integrating a character of a non-archimedean local field

2011-06-02T16:42:20Z

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.

integrating a character of a non-archimedean local field - MathOverflow

integrating a character of a non-archimedean local field