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I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions.

The notion of conductor is standard for characters $\chi$ of a $p$-adic number field $F=F_p$, and is defined as the least $r$ such that $\chi$ is trivial on $1+p^r\mathcal{O}_p$. Since characters of $GL(2, F)$ are $\chi \circ \det$ for characters $\chi$ on $F$, it is enough to define conductors for characters.

I am wondering about the analogous definitions for $PGL(2, F)$. Characters on $PGL(2, F)$ can be seen as characters on $GL(2, F)$ trivial on the center. Do we know their form, as $\chi \circ \det$ for specific kind of $\chi$ (trivial on squares?). What is the relation between the conductor of $\chi \circ \det$ and the one of $\chi$ in this setting? Is there a group such that $F^2(1+p^r\mathcal{O}_p)$ such that a character trivial on it is a character of $PGL(2)$ of conductor less than $r$?

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  • $\begingroup$ I'm curious about the notation $PGL(é)$: what does it mean? $\endgroup$
    – Qfwfq
    Commented Feb 27, 2018 at 16:47
  • $\begingroup$ @Qfwfq That I made a typo, now corrected :) $\endgroup$
    – Desiderius Severus
    Commented Feb 27, 2018 at 16:47
  • $\begingroup$ Ah okay, I suspected that, but.. who knows! :) $\endgroup$
    – Qfwfq
    Commented Feb 27, 2018 at 18:39
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    $\begingroup$ Somewhat related: mathoverflow.net/questions/292424/… $\endgroup$
    – D_S
    Commented Feb 27, 2018 at 21:30
  • $\begingroup$ I think also "a group such that $F^2(1 + p^r\mathscr O_p)$" is meant to be "a group such as $(F^\times)^2(1 + p^r\mathscr O_p)$". $\endgroup$
    – LSpice
    Commented Feb 28, 2018 at 21:01

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