Exercise 10.24

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

In Exercise 23 take $f$ to be a non trivial character $\chi$ and show that $\hat{\chi}(s) = (1/q) g_{-s}(\chi)$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof} By definition,
\begin{align*}
\hat{\chi}(s) &= \frac{1}{q} \sum_{t\in \F_q} \chi(t) \overline{\psi(st)}\
&= \frac{1}{q} \sum_{t\in \F_q} \chi(t) \zeta_p(-\mathrm{tr}(st))\
&= \frac{1}{q} \sum_{t\in \F_q} \chi(t) \zeta_p(\mathrm{tr}((-s)t))\
&=  \frac{1}{q} \,  g_{-s}(\chi).
\end{align*}

\end{proof}

Exercise 10.24

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Exercise 10.24

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