuous homomorphism from an Abelian topological group into the multiplicative group of complex numbers. In this setting is often the multiplicative group of some local field .

The restriction of a quasi-character to any compact subgroup of is a character of this subgroup (cf. Character of a group). In particular, if is a norm on and , then induces a character of the group , and is, in the non-Archimedean case, the same as the group of units of . If , then the quasi-character is said to be non-ramified. Any non-ramified quasi-character has the form

In the general case a quasi-character of the group has the form , where is a complex number and is a character of . The real part of is uniquely determined by the quasi-character and is called the real part of .

In the non-Archimedean case, for each quasi-character there is a positive integer such that

where is the maximal ideal in the ring of integers of . The smallest number with this property is called the degree of ramification of the quasi-character , and the ideal is called the conductor of .

References

[1]	S. Lang, "Algebraic numbers" , Addison-Wesley (1964)
[2]	I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)