# integrating a character of a non-archimedean local field

By way of motivation, this computation comes from a proof in Bump's book Automorphic Forms and Representations where he shows that the Weil index of the reduced norm of a four-dimensional central division algebra is $-1$.

Let $F$ be a non-Archimedean local field and $\psi$ a nontrivial smooth additive character of $F$. Choose a uniformizer $\pi \in \mathcal{O}_F$, write $q$ for the cardinality of the residue field, and $| \cdot |$ for the absolute value on $F$, normalized so that $|\pi| = q^{-1}$. Let $dx$ denote the additive Haar measure on $F$ such that $\mathcal{O}_F$ has measure $1$.

Bump claims that if $\pi^r\mathcal{O}_F$ is the conductor of $\psi$, meaning the largest fractional ideal on which $\psi$ is trivial, then we have \begin{equation*} \int_{|x| = q^{-s}} \psi(x) \ dx = \left\{ \begin{array}{rl} q^{-s}(1-q^{-1}) & \text{if } s \geq r, \\ -q^{-r} & \text{if } s = r-1, \\ 0 & \text{if } s < r-1. \end{array} \right. \end{equation*}

And here I get lost.

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Sorry about the broken code: it looks fine to me! Could someone who knows more about Latex on MathOverflow fix this? –  Justin Campbell Jun 2 '11 at 16:46
I would say this is more like an exercise and more appropriate for StackExchange (I personally don't care, but based on other comments I have seen). Here's a hint: first work out the $r=0$ case. Decompose $\mathcal O_F = \cup \varp^r \mathcal O_F^\times$ and compute the volme of $\varpi^r \mathcal O_F^\times$ with respect to the additive measure. –  Kimball Jun 2 '11 at 16:53
Whoops, the $\varp$ should of course be $\varpi$, which is just $\pi$ in your notation. –  Kimball Jun 2 '11 at 16:54
If G cpt gp, psi:G to C^* any group hom, then integral of psi(x) over G is either 0 or vol(G) according to psi nontrivial or trivial. This is all you need. –  Peter McNamara Jun 2 '11 at 17:28
@Peter,Kimball: Thanks, I figured it out from those hints. –  Justin Campbell Jun 2 '11 at 17:41