%15th draft \documentclass{amsart} %{llncs} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsgen} \usepackage{amsfonts} \usepackage{amscd} \usepackage{hyperref} %\usepackage{makeidx} % allows for index generation \usepackage{url} \usepackage{xspace} \usepackage{fancyvrb} %\usepackage{moreverb} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{axiom}{Axiom} \newtheorem{remark}{Remark} \newcommand{\comment}[1]{} \numberwithin{equation}{section} \def\ppp{{\mathbb{P}}} \def\aaa{{\mathbb{A}}} \def\fff{{\mathbb{F}}} \def\qqq{\mathbb{Q}} \def\rrr{\mathbb{R}} \def\ccc{\mathbb{C}} \def\zzz{\mathbb{Z}} \def\nnn{\mathbb{N}} \def\Aut{{\rm{Aut}}} \def\supp{{\rm{Supp}}} \def\Stab{{\rm{Stab}}} \def\ddiv{{\rm{div}}} \def\Div{{\rm{Div}}} \def\eval{{\rm{eval}}} \def\wt{{\rm{wt}}} \def\pf{\noindent {\bf Proof}:\ } \def\X{C} \def\Aut{\mbox{Aut} } \def\C{\mathbb C} \def\a{\alpha} \def\<{\langle} \def\>{\rangle} \def\e{\xi} \newcommand{\SAGE}{{\sf SAGE}\,} \newcommand{\sage}{\SAGE} \begin{document} \title{Bases for Riemann-Roch spaces of cyclic curves} \author{D. Joyner} \address{Mathematics Department, United States Naval Academy, Annapolis, MD 21402} \email{wdj@usna.edu} \author{A. Ksir} \address{Mathematics Department, United States Naval Academy, Annapolis, MD 21402} \email{ksir@usna.edu} \author{T. Shaska} \address{546 Science and Engineering Building, Department of Mathematics, Oakland University, Rochester, MI, 48309} \email{shaska@oakland.edu} \begin{abstract} Let $C$ denote an algebraic curve with equation $y^n=f(x)$ and $D$ an effective divisor on $C$. We give an explicit constructive method for computing a basis $B$ of $L(D)$. In particular, it is done in such a way that if $E3$. Then the discriminant of $h(x)$ is non-zero. We denote by $n$ be the projective degree of the curve. Let $\iota:C\rightarrow C$ denote the order $m$ normal automorphism such that $C_m = \< \iota \>$, sending a point $P=(x,y)\in C$ to $\iota(P)=P^*=(x, \e_m y)$, where $\e_m$ is the $m$-th primitive root of unity. Let $\rho:C\rightarrow \ppp^1= C / C^\iota$ denote the projection defined in affine coordinates by $(x,y)\longmapsto x$, so $\rho^{-1}(\rho(P))=\{P,\iota(P)\}$. A point $P$ on $C$ is ramified with respect to $\rho$ if and only if it is a fixed point of $\iota$. When we say ``ramification point'', we always mean ramification point with respect to $\rho$, and when we say a point is ``unramified'', we always mean that it is not a ramification point with respect to $\rho$. %%%%%%%% ketu If $h(x)$ has odd degree, then there is one ramified point at infinity, which we label $P_0$; if $h(x)$ has even degree then there are two unramified points at infinity, which we label $P_0$ and $P_0^*$. For simplicity of notation, let $P_0^*=P_0$ in the odd degree case. The following result summarizes some of what is known about the function field $F(C)$ and the affine coordinate ring $F[C]=F[x,y]/(y^m-h(x))$. \begin{proposition} \begin{itemize} \item[(a)] $F[C]$ is finitely generated as an $F[x]$-module and is integral over $F[x]$. Moreover, as an $F$-vector space \[ F(C) = F(x)\oplus y F(x). \] \item[(b)] $F[C]$ is finitely generated as an $F[y]$-module and is integral over $F[y]$. Moreover, as an $F$-vector space \[ F(C) = F(y)\oplus x F(y)\oplus \dots \oplus x^{d-1} F(y). \] \item[(c)] If $d$ is odd, the pole $x_0$ of $y \in F(y)$ has a unique extension $P_0$ to a totally ramified place of degree 1 of $C$ with ramification index $e(P_0/x_0)=d$. Moreover, \[ \ddiv_\infty(x)=2P_0,\ \ \ \ \ddiv_{P_0}(y)=dP_0. \] \item[(d)] If $d$ is even, the pole $x_0$ of $y \in F(y)$ has two extensions $P_0,P_0^*$ to unramified places of degree 1 of $C$ with ramification indices $e(P_0^*/x_0)=e(P_0/x_0)=d/2$. Moreover, \[ \ddiv_{\infty}(x)=P_0+P_0^*,\ \ \ \ \ddiv_{\infty}(y)=(d/2)P_0+(d/2)P_0^* . \] \item[(e)] If $d$ is odd, \[ L(rP_0)=Span\{x^iy^j\ |\ 2i+dj\leq r,\ 0\geq i,\ 0\geq j\geq d-1\}. \] \end{itemize} \end{proposition} \section{Functions with prescribed poles on cyclic curves }\label{sec:poles} We use the notation of the previous section. Let ${\mathcal P}=\{x_0,x_1,...,x_r,x_{r+1},...,x_{r+s}\}\subset \ppp^1$ denote a finite set, where $x_{r+1},...,x_{r+s}$ are branch points of $\rho$, $x_{1},...,x_{r}$ are not branch points, and $x_0=\infty\in \ppp^1$. For each $p\in \rho^{-1}({\mathcal P})$, let $e_p\geq 0$ be an integer. Let $D$ be the effective divisor on $C$ defined by \begin{equation*} D = \sum_{x\in {\mathcal P}} \sum_{p\in \rho^{-1}(x)} e_p\cdot p. \end{equation*} If $x = x_i\in {\mathcal P}$ and if $\rho^{-1}(x)$ contains two points, let $e_i^*= \min_{p \in \rho^{-1}(x)}e_p$ and let $e_i= \max_{p \in \rho^{-1}(x)}e_p$. If $\rho^{-1}(x_i)=\{p\}$ is a singleton, let $e_i^*=0$ and $e_i=e_p$. In this case, we have \begin{equation*} \begin{array}{c} D= e_0P_0+e_1P_1...+e_{r}P_{r} \ +e^*_0P^*_0\, +e_1^*P_1^*...+e^*_{r}P^*_{r}\\ \ \ \ +e_{r+1}P_{r+1}+ ...+e_{r+s}P_{r+s} \end{array} \end{equation*} where $P_1, \ldots, P_r$ are finite unramified points, $e_i \geq e_i^*$ for $i = 0, \ldots r$, and $P_{r+1} , \ldots P_{r+s}$ are finite ramified points. (Recall $e_0^*=0$ if $d$ is odd.) Let $(x_i, y_i)$ be the coordinates of $P_i$, $1 \leq i \leq r$. Note if $1 \leq i \leq r$ then $(x_i, -y_i)$ are the coordinates of $P_i^*$, and if $i > r$ then $y_i=0$. \begin{proposition} \label{prop:main} Let $C$ be as above. Let $D$ be an effective divisor on $C$ as above with the following properties: \begin{enumerate} \item $y_1, \ldots, y_r$ are all distinct \item $e_i \geq e_i^*$ for $i = 0, \ldots r$ \item $e_1 + \ldots + e_r \geq d/2$ \item $e_{r+1}, \ldots, e_{r+s}$, and $e_0$ if $d$ is odd, are either all even or all odd. \end{enumerate} Then there is a function $g$ with polar divisor $D$ which is a linear combination of functions of the form: (a) $\frac{y}{\prod_{i=0}^{s-1}(x-x_i)^{e_i} } $, (b) $\frac{1}{(x-x_i)^{m} }$ ($0 e_i^*$ for at least one unramified point $P_i$, let \[ a= \left\{ \begin{array}{ll} \sum_{i=1}^r e_i + \frac{e_0-d}{2}, & \ \ d \text{ odd and } e_0 \text{ even,}\\ \ &\ \\ \sum_{i=1}^r e_i + \frac{e_0+1-d}{2}, & \ \ d \text{ odd and } e_0 \text{ odd,}\\ \ &\ \\ \sum_{i=0}^r e_i - \frac{d}{2}, & \ \ d \text{ even,}\\ \end{array} \right. \] and let $p(x)$ be a polynomial of degree $a$ in $x$ with no roots among the $x$-coordinates of the points in the support of $D$. Let \[ \beta_i(x)= \left\{ \begin{array}{ll} (x-x_i)^{e_i}, &\ \ 1\leq i\leq r,\\ \ &\ \\ (x-x_i)^{e_i/2}, &\ \ i\geq r,\ e_i\ {\rm even},\\ \ &\ \\ (x-x_i)^{(e_i+1)/2}, &\ \ i\geq r,\ e_i\ {\rm odd.} \end{array} \right. \] We define a \textbf{seed function} as follows. Choose an initial index $i_0>0$ such that $P_{i_0}$ is an affine unramified point on $C$ and $e_{i_0} > e_{i_0}^*$\footnote{For future reference, by analogy with Gauss elimination/row reduction, we call the point $P^*_{i_0}$ a {\bf pivot point}.}. Let \[ f_{0,D }(x,y) =\frac{(y+y_{i_0})p(x)}{\prod_{i=1}^{r+s}\beta_i(x) }. \] Because of this, if $e_i$ is even for all ramified $P_i$ then $\ddiv_\infty (f_{0,D}(x,y))$ is \begin{eqnarray*} \ddiv_\infty (f_{0,D}(x,y)) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}-1) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} e_{r+i}P_{r+i}. \end{eqnarray*} If $e_i$ is odd for all ramified $P_i$, then $\ddiv_\infty (f_{0,D}(x,y))$ is \begin{eqnarray*} \ddiv_\infty (f_{0,D}(x,y)) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}-1) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} (e_{r+i}+1)P_{r+i}. \end{eqnarray*} Now if $e_{i_0}^* < e_{i_0} -1$, we will \textbf{pivot} until the pole at $P_{i_0}^*$ is $e_{i_0}^*$. Observe that the function \[ f_{i,m}(x,y) =\frac{1}{(x-x_i)^{m}} \] has \[ \ddiv_\infty (f_{i,m}(x,y)) = m P_i+m P^*_i. \] We call the $f_{i_0,m}$ the {\bf pivot functions} ($m>0$). It follows from an expansion in terms of a local coordinate or from the usual partial fraction decomposition of rational functions (see for example \S 5.11 in \cite{GG}) that there is a constant $c\not= 0$ such that the divisor of $g=f_{0,D}-cf_{i_0,e_{i_0}-1}$ is \begin{eqnarray*} \ddiv_\infty (g) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}-2) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} e_{r+i}P_{r+i}. \end{eqnarray*} if $e_{r+i}$ is even, or \begin{eqnarray*} \ddiv_\infty (g) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}-2) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} (e_{r+i}+1)P_{r+i}. \end{eqnarray*} if $e_{r+i}$ is odd. Thus we have reduced the multiplicity of the pole at $P^*_{i_0}$ by one. The constant $c$ is determined by the following calculation: choose a local parameter $t$ such that $t=0$ when $(x,y)=P^*_{i_0}$; for example $t=x-x_{i_0}$. Expand $f_{0,D}$ and $f_{i_0,e_{i_0}-1}$ as power series in $t$, so $f_{0,D}(x,y)=a\cdot t^{-e_{i_0}+1}+$ higher order terms, and $f_{i_0,e_{i_0}-1}(x,y)=b\cdot t^{-e_{i_0}+1}+...$ then $c=a/b$. Using the other pivot functions, we may compute constants $c_i$ such that the function \[ \begin{array}{ll} f_{1,D}(x,y)\ \ =&f_0(x,y)-c_1f_{i_0,e_{i_0}-1}(x,y)\\ &\ \ -c_2f_{i_0,e_{i_0}-2}(x,y)-...-c_{e_{i_0}}f_{i_0,0}(x,y) \end{array} \] has divisor \begin{eqnarray*} \ddiv_\infty (f_{1,D}(x,y)) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}^*) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} e_{r+i}P_{r+i}, \end{eqnarray*} if $e_{r+i}$ is even, or \begin{eqnarray*} \ddiv_\infty (f_{1,D}(x,y)) & = & e_0 P_0 (+e_0 P^*_0) + e_1 P_1 + e_1 P_1^* + \dots \\ & & +e_{i_0} P_{i_0}+(e_{i_0}^*) P^*_{i_0}+...+e_{r} P_{r}+e_{r} P^*_{r}\\ & & + \sum_{i=1}^{s} (e_{r+i}+1)P_{r+i}, \end{eqnarray*} if $e_{r+i}$ is odd. In other words, we have reduced the order of the pole at the pivot point $P^*_{i_0}$ from $e_{i_0}$ to $e^*_{i_0}$. Now, if there is one, choose a new index $i_1$, $1 \leq i_1 \leq r$, where $P_{i_1}$ is unramified and $e_{i_1}\not= e^*_{i_1}$. Using $f_{1,D}$ and proceeding as above, we may compute $f_{2,D}$, reducing the order of the pole at the new pivot point $P^*_{i_1}$ to $e^*_{i_1}$. Proceeding inductively, in this way we may reduce the orders of all of the poles at the finite unramified pivot points. If $d$ is even, and $e_0^* \not= e_0$, then we may use $P_0^*$ as a pivot point with powers of $x$ as the pivot functions. By this procedure, we find a function which we will call $f_{r,D}$ such that \begin{eqnarray*} \ddiv_\infty (f_{r,D}(x,y)) & = & e_0 P_0 (+e_0^* P^*_0) + e_1 P_1 + e_1^* P_1^* + \dots \\ & & +e_{r} P_{r}+e_{r}^* P^*_{r} + \sum_{i=1}^{s} e_{r+i}P_{r+i}, \end{eqnarray*} if $e_{r+i}$ is even, or \begin{eqnarray*} \ddiv_\infty (f_{r,D}(x,y)) & = & e_0 P_0 (+e_0^* P^*_0) + e_1 P_1 + e_1^* P_1^* + \dots \\ & & +e_{r} P_{r}+e_{r}^* P^*_{r} + \sum_{i=1}^{s}(e_{r+i}+1)P_{r+i}, \end{eqnarray*} if $e_{r+i}$ is odd. In the first case, we are done: $\ddiv_\infty (f_{r,D}(x,y)) = D$. In the second case, we pivot once at each of the ramified points $P_{r+1}, \ldots, P_{r+s}$ using the pivot function $f_{r+i,\frac{e_{r+i}+1}{2}}$ to remove the one excess pole. Summarizing, we have constructed an element $g_D\in L(D)$ with pole divisor $D$, as desired. \qed \section{Bases of Riemann-Roch spaces for cyclic curves}\label{sec:bases} Let $C$ be a smooth hyperelliptic curve $y^2=h(x)$, as in section \ref{sec:poles} and % \[ D =e_0P_0+e_1P_1...+e_{r}P_{r} (+e^*_0P^*_0)+e_1^*P_1^*... +e^*_{r}P^*_{r}(+e_{r+1}P_{r+1}) \] % be a divisor on $C$, where as in section \ref{sec:poles}, $P_1, \ldots, P_r$ are finite unramified points, and $e_i \geq e_i^*$ for $i = 0, \ldots , r$. In this section we only consider divisors $D$ with at most one ramified point in the support of $D$. Thus if the degree $d$ of $h(x)$ is even, we may have one finite ramified point $P_{r+1}$ in the support of $D$; otherwise if $d$ is odd, omit both terms in parentheses in the above expression. If $D$ satisfies the conditions of Proposition~\ref{prop:main}, then $D$ is non-special, so by the Riemann-Roch theorem, we have \begin{equation*} \dim L(D) = \deg(D) + 1 - g. \end{equation*} \begin{theorem}\label{thrm:main2} Let $C$ and $D$ be as above, and assume that $D$ satisfies the conditions of Proposition \ref{prop:main}. Then there is a basis of $L(D)$ consisting of functions which are linear combinations of (a) $\frac{y}{\prod_{i=0}^{s-1}(x-x_i)^{f_i} } $, (b) $\frac{1}{(x-x_i)^{m} }$ ($0