BouwWewers.tex

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\begin{document}

\title{Using Bouw-Wewers}
\author{Shahed Sharif}
\maketitle

The purpose of these notes is to compute the $L$-function and conductor of the superelliptic nonsingular curve with affine piece
\[
C: xy^r = (x+1)(x+t)
\]
over $t = 1$ and $t = \infty$.

\section{When $t = \infty$}
\label{sec:when-t-=}

Let $K = \F(t)$, where $\F$ is our finite field of characteristic relatively prime to $r$. Let $L$ be the splitting field of $X^r - t$ over $K$. Write $\tau$ for a fixed $r$th root of $t$. Let $E$ be the maximal subextension of $L/K$ which is unramified at $1/t$; observe that $E = K(\mu_r)$, where $\mu_r$ is the set of $r$th roots of unity. We first compute a semistable model for $C_L$ over $\tau = \infty$; this calculation is precisely that stated by Hall in the proof Lemma 6 of his $\ell$-torsion notes, with some notational changes.

\end{document}