View source for 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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebraic numbers" , Addison-Wesley  (1964) {{ZBL|0211.38501}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "The zeta-function" , Moscow  (1969)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> 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}}</TD></TR>
</table>

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