The Auto Igusa-Zeta function of a plane curve singularity is rational.
Andrew R. Stout
1506.02316.tex
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\begin{document}


\title{The Auto Igusa-Zeta function of a plane curve singularity is rational.}


\author{Andrew R. Stout}
\thanks{Subject Classification: 14H20 \& 14E18. The author was partially supported by a PSC-CUNY Grant, (Tradional A) cycle 46.}





\email{astout@gradcenter.cuny.edu}

\maketitle
%ABSTRACT


\section{Introduction}

The auto Igusa-zeta function $$\zeta_{X,p}(t)\in \grot{\Var{\kappa(p)}}[\bL^{-1}][[t]]$$ was introduced in \cite{Sch2}. In this paper, Schoutens showed that if $X=\bA_{k}^{d}$ where $k$ is an algebraically closed field and $p\in X$ is a closed point, then $$\zeta_{X,p}(t) = \bL^{-\dim(X)}(1-t)^{-1} . $$
In \cite{me}, I easily generalized this to obtain the following:

\begin{proposition}\label{first}
Let $k$ be any field and let $p$ be any point of $X$.
Let $f: X \to Y$ be a morphism of $k$-schemes which is \'{e}tale at $p \in X$, then 
$$\zeta_{X,p}(t) = \zeta_{Y,f(p)}(t) . $$
\end{proposition}

In particular, if $X$ is smooth at $p$, then
$$\zeta_{X,p}(t) = \bL^{-\dim{}_{p}(X)}(1-t)^{-1} \in \grot{\Var{\kappa(p)}}[\bL^{-1}][[t]] , $$
where $\dim{}_{p}(X):=\mbox{krull-dim}(\sO_{X,p})$.
 It is theorized further in loc. cit.  that the auto Igusa-zeta function $\zeta_{X,p}(t)$ is a perfect local invariant -- i.e., we propose the following conjecture 

\begin{conjecture} \label{conj1} Let $k$ be any field and let $X$ and $Y$ be any separated $k$-schemes of finite type.  If there are points $p\in X$ and $q\in Y$ such that $\zeta_{X,p}(t) = \zeta_{Y,q}(t)$, then $(X,p)$ is analytically isomorphic to $(Y,q)$. 
\end{conjecture}

In particular, a weak form of this conjecture\footnote{One also expect that if $[X]\zeta_{X,p}(t)=\Theta_{X,\fl}(t)$, then $X$ is smooth at $p$. Here, $\theta_{X,\fl}(t)$ is the classical motivic Igusa-zeta function.} will be that if $$\zeta_{X,p}(t) = \bL^{-\dim{}_{p}(X)}(1-t)^{-1}, $$ then $X$ is smooth at $p$. We show in \S \ref{proof} that this conjecture is true when $X$ is a plane curve. 

In \cite{me}, I explicitly computed the auto Igusa-zeta function in the following three cases: 1) $X$ is the node $xy=0$, 2) $X$ is the nodal cubic $y^2=x^3-x^2$, and 3) $X$ is the cuspidal cubic $y^2=x^3$. Furthermore, I indicated how to do such computations for more complicated singular points of algebraic curves. In summary, there is computational evidence that: 
\begin{itemize}
\item  Conjecture \ref{conj1} will be true.
\item $[X]\zeta_{X,p}(t)=\Theta_{X,\fl}(t)$ iff $X$ is a smooth.
\item $\zeta_{X,p}(t)$ is a rational function.
\item Every pole of $\zeta_{X,p}(t)$ is also a pole of $\Theta_{X,\fl}(t)$.
\end{itemize}

Indeed, we will show in this paper that each of these bullets will hold when $X$ is a curve on a smooth surface over an algebraically closed field of characteristic zero.
 
 \section{Auto-arc spaces as fibers of generalized arc spaces}

Let $k$ be any field and let $C$ be a variety\footnote{We take this to mean that $C$ is a separated scheme of finite type over $k$ which is also reduced}. Let $p \in C$ with residue field $\kappa(p)$ and consider the $n$-th jet $J_p^nC := \spec{\sO_{C,p}/\sM_p^n}$. For a separated $\kappa(p)$-scheme of finite type, we define $\nabla_{J_p^nC}X$ to be the scheme which represents the functor $\categ{Aff}_{\kappa(p)}\to \categ{Sets}$ defined by 
$$\spec{A} \mapsto \mbox{Mor}_{\sch{\kappa(p)}}(J_p^nC\times_{\kappa(p)} \spec{A}, X).$$
The canonical surjection of rings $\sO_{C,p}/\sM_p^n\onto \kappa(p)$, induces a morphism $\pi_n: \nabla_{J_p^nC}X \to X$.  When $X = J_p^nC$, we reserve the notion $\sA_n(C,p)$ for $ \nabla_{J_p^nC}J_p^nC$ and call it the {\it auto arc space of } $C$ {\it at the point} $p$ {\it at level} $n$. It is a basic example of an auto arc space which were introduced and studied \S 3, \S 8 and Appendix A of \cite{me}.

\begin{lemma}\label{lem1}
Let $C$ be any variety, $p$ a point of $C$, and $\kappa(p)$ the residue field at $p$. 
Let $U$ be the smooth locus of a variety $X$ over $\kappa(p)$, then $\pi_n: \nabla_{J_p^nC}X \to X$ is a piecewise trivial fibration over $U$. 
\end{lemma}

\begin{proof}
By Theorem 4.4 of \cite{Sch2}, $\pi_n^{-1}(U) \cong \nabla_{J_p^nC}U$. Using \'{e}tale charts of $U$, it is straightforward to show $\nabla_{J_p^nC}U \cong U \times \mathbb{A}_{\kappa(p)}^m$ where  $m = \dim{}_p(C)(\ell(J_p^nC)-1).$ 
\end{proof}

\begin{lemma}\label{lem2}
Let $C$ be a be a variety and let  $p$ be a point of $C$. Then, 
$$\pi_n^{-1}(p)^{\red} \cong \sA_n(C,p)^{\red} $$
where $\pi_n: \nabla_{J_p^nC}C \to C$.
\end{lemma}

\begin{proof}
This follows by applying reduction to the isomorphism given by Lemma 4.9 of \cite{Sch2}.
\end{proof}



\begin{lemma} \label{singlemma} Let $C$ be a variety with only one singular point $p\in C$. Then, 
$$[\nabla_{J_p^nC}C] = [C\setminus\{p\}]\bL^{m}+[\sA_n(C,p)]$$
in $\grot{\mathbf{Var}_{k(p)}}$ where $m = \dim{}_p(X)(\ell(J_p^nC)-1)$.
\end{lemma}

\begin{proof}
This is a direct concequence of Lemma \ref{lem1} and Lemma \ref{lem2}.
\end{proof}

The generalized motivic zeta function $\Theta_{C,\fx}(t)$ and the auto Igusa-zeta function\footnote{There it was denoted as $\bar\zeta_{C,p}(t)$ where the bar was written to clarify that it came from applying reduction to a more general type of zeta function.} $\zeta_{C,p}(t)$ were defined in \S 7 and \S 3 of \cite{me}, respectively.

\begin{theorem} Let $J_p^{\infty}C$ denote the formal scheme defined by $\colim_n J_p^nC$ where $C$ is a variety with only one singular point $p$. In this case, the motivic zeta function of $C$ with respect to $J_p^{\infty}C$ decomposes as $$\Theta_{C,J_p^{\infty}C}(t) = \bL^{-\dim{}_p(C)}\frac{[C]-1}{1-t} + \zeta_{C,p}(t)$$
in $\grot{\Var{\kappa(p)}}[\bL^{-1}][[t]]$.
\end{theorem}

\begin{proof}
This follows from the definition of the motivic zeta function (Definition 7.1 of \cite{me}) and Lemma \ref{singlemma}.
\end{proof}

\begin{example}
Let $C$ be the cuspidal cubic defined by $y^2=x^3$. This has one singularity at the origin $O=(0,0)$.
We immediately arrive at
\begin{equation*}
\Theta_{C,J_O^{\infty}C}(t) = \bL^{-1}\frac{[\mathbb{G}_m]}{1-t} + \zeta_{C,O}(t).
\end{equation*}
I showed in \S 7 of \cite{me} that $\zeta_{C,O}(t)$ is rational -- in fact, we arrived at the following explicit formula:
\begin{equation*}
\zeta_{C,O}(t)= \bL^{-1}+\bL t+\bL^2t^2+ \frac{(\bL^7-\bL^6)t^3+\bL^7t^4+\bL^7t^{7}}{(1-\bL t^3)(1-t)}
\end{equation*}
\end{example}


\begin{remark} We expect that $\Theta_{C,\fx}(t)$ is rational for any scheme $C$ and any filtered colimit of fat points $\fx$. In particular, we expect that the auto Igusa-zeta function $\zeta_{C,p}(t)$ is rational and that every pole of of  $\zeta_{C,p}(t)$ is also a pole of the classical motivic zeta function $\Theta_{C,\fl}(t)$ where $\fl = \colim_n \spec{\kappa(p)[t]/(t^n)}$.
\end{remark}

\section{Rationality of the auto Igusa-zeta function of a plane curve.}



In \cite{me}, we used a computational approach to find explicit formulas for the auto Igusa-zeta function of a few algebraic curves and arrived at a basic idea for how the auto Igusa-zeta function behaves for curves. Namely, it appears that finding the expansion of $\zeta_{C,p}(t)$ in terms of the classical motivic zeta function is closely related to its normalization. In this section, we will gain a better picture of why exactly that must be the case.


\begin{assumption} \label{ap}In this section, we work relative to the following set-up.
We let $C$ be a curve on a smooth surface over an algebraically closed field $k$. We let $\gamma: \bar C \to C$ be the normalization of $C$. We fix a point $p$ of $C$, and we assume that the characteristic of $k$ does not divide $\mbox{mult}_{q}(\bar C)$ for all $q \in \gamma^{-1}(p)$. Note that by a curve, we mean that $C$ is a reduced separated $k$-scheme of finite type which has pure dimension $1$. Also, note that the last condition means that $\mbox{char}(k)$ will not divide $\mbox{mult}_p(C)$.
\end{assumption}

\begin{remark}
If we start with a smooth point $p$ of $C$, then $\gamma: \bar C \to C$ will be an isomorphism away from the exceptional locus $\gamma^{-1}(C_{\mbox{sing}})$. In particular, $\hat\sO_{C,p} \cong \hat\sO_{\bar C, q}\cong k[[t]]$. This immediately implies that $J_p^{\infty}C \cong \mbox{Spf}(k[[t]])$ and for any $k$-scheme $X$, $\nabla_{\fl}X = \varprojlim_n\nabla_{J_p^nC}X$ is the usual arc space. Note that 
$\nabla_{\fl}X$ is most commonly denoted by $\sL(X)$  in the literature (cf. \cite{DL1}).  In this paper, we are mostly concerned with the case when $p$ is a singular point of $C$.
\end{remark}


\subsection{Preliminaries on the normalization of a curve.}

Now, consider its normalization $\gamma: \bar C\to C$. Let $p$ be a point of $C$. For each branch $q \in \gamma^{-1}(p)$, the Newton-Puiseux Theorem states that given local coordinates $(x,y)$ of the point $q$, we have
\begin{equation*}
\begin{split}
x &= t^r \\
y &= \sum_{i=1}^{\infty} a_{i}t^i
\end{split}
\end{equation*}
with $a_{i}\in k$. In fact, if we let $f(x,y)\in k[x,y]$ be such that $f(x,y) = 0$ defines $C$, then the image $F(x,y)$ of $f(x,y)$ in  $k[[x,y]]$ factors as
$F(x,y) = u(x,y) \prod_{j=1}^{s}G_j(x,y)$ where $u(x,y)$ is a unit and the $G_j(x,y)$ are distinct irreducible power series. The power series $G_j(x,y)$ are sometimes also referred to as the branches of $C$. In fact, using the Newton-Puiseux Theorem\footnote{This is where our assumptions that $k$ is algebraically closed and that $\mbox{char}(k)$ does not divide any $r_j$ are being used.} there is a ring homomorphism $\phi_j : k[[x,y]] \to k[[t]]$ such that
$$\phi_j(G_j(x,y)) = G_j(t^{r_j},\sum_{i=1}^{\infty} a_{i,j}t^i)=0  $$
which therefore induces an embedding $k[[x,y]]/(G_j(x,y))\into k[[t]]$. In other words, if we let $\hat\sO_{\bar C, q}$ be the completion of the local ring $\sO_{\bar C, q}$ with $q\in\gamma^{-1}(p)$ 
and let $\hat\sO_{C,p}$ be the completion of the local ring $\sO_{C,p}$, which is isomorphic to $k[[x,y]]/(F(x,y))$,  then it follows that there is an embedding of rings $$\hat \sO_{C,p} \into \hat\sO_{\bar C, q}$$ making $\hat \sO_{\bar C, q}$ a finite $\hat\sO_{C,p}$-algebra. 
 
 
In particular, if $C$ is unibranched (meaning that $s=1$), then 
 $\hat\sO_{C, p}$ embeds into $k[[t]]$ as a subring and this ring embedding is induced by $\phi_1$.  Moreover, if $s\geq 1$, we have an embedding 
 $\phi : \hat\sO_{C, p} \into \prod_{j=1}^{s} k[[t]]$ where $\phi = \prod_{j=1}^{s}\phi_j$. On the geometric side, $\gamma_n :  E_n \to \spec{\sO_{C,p}/\sM_p^n}$ where $E_n = \bar C\times_k  \spec{\sO_{C,p}/\sM_p^n}$. Taking filtered colimits in formal schemes, which is the same as completing the integral closure of $\sO_{C,p}$ with respect to the Jacobson radical, gives a formal scheme $$\hat E \cong \sqcup_{j=1}^{s} \mbox{Spf}(k[[t]])$$ and a surjective morphism $\hat\gamma : \hat E \to \mbox{Spf}(\hat\sO_{C,p})$ such that $\hat \gamma = \mbox{Spf}(\phi)$. The morphism $\hat\gamma$ will be called the {\it uniformization morphism of the analytic germ} $(C,p)$.
 
\begin{example}
Let $r> 1$ and $n> r$ with $n$ and $r$ coprime. Consider the curve defined by $f(x,y) = y^r - x^n = 0$, which has a singular point at the origin $O$. This is a unibranched curve, and the homomorphism $\phi_1 : k[[x,y]] \to k[[t]]$ defined by $\phi_1(x) = t^r$ and $\phi_1(y) = t^n$  induces an embedding $\hat\sO_{C,O} \into k[[t]]$. Moreover, one can show that $\gamma : \bar C \to C$ is bijective as it is given by $t \mapsto (t^r,t^n)$.
\end{example}



\subsection{Case of a unibranched curve.}

In this subsection, we focus entirely on the unibranched case as outlined above -- i.e., we let $C$ be a plane curve with only one singular point, which we may assume without loss of generality occurs at the origin $O$, such that the normalization $\gamma : \bar C \to C$ is bijective. Then, as outlined above, $\hat \sO_{C,O} \cong k[[x,y]]/F(x,y)$ embeds into $\hat\sO_{\bar C, \gamma^{-1}(O)}\cong k[[t]]$ via the morphism $\phi : k[[x,y]] \to k[[t]]$ defined $\phi(x) = t^{r}$ and $\phi(y) = \sum_{i=1}^{\infty}a_it^i$. 

By 1.89 and Theorem 1.96 of \cite{JK}, we have that $r=\mbox{mult}_{O}C$. Thus, without loss of generality, we may assume that $y^r$ occurs as a term in $F(x,y)$ with constant non-zero coefficient.  In other words, if we let $F_r$ be the homogeneous term of $F(x,y)$ of degree $r$, then $F_r$ is also the homogeneous term of $F(x,y)$ of minimal degree. 

The ring homomorphism $\phi: k[[x,y]] \to k[[t]]$ will induce a homomorphism $\phi_n : k[[x,y]]/(x,y)^n \into k[[t]]/(t^{rn})$. Note the only thing to check here is that $y^n \mapsto 0$ under $\phi_n$ but this is indeed the case as the first nonvanishing term of  $\sum_{i=1}^{\infty}a_it^i$ is $a_mt^m$ with $m\geq n$. This follows from the set-up in the preceding paragraph and Corollary 2.2.8 by \cite{EA}. We now have enough to prove the following theorem:

\begin{theorem}\label{unitheorem}
Let $X$ be a $k$-variety and let $(C,p)$ be a pointed unibranched curve satisfying Assumption \ref{ap}. Let $r=\mbox{mult}_p(C)$. Then 
\begin{itemize}
\item[(1)] The generalized arc space $\nabla_{J_p^nC}X$ is a closed subvariety of $\nabla_{\fl_{rn}}X$.
\item[(2)] The natural truncation morphism $\pi_n: \nabla_{J_p^nC}X\to \nabla_{J_p^{n-1}C}X$ is induced by restricting the natural truncation morphism $\nabla_{\fl_{rn}}X\to \nabla_{\fl_{r(n-1)}}X$.
\item[(3)] The projective limit $\nabla_{J_p^{\infty}C}X:=\varprojlim_n \nabla_{J_p^nC}X$ is a closed cylinder of $\nabla_{\fl}X$.
\item[(4)] There is an associated zeta function $Z_r(t)$ to $\nabla_{J_p^{\infty}C}X$.
\item[(5)] If $X$ has a resolution of singularities (e.g., $\mbox{char}(k) = 0$), then $Z_r(t)$ is rational -- i.e., $Z_r(t) \in \grot{\Var{k}}[\bL^{-1}](t)$.
\end{itemize}
\end{theorem}

\begin{proof}
 We have shown that there is a ring embedding $\sO_{C,p}/\sM_p^n \into k[t]/(t^{rn})$ for all $n>0$. This immediately implies $(1)$ and $(2)$. To prove $(3)$, it is enough to work with $k$ points. We see that the image of the ring embedding
 $\phi:\hat\sO_{C,p} \into k[[t]]$ can be described set-theoretically as the collection of arcs $a_0+\sum_{i\geq r} a_i t^i$ generated as a $k$-algebra by $1$, $\phi(x)$, and $\phi(y)$, with $\ord_t(\phi(x))=r$ and $\ord_t(\phi(y))=m$. Let $c$ be the conductor--i.e., the minimal natural number so that $(t^c)k[[t]] \subset \hat\sO_{C,p}$. It follows that for any $n\in \bN$ such that  $nr \geq c$, 
 $$\nabla_{J_p^{\infty}C}X = \pi_{nr}^{-1}(\nabla_{J_p^{n}C}X) . $$
For $(4)$, consider the generating function $$\bL^{-1}P(\bL^{-1}t) = \sum_{n=0}^{\infty}[\pi_{n+1}(\nabla_{J_p^{\infty}C}X)]\bL^{-(n+1)}t^n,$$
where $P(s)$ is the motivic Poincar\'{e} series of $\nabla_{J_p^{\infty}C}X$. This is a rational function  by Theorem 5.4 of \cite{DL2}. Thus, the sum of the $r$th terms of $\bL^{-1}P(\bL^{-1}t)$, which we denote here by $Z_r(t)$, is also rational. 
\end{proof}


 
\begin{corollary}
Let $C$ be a  unibranched curve satisfying Assumption \ref{ap}. Let $r=\mbox{mult}_p(C)$.  Then,
\begin{itemize}
\item[(1)] The auto-arc space $\sA_n(X,p)^{red}$ is a closed subvariety of $\nabla_{\fl_{rn}}C$.
\item[(2)] The natural truncation morphism $\pi_n: \sA_n(C,p)^{\red}\to \sA_{n-1}(C,p)^{\red}$ is induced by restricting the natural truncation morphism $\nabla_{\fl_{rn}}C\to \nabla_{\fl_{r(n-1)}}C$.
\item[(3)] The infinite auto-arc space $\sA(C,p):=\varprojlim_n (\sA_n(C,p))^{\red}$ is a closed cylinder of $\nabla_{\fl}C$.
\item[(4)] There is an associated zeta function $Z_r^{a}(t)$ to $\sA(C,p)$ which is rational -- i.e., $Z_r^{a}(t) \in \grot{\Var{k}}[\bL^{-1}](t)$.
\end{itemize}
\end{corollary}

\begin{proof}
Parts $(1), \ (2),$ and $(3)$ immediately follow from Lemma \ref{singlemma} and Theorem \ref{unitheorem}. Part $(4)$ is also the same except that we let $Z_r^a(t)$ denote the $r$th terms of $\bL^{-1}P(\bL^{-1}t)$ where $P(s)$ is now the motivic Poincar\'{e} series of $\sA(C,p)$.
\end{proof}


Let $C$ be any curve and let $p$ be a point of $C$. The Hilbert-Samuel function $\ell(\sO_{C,p}/\sM_p^n)$ is given by a linear polynomial $P(n) = e_0(C,p)n+e_1(C,p)$ for sufficiently large $n$, where $e_i(C,p)\in\bZ$. The polynomial $P(n)$ is sometimes called the Hilbert-Samuel polynomial of $C$ at $p$, and the number $e_0(C,p)$ is called the Hilbert-Samuel multiplicity of $C$ at $p$. It is well-known (e.g., see the beginning of Chapter $4$, Section $3$ of \cite{WF}) that $e_0(C,p)$ corresponds with the usual geometric notion of multiplicity used above --i.e., we have that $e_0(C,p)=r$ where $r$ is the number used in the work above. This immediately implies the following:

\begin{corollary} \label{stronglyrational}
Let $k$ be an algebraically closed field and let $X$ be a $k$-variety admitting a resolution of singularities.
Let $(C,p)$ be a pointed unibranched curve satisfying Assumption \ref{ap}.
Then,
 $\Theta_{X,J_p^{\infty}C}(t^r) -\bL^{-e_1(C,p)}Z_r(t)$ and $\zeta_{C,p}(t^r) -\bL^{-e_1(C,p)}Z_r^a(t)$ are polynomials (i.e., elements of $\grot{\Var{k}}[t]$) of degree at most $rc$, where $c$ is the conductor of $C$. In particular, $\Theta_{X,J_p^{\infty}C}(t)$ and $\zeta_{C,p}(t)$ are both  rational functions. 
 \end{corollary}

\subsection{The multibranched case.}

Let $X$ be any scheme. We will use the short hand $X^s$ for the fiber product $X\times_k\cdots\times_kX$ ($s$ times). Using the uniformization morphism and some facts about arc spaces, we arrive at the following analogue of Theorem \ref{unitheorem} for the multibranched case.

\begin{theorem} \label{multitheorem}
Let $X$ be a $k$-variety and let $(C,p)$ be a pointed  curve (not necessarily unibranched) satisfying Assumption \ref{ap}. Let $r_i=\mbox{mult}_{q_i}(\bar C)$ where $q_i$ are the branches of $C$ at $p$.  Then,
\begin{itemize}
\item[(1)] The generalized arc space $\nabla_{J_p^nC}X$ is a closed subvariety of $\prod_{i=1}^{s}\nabla_{\fl_{r_in}}X$.
\item[(2)] The natural truncation morphism $\pi_n: \nabla_{J_p^nC}X\to \nabla_{J_p^{n-1}C}X$ is induced by restricting the natural truncation morphism $\prod_{i=1}^{s}\nabla_{\fl_{r_in}}X\to \prod_{i=1}^{s}\nabla_{\fl_{r_i(n-1)}}X$.
\item[(3)] The projective limit $\nabla_{J_p^{\infty}C}X:=\varprojlim_n \nabla_{J_p^nC}X$ is a closed cylinder of $\nabla_{\fl}X^s$.
\item[(4)] There is an associated zeta function $Z_r(t)$ to $\nabla_{J_p^{\infty}C}X$. 
\item[(5)] If $X$ admits a resolution of singularities, then $Z_r(t)$ is rational --i.e., $Z_r(t) \in \grot{\Var{k}}[\bL^{-1}](t)$.
\end{itemize}
\end{theorem}

\begin{proof}
Let $C$ be a curve on a smooth surface and let $p$ be a point on $C$. Let $s$ be the cardinality of $\gamma^{-1}(p)$ where $\gamma: \bar C\to C$ is the normalization. Consider the uniformization homomorphism $\hat\sO_{C,p} \into \sqcup_{i=1}^{s}k[[t]]$ defined by $\sqcup_{i=1}^{s}\phi_i$ with $\phi_i(x) = t^{r_i}$ and $\phi_i(y) = \sum_{j=1}a_{j,i}t^j$. 
This induces the closed embedding of $(1)$ and $(2)$ quickly follows as well.

For $(3)$, note that 
$$\nabla_{\fl}X^s =\nabla_{\fl}(\prod_{i=1}^sX) \cong \prod_{i=1}^{s}\nabla_{\fl}X.$$
 Part $(3)$ follows immediately from the fact that $\varprojlim_n\nabla_{\fl_{r_in}}X \cong \nabla_{\fl}X$, the proof of Theorem \ref{unitheorem}, and the fact that a finite fiber product of closed cylinders is a closed cylinder. The proof for part $(4)$ follows from $(3)$ in exactly the same way as in the proof of Theorem \ref{unitheorem}. 
 
 For part $(4)$ and $(5)$, consider the generating function
 $$\bL^{-1}P(\bL^{-1}t) = \sum_{n=0}^{\infty}[\pi_{n+1}(\nabla_{J_{p}^{\infty}C}X)]\bL^{-(n+1)}t^n$$
 where $P(s)$ is the motivic Poincar\'{e} series of $\nabla_{J_{p}^{\infty}C}X$. This is a rational function by Theorem 5.4 of \cite{DL2} provided $X$ has a resolution of singularities. Thus, the sum of its $r$th terms, which we denote by $Z_r(t)$, is rational. More explicitly, we choose $r= \mbox{mult}_{p}(C) = r_1+\cdots+ r_s$. We have
 $$Z_r(t) = \sum_{n=m}^{\infty}[\nabla_{J_{p}^{(n+1)r}C}X]\bL^{-(n+1)r}t^{nr} + \mbox{lower order terms in }t$$
 for sufficiently large $m$. 
\end{proof}

\begin{remark}
Although we do not make use of it, one may also consider the generating function $$G(t) = \bL^{-s}P_1(\bL^{-1}t)P_2(\bL^{-1}t)\cdots P_s(\bL^{-1}t)$$
 where $P_i(w)$ is the motivic Poincar\'{e} series of $C_i$  where $C_i$ is the image of $\nabla_{J_{p}^{\infty}C}X$ under the natural projection onto the $i$th factor of $\prod_{j}^{s}\nabla_{\fl}X$.  This will also be rational provided $X$ has a resolution of singularities.
 \end{remark}
 
 

\begin{corollary}
Let $(C,p)$ be a pointed curve (not necessarily unibranched) satisfying Assumption \ref{ap}. Let $r_i=\mbox{mult}_{q_i}(\bar C)$ where $q_i$ are the branches of $C$ at $p$.  Then,
\begin{itemize}
\item[(1)] The auto-arc space $\sA_n(X,p)^{red}$ is a closed subvariety of $\prod_{i=1}^{s}\nabla_{\fl_{r_in}}X$.
\item[(2)] The natural truncation morphism $\pi_n: \sA_n(C,p)^{\red}\to \sA_{n-1}(C,p)^{\red}$ is induced by restricting the natural truncation morphism $\prod_{i=1}^{s}\nabla_{\fl_{r_in}}X\to \prod_{i=1}^{s}\nabla_{\fl_{r_i(n-1)}}X$.
\item[(3)] The infinite auto-arc space $\sA(C,p):=\varprojlim_n (\sA_n(C,p))^{\red}$ is a closed cylinder of $\nabla_{\fl}C^s$.
\item[(4)] There is an associated zeta function $Z_r^{a}(t)$ to $\sA(C,p)$ which is rational -- i.e., $Z_r^{a}(t) \in \grot{\Var{k}}[\bL^{-1}](t)$.
\end{itemize}
\end{corollary}

Following the proof of Theorem \ref{multitheorem} and the discussion immediately above Corollary \ref{stronglyrational}, we quickly obtain the multibranched version of Corollary \ref{stronglyrational}:

\begin{corollary}
Let $k$ be an algebraically closed field and let $X$ be a $k$-variety admitting a resolution of singularities.
Let $(C,p)$ be a pointed curve satisfying Assumption \ref{ap}.
Then,
 $\Theta_{X,J_p^{\infty}C}(t^r) -\bL^{-e_1(C,p)}Z_r(t)$ and $\zeta_{C,p}(t^r) -\bL^{-e_1(C,p)}Z_r^a(t)$ are polynomials (i.e., elements of $\grot{\Var{k}}[t]$) of degree at most $rc$, where $c$ is the conductor of $C$. In particular, $\Theta_{X,J_p^{\infty}C}(t)$ and $\zeta_{C,p}(t)$ are both  rational functions. 
 \end{corollary}

 
\subsection{Revisiting the conjecture of \S 1 } \label{proof} We may now quickly make part of Conjecture \ref{conj1} into a theorem:

\begin{theorem}
Let $C$ be a curve on a smooth surface $S$ over an algebraically closed field $k$ of characteristic zero. Then, $C$ is smooth at $p$ if and only if
$$\zeta_{C,p}(t) = \bL^{-1}\frac{1}{1-t}\ .$$
\end{theorem}

\begin{proof} The forward direction is an immediate corollary to Proposition \ref{first}.
The other direction is now (due to the rationality results in this section) also immediate. Indeed, it is impossible that $$\zeta_{C,p}(t)= \bL^{-1}\frac{1}{1-t}$$ for the mere reason that $(1-t)Z_r^a(t)$ will have a pole (in $t=\bL^{-v}$) by the Motivic Monodromy Conjecture, which was proved by Loeser (cf. \cite{L}) in the case of plane curves.
\end{proof}



\begin{thebibliography}{1}

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\newblock Singularities on Plane Curves.
\newblock London Mathematical Society Lecture Note Series (Book 276), Cambridge University Press, 2000.



\bibitem[DL1]{DL1}  J. Denef \& F. Loeser.
\newblock Motivic Igusa Zeta Functions.
\newblock In {\em J.
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\bibitem[DL2]{DL2} J. Denef \& F. Loeser.
\newblock  Germs of arcs of singular varieties and
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\newblock In {\em Inven. Math. 135}, pages 201-232, Springer, 1999.

\bibitem[WF]{WF} W. Fulton.
\newblock Intersection Theory.
\newblock Springer, 2nd edition, 1998.

\bibitem[JK]{JK} J. K\'{a}llor.
\newblock Lectures on Resolution of Singularities. 
\newblock Annals of Mathematics Studies (Book 166), Princeton University Press, 2007.

\bibitem[L]{L} F. Loeser. 
\newblock Fonctions d’Igusa p-adiques et polyn\^{o}mes de Bernstein. 
\newblock Amer. J. Math. 110 (1988), 1-22.

\bibitem[Sch2]{Sch2} H. Schoutens.
\newblock  Schemic Grothendieck Rings II. 
\newblock preprint available at {\it websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SchemicGrothendieckRingPartII.pdf} 2014.

\bibitem[St]{me} A. Stout. 
\newblock On the auto Igusa-zeta function of an algebraic curve. 
\newblock Preprint, arxiv.org (2014)


\end{thebibliography}

\bigskip

\bigskip

\noindent\address{Andrew R. Stout\\
 Dept. of Math.  \& Comp. Sci., \\
Bronx Community College, CP 315\\
2155 University Avenue\\
Bronx, NY 10453}
\end{document}


1506.02316.extracted_bib
[EA]{EA E. Casas-Alvero. \newblock Singularities on Plane Curves. \newblock London Mathematical Society Lecture Note Series (Book 276), Cambridge University Press, 2000.
1506.02316.extracted_bib
[DL1]{DL1 J. Denef \& F. Loeser. \newblock Motivic Igusa Zeta Functions. \newblock In {\em J. Algebraic Geometry 7 no. 3, pages 505 - 537. 1998.
1506.02316.extracted_bib
[DL2]{DL2 J. Denef \& F. Loeser. \newblock Germs of arcs of singular varieties and motivic integration. \newblock In {\em Inven. Math. 135, pages 201-232, Springer, 1999.
1506.02316.extracted_bib
[WF]{WF W. Fulton. \newblock Intersection Theory. \newblock Springer, 2nd edition, 1998.
1506.02316.extracted_bib
[JK]{JK J. K\'{allor. \newblock Lectures on Resolution of Singularities. \newblock Annals of Mathematics Studies (Book 166), Princeton University Press, 2007.
1506.02316.extracted_bib
[L]{L F. Loeser. \newblock Fonctions d’Igusa p-adiques et polyn\^{omes de Bernstein. \newblock Amer. J. Math. 110 (1988), 1-22.
1506.02316.extracted_bib
[Sch2]{Sch2 H. Schoutens. \newblock Schemic Grothendieck Rings II. \newblock preprint available at {websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SchemicGrothendieckRingPartII.pdf 2014.
1506.02316.extracted_bib
[St]{me A. Stout. \newblock On the auto Igusa-zeta function of an algebraic curve. \newblock Preprint, arxiv.org (2014)