View source for Quasi-character

A continuous [[Homomorphism|homomorphism]] from an Abelian [[Topological group|topological group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763901.png" /> into the multiplicative group of complex numbers. In this setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763902.png" /> is often the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763903.png" /> of some local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763904.png" />.

The restriction of a quasi-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763905.png" /> to any compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763906.png" /> is a character of this subgroup (cf. [[Character of a group|Character of a group]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763907.png" /> is a norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q0763909.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639010.png" /> induces a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639011.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639013.png" /> is, in the non-Archimedean case, the same as the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639015.png" />, then the quasi-character is said to be non-ramified. Any non-ramified quasi-character has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639016.png" /></td> </tr></table>

In the general case a quasi-character of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639017.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639019.png" /> is a complex number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639020.png" /> is a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639021.png" />. The real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639022.png" /> is uniquely determined by the quasi-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639023.png" /> and is called the real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639024.png" />.

In the non-Archimedean case, for each quasi-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639025.png" /> there is a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639026.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639028.png" /> is the [[Maximal ideal|maximal ideal]] in the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639029.png" />. The smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639030.png" /> with this property is called the degree of ramification of the quasi-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639031.png" />, and the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639032.png" /> is called the conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076390/q07639033.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebraic numbers" , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "The zeta-function" , Moscow  (1969)  (In Russian)</TD></TR></table>