Quasi-character

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

The restriction of a quasi-character $c$ to any compact subgroup of $G$ is a character of this subgroup (cf. Character of a group). In particular, if $\Vert\cdot\Vert$ is a norm on $K$ and $U = \{ a \in K^*\ :\ \Vert a \Vert=1 \}$, then $c$ induces a character of the group $U$, and $U$ is, in the non-Archimedean case, the same as the group of units of $K$. If $c(U) = \{1\}$, then the quasi-character is said to be non-ramified. Any non-ramified quasi-character has the form $$ c(a) = \Vert a \Vert^s = e^{s \log \Vert a \Vert} \ . $$ In the general case a quasi-character of the group $K^*$ has the form $c = c_1\Vert a \Vert^s$, where $s$ is a complex number and $c_1$ is a character of $K^*$. The real part of $s$ is uniquely determined by the quasi-character $c$ and is called the real part of $c$.

In the non-Archimedean case, for each quasi-character $c$ there is a positive integer $f$ such that $$ c(1+\mathfrak{m}^f) = 1 $$ where $\mathfrak{m}$ is the maximal ideal in the ring of integers of $K$. The smallest number $f$ with this property is called the ramification degree of the quasi-character $c$, and the ideal $\mathfrak{m}^f$ is called the conductor of $c$.

References

[1]	S. Lang, "Algebraic numbers" , Addison-Wesley (1964) Zbl 0211.38501
[2]	I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)
[a1]	J. Tate, “Fourier analysis in number fields and Hecke’s $\zeta$-functions" (Princeton, 1950), reprinted in Cassels, J.W.S., Fröhlich, A. (edd.) "Algebraic number theory". Academic Press (1967) Zbl 0153.07403

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