Exercise 8.5

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}}

If $\chi^2 
e \varepsilon$, show that $g(\chi)^2 = \chi(2)^{-2}J(\chi,ho)g(\chi^2)$. [Hint: Write out $g(\chi)^2$ explicitly and use Exercise 4.]

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}
Let  $\zeta = e^{2i\pi/p}$.
Using the result of Ex. 8.4, we obtain
\begin{align*}
g(\chi)^2 &= \left(\sum_t \chi(t) \zeta^t ight)\left(\sum_s \chi(s) \zeta^sight)\
&=\sum_{s,t} \chi(t)\chi(s) \zeta^{t+s}\
&=\sum_k\left(\sum_{s+t=k} \chi(t) \chi(s)ight)\zeta^k\
&=\sum_k\left( \sum_t \chi(t(k-t) ight) \zeta^k\
&= \chi(-1)\sum_t \chi(t^2) + \sum_{k
eq0} \chi(k^2/2^2) J(\chi,ho) \zeta^k\
&= \chi(-1)\sum_t \chi^2(t) + \chi(2)^{-2}  J(\chi,ho) \sum_{k
eq0} \chi^2(k)  \zeta^k\
\end{align*}

If $\chi^2 
eq \varepsilon, \sum_t \chi^2(t)=0$, so

$$g(\chi)^2 = \chi(2)^{-2}  J(\chi,ho) g(\chi^2).$$
\end{proof}

Exercise 8.5

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Exercise 8.5

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