\input amstex
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\magnification=1200
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\document
\topmatter

\title
Reduction Theory, Elliptic Solitons and Integrable Systems
\endtitle

\author{Emma Previato}\endauthor
\address
Dept. of Mathematics and Statistics, Boston University, Boston, 
MA 02215, USA
        \endaddress
\email ep\@bu.edu
\endemail % use \@ for the @


\thanks{Research at MSRI is supported in part by NSF grant DMS-9701755.
Moreover, NSF support under Grant DMS-9971966
is gratefully acknowledged.}
\endthanks

\abstract
One goal of this paper is to connect two of the 
thesis topics of Sophie Kowalevski, namely reduction theory
for abelian functions and integrability of a dynamical system.
Part I is a digest of classical and current research
on splittable Jacobians and curves with automorphisms.  
Part II is a collection of applications of the splittable
situation to the explicit integration of Hamiltonian systems
of finite and infinite dimension.  Both fields are full of open
questions, and the last section in each part is a collection of such.
\endabstract

\endtopmatter

\def \bbZ {\Bbb Z}
\def\bbR {\Bbb R}
\def\bbC {\Bbb C}
\def\bbH {\Bbb H}
\def\bbQ {\Bbb Q}
\def\bbN {\Bbb N}
\def\bbZ {\Bbb Z}
\def\bbF {\Bbb F}
\def\bbQ {\Bbb Q}
\def\bbP {\Bbb P}


\document

\specialhead Introduction\endspecialhead

This article will be divided in two parts, each
having to do with work of S. Kowalevski's.
Kowalevski devoted part of her thesis to the study of abelian
surfaces whose function theory reduces to elliptic functions.
She also spent time pursuing the question
of integrability of a rotating body, which had been proposed by
her mentor, K. Weierstrass, but didn't obtain her striking
results until later, when they earned her the Bordin prize.
In the light of more recent discoveries in soliton
equations and completely integrable systems, these two
apparently unrelated areas of her work have come in 
close connection, as was remarked already in [BBME]. 
We give a  survey of classical and recent questions of
splittability for abelian varieties and attempt to connect them
with related questions arising for differential equations.
Indeed, it is an established feature of this area of mathematics
that connections prompt unexpected progress.
In the theory of the KP equation, the question of
reduction becomes that of expressing
KP solutions algebraically by means of elliptic functions. 
The question of characterizing solutions to the KP equation that
are elliptic in the first variable is surprisingly connected
with an important hamiltonian system.
It was solved by Krichever, who
defined an algebraically completely integrable hamiltonian
system parametrizing them all.  This system was identified as a rich
algebro-geometric object by Treibich and Verdier, and it was also
identified as a special case of a (generalized) Hitchin system
by Donagi and Markman.  We describe the system and several generalizations
of it, along with its salient features, including the question
of reduction, where the word has a different meaning than in Part I.
The KP solution reduces to, e.g., a KdV or Boussinesq solution
according to certain flows being stationary,
and the question of non-emptiness of these loci is quite difficult.

I am deeply indebted to the organizers of the Kowalevski meeting,
for an opportunity of mathematical excitement and
also for the moving exposure to details of her life in Russia.
It was a special event, and it is the more special
for me to have been asked to write a survey of problems that
stemmed from her work.  I am also thankful
to my collaborators, J.C. Eilbeck and V.Z. Enolskii,
who asked me to present on their behalf our joint work on splittability
of certain genus-3 varieties of elliptic solitons
(section II-3),
despite being among the participants: it was an honor and a pleasure
for me.

\bigskip
\specialhead Part I: Reduction for abelian integrals\endspecialhead

\head 1. Weierstrass, Kowalevski, Poincar\'e\endhead
Sophie Kowalevski spelled her name thus, when she published her paper
on reduction of abelian integrals [Ko1]. She found a necessary and
sufficient condition for a curve of genus 3 to cover an elliptic
curve. This was one of the three results each of which Weierstrass
``without any hesitation (...) would have accepted (...) as a doctoral
dissertation.'' [Ke, Appendix C] Part of the depth of the work is by reason
of the interplay between algebra and analysis; indeed, the condition
can be stated in several different ways, and several variants of the
question could be posed. Some of these variants are still actively
pursued in genus 3, besides being wide open in higher genus, and the
applications of reduction theory to another area of Kowalevski's work,
integrable systems, are also the object of current research.

\medskip
Weierstrass phrased the problem of reduction of abelian integrals as
follows [Ke, Appendix C]:

Give a condition that an algebraic function $f(x)$ must satisfy if
among the integrals
$$\int F(x,f(x))dx,
$$
where $F$ is a rational function of $x$ and $f(x)$, there exist some
that can be transformed into elliptic integrals. This means that the
curve $X$ whose function field is $\bbC (x,f(x))$ has a holomorphic
differential which is elliptic, or equivalently Jac $X$ contains an
elliptic curve. Weierstrass gave the following answer: the required
condition is the existence of a standard homology basis for $X$
(namely, cycles $\alpha_1,\ldots,\alpha_g,\ \beta_1,\ldots,\beta_g$
whose intersection matrix is $\left[ \matrix 0 & I \\
-I & 0\\
\endmatrix\right]$) such that the corresponding basis of holomorphic
differentials has period matrix $\tau_{ij},\ 1\le i,\ j\le g$, with
$\tau_{12} ={m\over n},\ \tau_{13}=\ldots =\tau_{1g}=0$ for some
integers $m,n$. The algebraic side of this condition is then the
determination of the moduli of the curves that satisfy it;
L. K{\"o}nisberger treated the case $g=2, n=2$ and Kowalevski the case
$g=3,n=2$. In her case, the algebraic condition is that the canonical
model of the curve, namely a plane smooth quartic, should have
bitangents meeting at one point [B, \S76].

\medskip
In order to formulate a more general problem, we briefly recall some
terminology; more details and proofs can be found in a standard text
like [LB].
\definition{Definition} 
(1) An abelian variety $A$ of
dimension $g$ is a complex
torus $\bbC /\Lambda$, where $\Lambda$ is a lattice of rank $2g$,
which is also an algebraic variety. If $L$ is a period matrix for the
torus, namely a $g\times 2g$ complex matrix whose columns span
$\Lambda$, then a principal polarization of $A$ is determined by a
skew-symmetric matrix $P$ in $SL(2g,\bbZ )$ such that $LP {}^tL=0$ and
the hermitian matrix $iL P {}^t\overline{L}$ is positive definite. The
$g\times 2g$ complex matrices arising this way are called Riemann
matrices. 
%Equivalently, there exists a skew symmetric $2g\times 2g$
%rational matrix $P$ such that $LP^tL=0$ and $iLP {}^t\overline{L}$ is
%positive definite.

(2) The algebraic counterpart of (1) is the
following: a principal polarization is an ample line bundle ${\Cal L}$
over $A$, such that the homomorphism
$$\phi_{\Cal L} : A\rightarrow {\text\rm Pic}^0 A\qquad a\mapsto 
T_a^\ast {\Cal L}
\otimes {\Cal L}^{-1}
$$
is an isomorphism. 

The link between (1) and (2) is given by Riemann's
theta function $\vartheta$: if $A$ has a principal polarization, then
$L$ can be normalized to $[I\ \Omega ]$ with $\Omega$ a $g\times g$
symmetric matrix and Im $\Omega$ positive definite, so that for
$z\in\bbC^g$
$$\vartheta (z,\Omega )=\sum_{n\in\bbZ^g}\exp (\pi i {}^tn\Omega n+2\pi
i {}^tzn)
$$
is an analytic function. ${\Cal L}$ corresponds to Riemann's theta
divisor
%\hfil\break 
$\Theta = \{ z|\vartheta (z,\Omega )=0\}$.

(3) Two complex tori $A$ and $A^\prime$ with period matrices $L$ and
$L^\prime$ are said to be isomorphic if  and only if there are
matrices $C\in GL(n,\bbC )$ and $T\in GL(2n,\bbZ)$ such that
$CL=L^\prime T$. Notice that $T$ can be taken to be in $SL(2n,\bbZ)$
by changing $C\mapsto C\cdot \det T$. Then, if $\phi
:A^\prime\rightarrow A$ is the isomorphism, the pull-back of a
principal polarization $P$ gives a principal polarization $P^\prime
=TP {}^tT$ for $A^\prime$.

\enddefinition

\medskip
\remark{Remark}  %{remark 1.1}
By the previous definition then, an abelian variety of dimension one
(also called an elliptic curve) has a unique principal polarization, 
whereas for $g>1$ it may have none -- infinitely
many examples of 2-dimensional abelian varieties that do not are given
in [E3] -- or several, as the following theorem illustrates.

\endremark

\proclaim{Theorem} 
{\rm [L2, Th. 1.5]}  If the abelian variety $A$ has a principal
polarization, then the set of isomorphism classes of principal
polarizations is in one-to-one correspondence with the set of
automorphisms of $A$, which are symmetric w.r.t. any fixed principal
polarization and have positive eigenvalues, modulo the natural action
of Aut$A$.
\endproclaim


In particular, the set of non-isomorphic principal polarizations is
finite, taking care of a question posed by Martens [Mar1].

Before returning to Weierstrass' question let's recall two classical
facts:

\proclaim{Theorem} 
A Jacobian 
{\rm $A={\text Jac}(X)$} 
has a canonical principal polarization
given by the injective homomorphism $H_1(X,\bbZ) \cong
\bbZ^{2g}\hookrightarrow  H^0 (X,\Omega^1)^\ast$ (integration over a
cycle). The corresponding theta divisor is only determined up to
translation, giving isomorphic principally polarized abelian
varieties, and is irreducible, i.e., the Jacobian with its canonical
principal polarization is indecomposable.
\endproclaim

\proclaim {Theorem}
\hbox{\rm (Poincar{\'e}'s Complete Reducibility)} If $X$
is an abelian variety and $Y$ an abelian subvariety of $X$, there
exists an abelian subvariety $Z$ of $X$ such that $Y\cap Z$ is finite
and $Y+Z=X$.
\endproclaim

\remark{Remark}
(1) The statement of this theorem is equivalent to saying that $X$ is
isogenous to the product $Y\times Z$, where an isogeny is a surjective
group homomorphism with finite kernel.

(2) This statement is the algebraic version. Analytically, it could be
stated as follows:


\proclaim{Theorem}
{\rm (Weierstrass-Poincar{\'e}, [Mar2])}.  Let $[E\ Z]$ be a
$g\times 2g$ Riemann matrix admitting reduction: $H\cdot [E\
Z]=\Pi\cdot M$, where $H$ is a maximal-rank $p\times g$ complex
matrix, $\Pi$ is a $p\times 2p$ complex matrix and $M$ is a
maximal-rank $2p\times 2g$ integral matrix, with $1\le p\le g$. Then
there exist matrices $A\in GL(g,\bbC),\ T\in \hbox{\rm Sp}  (2g,\bbC )$ such
that
$$[E\ Z]\cdot T=A\cdot
 \left[
%\vbox{\settabs 20\columns
{{E_1\quad 0\quad Z_1\quad Q}
\atop
{{\ 0 \quad E_2\quad {}^tQ\quad Z_2}}}\right]
$$
%\+ $E_1$ & 0 & $Z_1$ & $Q$\cr
%\+ 0 & $E_2$ & ${}^tQ$ & $Z_2$\cr}\right
%$$
where $Z_1,Z_2$ are $p\times p$ and $(g-p)\times (g-p)$ Riemann
matrices, and $Q$ is a $p\times (g-p)$ rational matrix, whose only
possible non-zero entries occur at the beginning of the main
diagonal.
\endproclaim
%\remark{Remark}
(3) The analytic version of the result has the practical consequence that
Riemann's theta function reduces to theta functions of lower
dimension. Indeed (cf. [Mar2]), for a period matrix $[I\ Z]$ with
$$Z=\left[
%\vbox{\settabs 20\columns
{{Z_1\quad Q}\atop 
{{ {}^tQ \quad Z_2}}}\right]
%\+$Z_1$ &  $Q$\cr
%\+ ${}^tQ$ & $Z_2$\cr}\right]
$$
as above,
$$\vartheta (z; Z)=\sum_{m_2} \vartheta (z_1+Qm_2; Z_1)\vartheta
\left[   %\vbox{\settabs 20\columns
{{D^{-1}\  m_2}\atop 0}\right](Dz_2;DZ_2D)
%\+$D^{-1}$ &  $m_2$\cr
%\+ &\ 0\cr}\right](Dz_2; DZ_2 D)
$$
where $z= \left[ {{z_1}\atop {{z_2}}}\right]$, $D=$
diag$(d^1,\ldots,d^{g-p})$ is an integral matrix such that $QD$ is
integral, we rewrite any integral vector of length $g-p$ as
$n_2=m_2+Dk_2$ with $0\le m_2^j <d^j$, and $\vartheta \left[ {{D^{-1} 
m_2}\atop 0}\right]$ $(Dz_2; DZ_2D)$ is the theta function with
characteristics, defined more generally for real $g$-vectors $r$ and
s, in dimension $g$,
$$\vartheta \left[ {r\atop s}\right] (z;Z)=\sum_m \exp (\pi i
{}^t(m+r)(Z(m+r)+2(z+s))).
$$
\endremark
It is now apparent that we can ask several important types of
reduction questions, with practical applications to solving integrable
systems (of finite and infinite dimension!); that there are advantages
in formulating both the algebraic and the analytic versions of the
questions; and that, even if we decide to restrict our attention to
Jacobians, more general abelian varieties may come into play, as we
will see next.



\head 2. Automorphisms\endhead
By Poincar{\'e}'s complete reducibility, if the abelian variety $A$
contains an abelian subvariety $B$, it is isogenous to a product
$B\times C$ (with $C$ isogenous to $A/B$). One question is, when is
$A$ actually isomorphic to a product of lower-dimensional abelian
varieties? A second question is whether $A$ is isomorphic to a product
of lower-dimensional abelian varieties that admit a principal
polarization; and if so, whether the product polarization induces a
given principal polarization on $A$. These three questions are
actually inequivalent. Let's follow Ries [R] and call a principally
polarized abelian variety ``splittable'' if it is isogenous or
isomorphic to a product of lower-dimensional abelian varieties. One
way, though not the only one, to find splittable Jacobians is to
consider curves with automorphisms.

Hurwitz [Hu2] proved that a curve of genus $g$ can have at most 84$(g-1)$
automorphisms. Klein's quartic, in homogeneous coordinates
$X_0X_1^3+X_1X_2^3+X_0^3X_2=0$, is the only curve of genus 3 for which
the Hurwitz bound is attained. As a consequence, it is actually
possible to calculate the period matrix, a rare situation for a curve
given algebraically ([BT], [LB \S11.7], [RL]), because it is preserved
by many symmetries:
$$Z=\left[
\matrix  %\vbox{\settabs 9\columns
-{1\over 8}+{{3\sqrt 7 i}\over 8} & 
  -{1\over 4}-{{\sqrt 7 i}\over
4}  &
-{3\over 8}+{{\sqrt 7 i}\over 8}\\
-{1\over 4}-{{\sqrt 7 i}\over 4} & 
\ \  {1\over 2}+{{\sqrt 7 i}\over
2} & -{1\over 4} -{{\sqrt 7 i}\over 4}\\
-{3\over 8}+{{\sqrt 7 i}\over 8} & 
 -{1\over 4}-{{\sqrt 7 i}\over
4} &\ \  {7\over 8} +{{3\sqrt 7 i}\over 8}\\
\endmatrix\right] .
$$
As observed in [RL], all entries lie in the field generated (over 
the field $k$ of definition of the curve, $k=\bbQ$, e.g.) by the
character of the representation induced on the differentials of the
first kind by the automorphism group of the curve. But another
interesting phenomenon occurs: Jac$(X)=\bbC^3/\Lambda$, where $\Lambda$
is the lattice corresponding to $[I\ Z]$, is actually isomorphic to
the product of 3 elliptic curves.

Baker [B, \S76] shows that 
$$
\left[
\matrix
1\ 0\ 0 & {{1+i\sqrt 7}\over 4} & 0 &  0\\
 0\ 1\ 0 &\ \  0 & 2\ {{1+i\sqrt 7}\over 4} & 0 \\
0\ 0\ 1 &\ \ 0 &  0 & 2 {{1+i\sqrt 7}\over 4}\\
\endmatrix \right]
$$
is a matrix of abelian integrals that gives an isomorphic abelian
variety. Since Jacobians are indecomposable, however, this choice of
homology basis cannot be canonical, or the matrices would determine
isomorphic polarizations, when a polarization is viewed as a bilinear form:
$H=H((0,x_2),(0,y_2))={}^tx_2(\hbox{\rm Im}Z)^{-1}y_2$ on the complex
structure $x=\Omega x_1+x_2$ on $\bbR^{2n}=\{ (x_1,x_2)\in\bbR^n\times
\bbR^n\}$, according to the dictionary given in [Mu, Ch.II \S4].

The point of this example is to exhibit a Jacobian that is not only
isogenous to a product, as predicted by Poincar\'e's Reducibility, but
actually isomorphic to one. Earle [E1] constructs 1-dimensional
families of Jacobians ${\text{Jac}}(X)$ 
of any even genus $2k$ which are both
isogenous and isomorphic to ${\text{Jac}}(Y_1)\times {\text{Jac}}(Y_2)$, 
with $Y_i$ quotients
of $X$ by an automorphism; $X$ is the hyperelliptic curve given by
$$y^2=(x^n-1)(x^n-t),\ n=2k+1,\ t\in\bbC \backslash \{ 0,1\}.
$$
It is then a natural question whether, for a Jacobian isomorphic to a
product of lower-dimensional tori, these tori ``will necessarily be
Jacobians themselves, or at least will admit principal polarization''
[E3]. Earle replies ``emphatically no'' by constructing perturbations
Jac$(X)$ of the Klein Jacobian and showing that Jac$(X)$
admits to isomorphisms onto the product of two complex tori of
dimensions 1 and 2, and for any such isomorphism the 2-dimensional
torus admits no principal polarization.

Since $X\hookrightarrow$ Jac($X$) projects to each elliptic curve
$E_i$, there exist finite maps $X\rightarrow E_i$; in fact each of
them is obtained by moding out by an automorphism of order 2, as
illustrated in [E2]. First, we connect the actual period matrix of
Jac$(X)$ with Baker's matrix:

\medskip
\qquad $C[I\ Z]=[I\ Z^\prime ]N$

\medskip
\noindent where $Z^\prime =\left[ 
\matrix
\tau & 0 & 0\\
0 & 2\tau & 0\\
0 & 0 & 2\tau\\
\endmatrix \right],\ \ C=\left[ 
\matrix
1 & 0 & -1\\
2\tau -1 & -1 & 0\\
-2\tau & 0 & 1\\
\endmatrix \right]$ and the unimodular matrix
$$N=\left[
\matrix
1 & 0 & -1 & 0 & 0 & -1\\
-1 & -1 & 0 & -1 & 1 & 0\\
0 & 0 & 1 & 1 & -1 & 1\\
0 & 0 & 0 & 1 & 0 & -1\\
1 & 0 & 0 & 0 & -1 & 0\\
-1 & 0 & 0 & 0 & 0 & 1\\
\endmatrix \right]$$
is not symplectic, since the polarization of Jac$(X)$ is
indecomposable. But then the three resulting equations
$$[1\ 0\ -1][I\ Z]=[1\ \tau ]
\left[
\matrix
1 & 0 & -1 & 0 & 0 & -1\\
0 & 0 & 0 & 1 & 0 & -1\\
\endmatrix \right]
$$
$$[2\tau -1\ -1\ 0][I\ Z]=[1\ 2\tau ]
\left[
\matrix
-1 & -1 & 0 & -1 & 1 & 0\\
1 & 0 & 0 & 0 & -1 & 0\\
\endmatrix \right]
$$
$$[-2\tau\ 0\ 1][I\ Z]=[1\ 2\tau ]
\left[
\matrix
0 & 0 & 1 & 1 & -1 & 1\\
-1 & 0 & 0 & 0 & 0 & 1\\
\endmatrix \right]
$$
each describe a map of Jac$(X)$ to the corresponding elliptic curve
$E_i$; the corresponding maps $X\rightarrow E_i$ have degree $a\cdot
d-b\cdot c$ ($=2$, in our case) where $\left[
\matrix
a & b\\
c & d\\
\endmatrix \right]$ is the matrix of the induced map
$H_1(X)\rightarrow H_1(E_i)$ with respect to a canonical homology
basis. Since all elements of order $2$ in Aut$X$ are conjugate, the
three  elliptic curves must be isomorphic, as seen by the equation
$$\tau [1\ 2\tau ]=[1\ \tau ]
\left[
\matrix
0 & -1\\
1 & 1\\
\endmatrix \right].
$$
We note that the elliptic curve $E=E_i$ does not have extra
automorphisms. However, it does have complex multiplication, in fact
Ries [R] proves that $E$ is the only elliptic curve with complex
multiplication by $\tau$ and that End $E=\bbZ [\tau ]$.


Again the interplay of analysis and algebra has a role in some of the
many, still wide-open questions that have been asked about the
automorphism groups of Riemann surfaces. Hurwitz besides finding the
bound $84(g-1)$ proved that it is achieved if and only if the group
can be generated by two elements, $t,u$ such that $t^2=u^3=(tu)^7=1$
[Hu2]. Later it was determined that the maximum possible order of an
automorphism is $2(2g+1)$ (and this is optimal). Since I have not
read the original papers I refer instead to the survey [M] where they
are cited. Call $N(g)$ the maximum order of Aut$X,\ X$ a curve
of genus $g$. Accola [A1] proved that $8(g+1)\le N(g)$ and that
equality obtains infinitely often, and on hyperelliptic curves, however
Maclachlan independently verified the bound on a disjoint family of
curves [Ma]. What seems to be more elusive is an understanding of the
subvariety of moduli space where on particular group acts (or even
its dimension!), except for the fact that the bound $84(g-1)$ can only
be achieved on isolated curves (but  we know now, not a unique one as
is the case for the first two genera, $g=3$ and 7 [M \S7]) despite
the importance of the locus of curves with automorphisms, in view of
the fact that it is the branch locus of the covering map
${\Cal T}_g\rightarrow {\Cal M}_g$ from 
Teichm\"uller space (under the action of the
mapping class  group). As noted in [R], we can view the question of
characterizing the family of principally polarized abelian varieties
with the action of a given group $G$ of polarization-preserving
automorphisms as a Schottky-type problem. For example, Ries proves
that the Klein curve belongs to a 1-parameter family of principally
polarized abelian varieties which
split completely and whose automorphism group contains $SL(2,2^3)$; he
uses the automorphism group and the attendant group ring in End$A$ to
produce a splitting of $A$.

{}Finally Accola [A2], starting with an analytic property of theta
functions, constructs a plane model of the curve whose properties
allow him to prescribe certain automorphisms of the curve and to
calculate the dimension of the locus of such curves in Teichm\"uller
space.

The lessons we learned in this section are that it is rare for a curve
to have automorphisms, and that even if it has many, the question of
splitting its Jacobian is by no means settled, in particular as
regards polarization. 
%Earle's examples of a 2-dimensional family of
%splittable Jacobians in dimension 3 shows that not all splittings come
%from symmetries, since the maximal dimension of curves of genus 3 with
%automorphisms is cf. W. Baily.

Before leaving this section we mention another classical problem,
having to do with ordinary differential equations, where the Klein
curve stars again. We use it as motivation a
sketch  of differential Galois theory which we take from [SU].

The third-order equation
$$H(y)=x^2(x-1)^2
y^{\prime\prime\prime}+(7x-4)x(x-1)y^{\prime\prime}+$$

$$\left( {{72}\over
7}(x^2-x)-{{20}\over 9} (x-1)+{3\over 4}x \right) y^\prime 
+\left(   {{792}\over
{{7^3}}}(x-1) +{5\over 8} +{2\over {{63}}}\right)y
$$
has regular-singular points at $x=0$, 1, $\infty$. Hurwitz [Hu1] proved that
its solutions are algebraic functions defined on Klein's curve. But
more is true. The field generated by a basis of solutions for 
a linear differential equation $L(y)=0$ is
a differential field $E$, called a Picard-Vessiot extension. The
differential Galois group ${\Cal G}(L)$ of $E/\bbC (z)$ consists of all
automorphisms of $E$ that fix $\bbC (x)$ elementwise and commute with
differentiation. The monodromy group is always a subgroup of the
Galois group. In fact, let $c_1,\ldots,c_n$ be the singular points of
$L(y)=0$ (including infinity if it is a singular point) and let $c_0$
be an ordinary point of the equation. Consider these points as lying
on the Riemann sphere $S^2$. Let $\{ y_1,\ldots, y_n\}$ be a
fundamental set of solutions of $L(y)=0$ analytic at $c_0$ and let
$\gamma$ be a closed path in $S^2-\{ c_1,\ldots, c_n\}$ that begins
and ends at $c_0$. One can analytically continue $\{ y_1,\ldots,y_n\}$
along $\gamma$ and get new fundamental solutions $\{
\overline{y}_1,\ldots, \overline{y}_n\}$ analytic at $c_0$. These two
sets must be related via
$(\overline{y}_1,\ldots,\overline{y}_n)^T=M_\gamma (y_1,\ldots,
y_n)^T$ where $M_\gamma\in GL(n,\bbC )$. One can show that $M_\gamma$
depends only on the homotopy class of $\gamma$ and that the map
$\gamma\mapsto M_\gamma$ defines a group homomorphism from
$\pi_1(S^2-\{ c_1,\ldots,c_n\})$ to ${\Cal G}(L)$. The image of this
map depends on the choice of $c_0$ and $\{ y_1,\ldots,y_n\}$ but is
unique up to conjugacy and is called the { monodromy group of}
$L(y)$. In general the image of this group will be a proper subgroup
of ${\Cal G} (L)$ but when $L(y)$ is fuchsian, the Zariski closure of
this group will be the full Galois group ${\Cal G}(L)$. 
Thus, the Galois group of $H(y)$
coincides with the automorphism group of the Klein curve and
Singer and
Ulmer [SU] check this computationally.


\head 3. Poncelet\endhead
As we saw in the previous sections, the reduction problem can be
approached in two different ways: by cover, or by symmetries. The
question of whether a Jacobian is splittable is still more general,
but these are sufficient conditions:


$\bullet$ if a curve $\Gamma$ of genus $h$ covers a curve $X$ of genus
$g$ then Jac$\Gamma$ is splittable, more precisely it is isogenus to
$B\times C$ with $B=$ Jac$(X)$,

$\bullet$ if $\sigma$ is an automorphism of a curve $\Gamma$ then
$\Gamma$ covers $X=\Gamma /\sigma$.


However, the question of whether Jac$\Gamma$ contains an elliptic curve
is equivalent to the fact that $\Gamma$ covers an elliptic curve. For
$h(\Gamma )=2$, several results are known and work is in
progress. Lange [L1] proved that the set ${\Cal M}_2(d)$ of curves of genus 2
that admit a morphism of degree $n$ to an elliptic curve is a
subvariety of the moduli space ${\Cal M}_2$ that has dimension 2 for each $n$
divisible by 2, 3 or  5, and dimension $\le 2$ for any $n\le 2$. He
obtains algebraic equations for this subvariety by defining the cover
explicitly and using the fact that $\Gamma$ covers an elliptic curve
$E$ if and only if the surface $\Gamma\times E$ has a nontrivial
section. However, in [Kuh] Kuhn produces a more combinatorial
construction which shows (by Riemann's existence theorem) that the
dimension of ${\Cal M}_2(d)$ is 2 for every $d$
and from which one could
presumably calculate the number of its connected components (as [L1]
notices, ${\Cal M}_2(2)$ is connected). But a difficult question, which has
arithmetic significance, remains: what is an algebraic equation for a
complementary elliptic curve $J(\Gamma )/E$? Even ignoring the fact
that a complementary curve is not canonical (cf. [Kuh]) and eliminating
all the interesting arithmetic features that occur in finite
characteristic, this question remains challenging. Kuwata [Kuw] had the
idea of interpreting it in the framework of Poncelet's theorem and we
sketch his construction in view of the significance of this theorem in
dynamics, where it can be generalized to hyperelliptic curves of any
genus $g>1$ in place of $E$ [P].

\medskip
{}First we describe Kuhn's combinatorial parametrization.
\proclaim {Theorem} {\rm [Kuh]} Let $\Gamma$ be a hyperelliptic curve of
genus $g=2$ covering a curve $E$ of genus 1 by a map $f:\Gamma
\rightarrow E,\ \iota$ the hyperelliptic involution on $\Gamma,
C_1,\ldots,C_6$ the Weierstrass points of $X$ and $D_1,\ldots, D_4$
the ramification points of $\pi_E$ in the following commutative
diagram (where superscript $\iota$ means modulo $\iota$):
$$\matrix
\ \ \ \ \ \ \Gamma &\ \  \buildrel f\over\longrightarrow & E\\
 \pi_\Gamma\ \downarrow & \hfil & \ \ \ \ \ \downarrow\ \pi_E\\
\ \ \ \ \Gamma^\iota &\ \  \buildrel {f^\iota}\over\longrightarrow &\  E^\iota\\
\endmatrix
$$
Let $c_j$ and $d_i$ be the images of $C_j$ and $D_i$ in $\Gamma^\iota$ and
$E^\iota$; then for all $i$, $f^{\iota\ast} (d_i)$
%=\Sigma
contains an odd number of $c_j$'s 
(counted with multiplicity) and 
any other point of $\Gamma$ with even multiplicity;
every such ramification configuration $f^\iota:
\bbP^1\rightarrow \bbP^1$ lifts to a morphism $f:\Gamma\rightarrow E$. For
a given hyperelliptic curve $\Gamma$, the existence of a lifting of
$f^\iota: \Gamma^\iota\rightarrow\bbP^1$ to a map from $\Gamma$ to an elliptic
curve is an algebraic condition in terms of the points $c_j$, and it
falls into two different cases:

(a) (generic) If $f$ is unramified over the $D_i$, the ramification of
$f^\iota$ over the $d_i$ is as follows (with one more point of
$\Gamma^\iota$ where $f^\iota$ is doubly ramified), with the symbol --
representing an unramified point of $\Gamma^\iota$ over one of the $d_i$,
which is therefore one of the $c_j$, and $\ast$ representing a doubly
ramified point:

%\vfil\eject
%\bigskip
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \hbox{\rm case 1}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \    
 \hbox{\rm case 2}\ \ \ \ \ \ \ 
 \ \ \ \ \ \ \ \ \ \ \  \ \  \ \ \ 
\hbox{\rm case 3}

$$\hbox{\rm even $d$:}\ \ \ 
\matrix
\ast & \ast & \ast & \ast\\
\vdots & \vdots & \vdots & \vdots \\
\ast & \ast & \ast & \ast \\
\ast & = & = & = \\
d_1 & d_2 & d_3 & d_4\\
\endmatrix
\ \ \ \ \ \ \ \ 
\matrix
\ast & \ast & \ast & \ast\\
\vdots & \vdots & \vdots & \vdots\\
\ast & \ast & \ast & \ast\\
\ast & = & \ast & \ast\\
= & = & \ast & \ast\\
d_1 & d_2 & d_3 & d_4\\
\endmatrix 
\ \ \ \ \ \ \ \ \ \ \ 
\matrix
\ast & \ast & \ast & \ast\\
\vdots & \vdots & \vdots & \vdots\\
\ast & \ast & \ast & \ast\\
= & \ast & \ast & \ast\\
= & \ast & \ast & \ast\\
= & \ast & \ast & \ast\\
d_1 & d_2 & d_3 & d_4\\
\endmatrix
$$

\vfil\eject
\ \ \ \ \ $\hbox{\rm odd $d$:}\ \ \ \ \ 
\matrix
\ast & \ast & \ast & \ast\\
\vdots & \vdots & \vdots & \vdots\\
\ast & \ast & \ast & \ast\\
= & \ast & \ast & \ast\\
- & - & - & - &\\
d_1 & d_2 & d_3 & d_4\\
\endmatrix
$

\medskip
\noindent (b) {(special)} If $f$ is  ramified over some $D_i$, then the
ramification of $f^\iota$ over the $d_i$ is as shown in (a) except
that either one of the $c_j$ has ramification of degree 3, or there is
a unique point, not one of the $c_j$, with ramification of degree 4.
\endproclaim

\bigskip
We assume from now on that $d=3$. There are two possible cases for the
map $f:\Gamma \rightarrow E$, after normalizing the choice of origin
in $E$ so that the induced involution $\iota$ is $\pm 1$:

\medskip
case 1.\ \ $f$ ramifies at two distinct points $R$ and $\iota (R)$;

case 2.\ \ $f$ ramifies at a Weierstrass point $C_1$.

\medskip
After relabeling, we have the following situation:
$$f^{\iota\ast} (d_i)\ni c_i, e_i\ (i=1,2,3);\ \ \ \
f^{\iota\ast} (d_4) =\{ c_4,c_5,c_6\}$$
and the ramification points of $f^\iota$ are

\medskip
case 1.\ \ $r=\pi_\Gamma(R)$ and the $e_i,\ i=1,2,3$;

case 2.\ \ $e_1,e_2$ and $c_3=e_3$ with multiplicity 2.

\bigskip
Now we recall Poncelet's result in the briefest possible terms, only
for the case of triangles and for a generic situation (the conics in
question are assumed to meet at four distinct points, e.g.)
\proclaim {Theorem} If there is a triangle inscribed in a conic $Q_1$
and circumscribed to a conic $Q_2$, then there is a 1-parameter family
of such triangles, obtained by moving the divisor $D=P_1+P_2+P_3$
(where $P_i$ are the tangency points of the sides of the
triangle to $Q_2$) in a linear equivalence class. This family of
triangles corresponds to a curve of genus 1, $I=\{ (P,\ell )|P\in
Q_1,\ \ell\in Q_2^\ast,\ P\in\ell \}$ whose Jacobian $\hbox{
Pic}^0(I)$ is therefore an elliptic curve; and to the point of order
three ${\Cal D} =$ the divisor class $[\tau (P,\ell )-(P,\ell )]$ where
$\tau$ is the fixed-point free involution that sends a pair $(P,\ell
)$ to the pair $(Q,m)$, $(P,Q)$ being the side of a triangle.
\endproclaim

We now return to the map $f:\Gamma\rightarrow E$, case 1. We choose a
conic $Q_2\cong \bbP^1\cong \Gamma^\iota$. This is then Kuwata's
observation \hbox{\rm [Kuw, Lemma 3.4]}

\proclaim {Proposition}
Let ${\Cal D}$ be the divisor class determined by $f^{\iota\ast}$. Then the
corresponding elliptic curve $E^\prime =\hbox{\rm Pic}^0(Q_1,Q_2)$ can
be viewed as a double covering of $Q_2$ branched over $c_1,c_2,c_3$
and $r^\prime$, where $rr^\prime$ is the side of a triangle, while
$r$ is the image of ${\Cal D}$ under the projection
$E^\prime\rightarrow Q_2$. If $\overline{E}^\prime$ is the quotient of
$E^\prime$ by the subgroup $\langle {\Cal D}\rangle$ of order 3, then
$\overline{E}^\prime$ can be viewed as a double covering of $E^\iota$
ramified at $f^\iota (r),\ q_1,q_2,q_3$, as in the following
commutative diagram:
$$\matrix
\ \ \ \Gamma & \buildrel f\over\rightarrow & E\\
\pi_\Gamma\ \downarrow & \hfil &\ \ \  \downarrow\ \pi_E\\
\ \ \ \Gamma^\iota & \buildrel {f^\iota} \over\rightarrow & E/\{ \pm 1\} \\
\ \ \ \uparrow & \hfil & \uparrow\\
\ \ \ E^\prime & \buildrel \phi\over\rightarrow & \overline{E}^\prime\\
\endmatrix
$$
\endproclaim


With this motivation, Kuwata constructs explicitly a family of curves
$\Gamma$, para-\hfill\break
\noindent metrized by an open set $U$ of $\bbP^1$, that admit
a morphism of degree 3 to a fixed elliptic curve $E$. Let $E$ be given
in Weierstrass form: $y^2=x^3+ax+b$, $\delta =4a^3+27b^2$ and for
$u\in\bbC$ let
$$\matrix
d(u) &= & & 27bu^3-18a^2u^2-27abu-2a^3-27b^2\\
c(u) &=& &3 au^4+18bu^3-6a^2u^2-6abu-a^3-9b^2\\
f_u(t) &=& & 4c(u)t^3-3\delta (3u^2+a)t-3\delta (2au+3b)\\
g_u(t) &=& & (u^3+au+b)t^3-(3au^2+9bu-a^2)t^2+\delta\\
\endmatrix
$$
\proclaim {Proposition} The set of all curves $\Gamma$ of genus 2
that admit a morphism $\Gamma\rightarrow E$ of degree 3 can be
parameterized by the open set $U=\{ u\in\bbP^1:\ c(u)d(u)\not= 0\}$,
with the curve $\Gamma_u$ given by equation: $-\delta
d(u)s^2=f_u(t)g_u(t)$. The morphism $f:\Gamma\rightarrow E$ is given
by the formula:
$$f(t,s) =\left( {{\delta d(u)(3(u^3+au+b)t+3au^2+9bu-a^2)}\over
 {{c(u)f_u(t)}}}\right.
$$
$$\left. +{{9bu^4-8a^2u^3-18abu^2-18b^2u+a^2b}\over
{{c(u)}}}, {{\delta d(u)g_u(-2t)}\over {{f_u(t)^2}}}s\right)
$$
and is in case 2 if and only if $u^3+au+b=0$.
\endproclaim
The reason for such polynomial expressions in the parameter is for the
result to hold over any field of definition $k$ (except the formulas
need to be modified in characteristic 3). So, the point of order 3
$\left( x(u)={{-2au+3b}\over {{3u^2+a}}},\ y(u)\right)$ on the curve
$\overline{E}^\prime$ will be defined over $k$, with
$\overline{E}^\prime$:
$$ \matrix
d(u)y^2 &=(x-e(u))(x^3+ax+b)\\
e(u) &\ \ \ \ \ ={{9bu^4-8a^2u^3-18abu^2-18b^2u+a^2b}\over {{c(u)}}}\\
\endmatrix
$$
as well as an explicit isogeny of degree 3: $E^\prime\rightarrow
\overline{E}^\prime$. $\Gamma_u$ is then $E^\prime /\{ \pm 1\}
\times_{\overline{E}^\prime /\{ \pm 1\}}E$.

Poncelet's theorem now gives a way of writing an algebraic equation
for the second elliptic curve $F=$Jac$(\Gamma_u)/E$, which in this case
is canonical.
\proclaim {Theorem} $F_u=$\hbox{\rm Jac}$(\Gamma_u)/E$ can be given
by the equation:
$$3\delta y^2=x^3-3\delta (3u^4+6au^2+12bu-a^2)x-6\delta
(9bu^6-12a^2u^5-45abu^4
$$
$$-90b^2u^3+15a^2bu^2-(4a^4+18ab^2)u-3a^2b-18b^3).
$$
\endproclaim
The morphism $\Gamma\rightarrow F$ and the $j$-invariant of $F$ are
also computed explicitly in [Kuw]. The fact that allows $F$ to be
identified is the following:
\proclaim {Proposition}
{\rm [Kuw, 5.2]}
 Let $Q_2$ be a conic in $\bbP^2$ and let $w_1,\ldots,w_6$ be mutually
 distinct points in $Q_2$. We denote by $\ell_i$ the tangent line to
 $Q_2$ at $w_i$ and we set $p_{ij}=\ell_i\cap\ell_j$. Then the
 following two conditions are equivalent.

(1) There exists a conic $Q^\prime$ passing through $p_{12},p_{23}$
    and $p_{31}$, and tangent to $\ell_4,\ell_5$ and $\ell_6$.

(2) There exists a conic $Q^{\prime\prime}$ passing through
    $p_{45},p_{56}$ and $p_{64}$, and tangent to $\ell_1,\ell_2$ and
    $\ell_3$.

Let $\Gamma$ be a curve of genus 2 that is a double covering of $Q_2$
ramifying at $w_1,\ldots,$
$w_6$. If one of the above conditions is
satisfied, then the Jacobian $\hbox{\rm Jac}(\Gamma )$ of $\Gamma$ is
isogenous to a product of two elliptic curves $E$ and $F$.
\endproclaim

The key to this ``duality'' is the following lemma due to Kuhn, which
generalizes Jacobi's description of all the degree-2 morphisms from a
curve of genus 2 to an elliptic curve: given an involution $\alpha$ of
$\bbP^1$ and three points $a_1,a_2,a_3$ not fixed by the
involution, the curve of genus 2 ramified over $a_1,a_2,a_3,\ \alpha
(a_1),\alpha (a_2),\alpha (a_3)$ maps to the two elliptic curves
ramified over $\bbP^1 /\alpha$ at the images of the $a_i$ and one of
the two fixed points of the involution.
\proclaim {Lemma} \hbox{\rm [Kuh, p.47]} If $f:\Gamma\rightarrow E$ and
$g:\Gamma\rightarrow F=\hbox{\rm Jac}(\Gamma )/E$ are a given morphism
of degree 3 and the canonically associated one, there is a labeling
of the branchpoints of $\Gamma$ such that $f^{\ast\iota}(d_i)\ni c_i,\
i=1,2,3,\ f^{\iota\ast}(d_4)=\{ c_4,c_5,c_6\}$ and
$g^{\ast\iota}(b_i)\ni c_{i+3},\ i=1,2,3$, $g^{\ast\iota}(b_4)=\{
c_1,c_2,c_3\}$ where $b_1,\ldots, b_4$ correspond to the points of
period 2 of $F$, and $d_4,b_4$ correspond to the origins of $E,F$:
$$\matrix
 F & \buildrel g\over\leftarrow & \Gamma & \buildrel f\over\rightarrow
& E\\
\downarrow & \hfil & \downarrow & \hfil &\downarrow\\
F^\iota & \leftarrow &\Gamma^\iota &\rightarrow & E^\iota\\
\endmatrix
$$
\endproclaim
\remark {Remark} There are exactly two conics (up to projective
equivalence) that are both 3-inscribed in and 3-circumscribed to a
given conic $Q^\prime$, according to Barth and Michel [BM]. They
correspond to ``control points'': $[1,\omega,\omega^2]$ and
$[1,{{1+\sqrt 5}\over 2}, {{1-\sqrt 5}\over 2}]\in\bbP^2$, resp. Since
Kuwata points out that $Q^\prime,Q^{\prime\prime}$ in the previous
proposition are mutually 3-circumscribed and the corresponding
Poncelet elliptic curve has an automorphism of order 3, it seems
reasonable that it must correspond to the solution
$[1,\omega,\omega^2]$ (checking this would be a straightforward
calculation).
\endremark

\head 4. Problems\endhead
In this part we have seen that many beautiful classical questions
having to do with special abelian varieties are still quite open, and
in fact despite the progress of algebraic geometry no particularly
powerful techniques have been introduced, as opposed to a classical
area such as enumerative geometry, which has received so much impetus
lately from the theory of Gromov-Witten invariants. While it is often
the case that in group theory the methods are {\it ad hoc} (cf. [A1]),
still I would pose some questions that seem both tractable and useful,
because they have arisen in several contexts, one of which will be the
theme of the next part.

\bigskip
{\bf 4.1} Genus 2, splittable case. We refer to the notation of section
  3. Use the invariant theory of 6 points on a line, and the topology
of finite covers of this moduli space, to investigate the number of
connected components of ${\Cal M}_2 (d)$.

\bigskip
{\bf 4.2} Genus 2, splittable case. Pursue Kuwata's idea and apply
  Poncelet's theorem to describe ${\Cal M}_2(d)$, especially the
  complementary elliptic curve, which is canonical for an ``optimal''
  cover [Kuh], namely one that doesn't factor non-trivially through
  another elliptic curve. In view of Kuhn's statement, to the effect
  that if $\Gamma\rightarrow E$ is optimal of degree $d$, there exists
  a unique $E^\prime$ and an optimal morphism $\Gamma\rightarrow
  E^\prime$ also of degree $d$, with Jac$(\Gamma )\sim E\times
  E^\prime$, these pairs $(E,E^\prime )$ should again give examples of
  the inscribed and circumscribed pairs that Barth and Michel suggest
  are worth understanding [BM, \S6]

\bigskip
{\bf 4.3}  Higher genus hyperelliptic $\Gamma$ and Poncelet's theorem. There
  is a higher-dimensional generalization of Poncelet's theorem,
  together with its interpretation as one of the aci's of II.2
  below [P], and here closure holds only in the rare situation
  that the curve has a point $P$ with ${\Cal O}(P-\iota P)$ of finite
  order. But this might very well be related to the fact that $\Gamma$
  admits a morphism to an elliptic curve, with some restrictions on
  the inverse image of the origin. This is only speculative at this
  stage, but concrete calculations in genus 2 and 3 should provide a
  good indication.

\bigskip
{\bf 4.4} Genus 3. Kowalevski's curve:
$$(z^2-f_2)^2=4xy(ax+by)(cx+dy)
$$
in projective coordinates, with $f_2$ a form of degree 2 in $x$ and
$y$, has genus 3 and admits a 2:1 morphism to an elliptic curve, yet
is not hyperelliptic. She proved that a canonical quartic has
splittable Jacobian if and only if it has two concurrent bitangents,
as recalled above. This point of view seems to me worth pursuing in
two directions. One question is the dimension of the subvariety ${\Cal
M}_g(d)$ of curves of genus $g$ that admits a degree $d$ morphism to
an elliptic curve, then its topology and defining equations. Even the
dimension question was nontrivial in $g=2$ [L1,Kuh]  and I am advocating
finding geometric criteria on the canonical curve equivalent to the
property of being in  ${\Cal M}_g(d)$. The technique goes back to
Riemann and has been very fruitful, namely translating analytic
properties of the theta function (with characteristics) into
properties of linear series on the curve; as an example, again
recalled above, Accola infers from the  vanishing of $\vartheta$ with
1/2-period characteristics the existence of an automorphism of order 2
on the Riemann surface; then on the geometric side he uses the same
vanishing to produce a plane model of the curve with given properties;
by putting the two together, he derives the dimension of moduli of
curves with an automorphism of order 2. In the second direction,
geometric properties of (nonhyperelliptic) canonical curves reflect of
course on the moduli space of rank 2-bundles with trivial determinant
${\Cal M}_\Gamma (2,{\Cal O})\hookrightarrow |2\Theta_\Gamma |^\ast
=\bbP^{2^g-1}$; for genus 3, Coble's analysis [C] highlights important
features of this variety, with many questions remaining open. The
special case corresponding to curves with splittable Jacobians seems
fertile ground to plow. Lastly, it seems to be unknown whether the
Jacobian of Kowalevski's curve is actually isomorphic to a product of
three elliptic curves; number-theoretic applications of such a
feature  would include current techniques of coding theory and
cryptography.
 
\bigskip
{\bf 4.5} Differential Galois Theory. L. Greenberg [Gr] proved that for any
  finite group $G$ and non-negative integer $g$ there is a compact
  Riemann surface $X$ with Aut $X\cong G$ and $X/G$ of genus $g$
  (cf. [TT]; as those authors note, the genus of $X$ is not controlled
  by this statement). Given the recent advances in differential Galois
  theory, particularly in computation, it would seem worth pursuing
  the question: when $X/G$ is $\bbP^1$, G can be viewed as the
  monodromy group of a differential equation with coefficients in
  $\bbC (z)$; find an equation explicitly and use the coefficients
to  gain insight on the moduli space of curves with Aut$(X)=G$.

\medskip
I won't mention here any other of the many fascinating problems on
automorphisms because I can't offer any suggestion of a method (that's
why I included some at the end of Section 2); it should be said that
automorphisms of the curve $X$ have also been used in the construction
and investigation of Prym varieties with applications to
integrable systems (cf. [K], [D], and the recent [LR] for progress on
the geometric side). In concluding, I will stress again the
`interdisciplinary' nature of the reduction problem; to wit,  (4.1-4.3)
are geometric, 4.4 links analysis and algebra (group theory), 4.5 is
differential calculus; and since I am unable to touch at all on number
theory I will cite a question in [TT] 
as an example: are the entries of the period
matrix for a curve with $84(g-1)$ automorphisms algebraic numbers?



\specialhead Part II: Elliptic solitons and integrable systems\endspecialhead

\head 1. Calogero-Moser-Krichever: Aspects\endhead
Ever since Dubrovin and Novikov [DN] obtained the superposition of
`periodic solitons',
$$2\wp (x-x_1(t))+2\wp (x-x_2(t))+2\wp (x-x_3(t))
$$
the question of elliptic solutions of
the KdV or KP equation has been with us. There are two (inequivalent)
properties that one may seek in this regard; consider to begin with an
`algebro-geometric solution of the KP hierarchy' by which what is meant
is a (complex-valued) function of $g$ (complex) variables
$\underline{t} =(t_1=x,
t_2=y, t_3=t,\ldots,t_g)$ that is produced from algebro-geometric
data (roughly speaking, a curve $X$ of genus $g$ with a choice of
standard homology basis, a (smooth) point $\infty\in X$ and a divisor
$D=P_1+\ldots +P_g$ on $X$ with $h^0(D-\infty )=0$)
$$u(\underline{t} )=2{{d^2}\over {{dx^2}}}\log 
{\vartheta \left( \Sigma_{j=1}^g \int_{P_0}^{P_j} \vec
\omega +
\Sigma t_iU_i-\Delta \right)}+{\text{const.}}$$
where $U_j\in\bbC^g$ and $\Delta$ are constants that depend on
the curve (as well as the additive const.).
%-{{d^2}\over {{dz^2}}}\log e^{\Sigma t_i\int_{P_0}^P
%\eta_i-\Sigma t_ia_{0i}}\ {{\vartheta \left( \int_{P_0}^P \vec
%\omega+\Sigma t_iU_i-\Delta \right)\vartheta\left( \int_{P_0}^\infty
%\vec\omega -\Delta \right)}\over {{\vartheta\left( \int_{P_0}^P
%\vec\omega -\Delta\right)\vartheta\left( \int_{P_0}^\infty \vec\omega
%+\Sigma t_iU_i-\Delta\right)}}}
%$$
%where $a_{0i}$ and $U_j\in\bbC^g$ are constants that depend on
%the curve (cf. [SW], e.g.).
 
This function could have one of two
properties:

(a) $u(\underline{t})$ is elliptic (i.e., doubly periodic) in the
first variable $x$;

(b) $u(\underline{t})$ is an algebraic expression in $g$ elliptic
functions $\wp_1,\ldots,\wp_g, \wp_1^\prime ,\ldots,\wp_g^\prime$
where each $(\wp_i,\wp_i^\prime )$ corresponds to a given lattice
$\Lambda_i$.

The first question, where knowledge is much more advanced, also arises
in an another way: suppose that one asks what is the dependence on
time $(t=t_3)$ of the poles in $x$ of a solution of the KdV
equation. It turns out [AMcKM] that they obey an ``algebraically
completely integrable hamiltonian system'' (aci for short) called the
Calogero-Moser system, with hamiltonian:
$$H={1\over 2}\sum \dot x_i^2 -2\sum_{i\not= j} {1\over
{{(x_i-x_j)^2}}}.
$$
Inspired by this formula, Krichever [Kr] showed that the hamiltonian
system with
$$H={1\over 2}\sum \dot x_i^2 -2\sum_{i\not= j} \wp (x_i-x_j)
$$
likewise is aci, and moreover that any algebro-geometric KP solution
that is elliptic in $x$ has the form: $u(\underline{t})=\sum_{j=1}^g
\wp (x-x_j(\underline{t}))$, with $x_j$ depending on $t_2,t_3,\ldots, t_g$
according to this hamiltonian, which is called the
Calogero-Moser-Krichever system. To link these problems with the
previous part, one needs only to observe that the formula 
given for $u(\underline{t})$ linearizes
the KP flows on Jac $X$, therefore periodicity in $x$ is equivalent to
requiring that an elliptic curve $E$, tangent at the origin of motion to
$U_1$, be contained in Jac$X$. Following this point of view, Treibich
and Verdier in a series of papers gave a geometric description of what
they named ``elliptic solitons''; Markman interpreted the locus as a
generalized Hitchin system (cf. [DM] for a survey).

As regards question (b), however, nothing is known apart from a few
examples. First of all, since 
the functions $\displaystyle{
{{d^k}\over {{dx}^k}}
\log 
%e^{a_0+a_1x}
\vartheta \left( \Sigma_{j=1}^g \int_{P_0}^{P_j} \vec
\omega
+ xU_1-\Delta \right)
}$, $k\ge 2$, generate the function field of
Jac$X$
and the $t$-flows span the tangent space to
Jac$X$, in order for such a basic function to be elliptic Jac$X$ would have
to be splittable as an $E_1\times\ldots \times E_g$, secondly, the time flows
would have to be tangent to tori, which seems to be a more subtle
question. We make this observation more precise in the following:

\remark{Remark} 
 The locus of elliptic solitons of genus $g$ has dimension $g$;
   while the locus of abelian varieties of dimension $g$ isogenous to a
   product of $g$ elliptic curves has the same
   dimension, it is very unlikely that the dimension of the Jacobian
   sublocus is the same. Thus, in principle the locus of elliptic
   solitons does not coincide with
%contains properly
 the locus of elliptic solutions.
\endremark

We will give some concrete details on elliptic solitons in the next
section. Then section 3 is related to both problems in yet another
way, and harks back to classical questions. We conclude this section with
the examples of (b) that we know of.


\example{Example 1} In [BBM1,2] the authors implement a theorem of Appell
and by suitable analytic conditions on the period matrix they produce
elliptic solutions for integrable hierarchies; they give examples of
hyperelliptic and trigonal curves where the condition is satisfied by
choosing suitable homology bases. In [BBME], the more general
 reduction criterion due to Weierstrass is applied.
\endexample

\example{Example 2} In [BE1-3], the authors work out explicit
expressions for elliptic solutions in special cases, as well as
equations for the locus of genus-2 curves which allow reduction
(``Humbert surface'').
\endexample

\example{Example 3} In [Le] and in [MS] the authors calculate the
expression of elliptic solutions for the KdV, Boussinesq equations,
resp., coming from the curve: $y^2=x(x^4-\alpha^4)(x^4-\beta^4),\
y^4=(x-\alpha )(x-\beta )(x-\gamma )(x-\delta )$, resp.
\endexample

More examples will be given in section 3.

\head 2. Elliptic and abelian solitons \endhead

Perhaps the most remarkable fact about Krichever's aci is that for the
moduli space of spectral curves $\Gamma$ that gives the base of his
system, Jac $\Gamma$ are all splittable. We provide a different
interpretation of such moduli space, due to Treibich and Verdier,
which allows us briefly to survey the Hitchin-Markman system, a
`non-abelian' version of aci in the sense that the motion takes place
on (the cotangent bundle to) a moduli space of vector bundles over a
fixed curve $X$. To begin with, we take $X=E$, our fixed elliptic
curve.

\definition{Definition}  Let $S$ be the projectivization of the total
space of the semistable rank-2 bundle with trivial determinant over
$E$. $S$ is the ruled surface $\pi_S:S\rightarrow E$ characterized by
the existence of a unique section $C_0$ with zero self-intersection. 
Let's denote by $S_q$ the fibre, $q\in E$.
\enddefinition

\definition{Definition} A finite pointed morphism $\pi: (\Gamma,
p)\rightarrow (E,q)$ is called a tangential cover when $\pi^\ast (E)$
is tangent to $A_\Gamma (\Gamma )$, with $A_\Gamma :r\mapsto {\Cal
O}(r-p)$ the Abel map. It is called primitive if it does not
factor nontrivially through another tangential cover.
\enddefinition

\proclaim {Theorem} {\rm [TV1]} Given a tangential cover $\pi :(\Gamma,
p)\rightarrow (E,q)$ of degree $n$, there exists a morphism $\rho
:\Gamma\rightarrow S$ of degree 1 that sends $p$ to $p=C_0\cap S_q$
and $\Gamma$ to a member of the linear system $|n C_0+S_q|$. Any two
such morphisms $\rho_1,\rho_2$ are related by $\rho_1=\sigma\circ
\rho_2$, $\sigma\in$ Aut$(S/E)=\bbC$. Conversely, any irreducible
member of $|nC_0+S_q|$ with the natural projection to $E$ is a
tangential cover of degree $n$. The map $\Gamma\mapsto\rho (\Gamma )$
gives a 1:1 correspondence between primitive tangential covers and the
open affine set $V(n,E)$ of irreducible divisors in the linear system
$|nC_0+S_q|$, which has dimension $n$ and only one fixed point, $p$;
the general member of $V(n,E)$ is smooth of genus $n$.
\endproclaim

\definition{Definition} [DM, 4.6-4.9] 
Let  $\hbox{\rm Higgs}_X^{sm}(r,d,\omega (D))$ be
the component of the moduli space parametrizing semistable Higgs pairs
$(E,\phi )$ that have generically irreducible and reduced spectral
curves. Here $E$ is a rank-$r$, degree-$d$ vector bundle on the curve
$X,\ \omega$ is the canonical line bundle of $X,\ D$ is a divisor on
$X$ (of sufficiently large degree) and $\phi :E\rightarrow
E\otimes\omega (D)$ is a twisted endomorphism. Let $b_i=(-1)^i$ trace
$(\Lambda^i \phi )$ and let the hamiltonian be the characteristic
polynomial map:
$$H:\ \hbox{\rm Higgs}_X^{sm}(r,d,\omega (D))\rightarrow B_\omega:
\bigoplus_{i=1}^r H^0 (X,\omega (D)^{\otimes i}),\ \ \phi\mapsto (b_i).
$$
Let also $y\in H^0 (X,\pi^\ast \omega (D))$ be the tautological
section of the total space $\pi :\omega (D)\rightarrow X$. Then the
spectral curve of $\phi$ is the inverse image $\Gamma$ under char$(\phi
)=y^r-tr(\phi )
y^{r-1}+\ldots +(-1)^r \hbox{\rm det}(\phi ):\ \omega (D)\rightarrow
\omega (D)^{\otimes r}$, in the total space of $\omega (D)$, of
the zero section of $\omega (D)^{\otimes r}$.
\enddefinition


\proclaim {Theorem} {\rm [DM, 4.9]} The Hamiltonian map $H$ is an
algebraically completely integrable system. The generic (Lagrangian)
fibre is the Jacobian of a smooth spectral curve of genus
$r^2(g-1)+1+(\hbox{\rm deg}D) {{r(r-1)}\over 2}$.
\endproclaim

\proclaim {Theorem} {\rm [DM, 6.14]} The variety of elliptic KP
solitons of degree $n$ corresponding to a fixed pointed elliptic curve
$(E,q)$ is birational to the divisor of traceless Higgs pairs in
the ($2n$-dimensional) symplectic leaf of $\hbox{\rm Higgs}^{sm}_E
(n,0,\omega (q))$ that have residue
$$\left[
\matrix
-1 & 0 & \ldots && 0\\
0 & -1 & \ldots &\hfil\\
\vdots & \hfil & \hfil& & \vdots\\
& & & -1& 0\\
0 & \ldots& & 0 & n-1\\
\endmatrix \right]
$$
over $q$.
\endproclaim  

\medskip
It is now a natural question whether there can be a curve $X$ of genus
$g>1$ in this ``tangential position'' to a family of solitons, more
specifically an integrable system equivalent to a Lax pair whose
entries are rational functions over a curve of genus $>1$. The answer
must be negative for dimensional reasons, as noted in [DKN, 2.1]
(comment on Lax pairs whose coefficients are sections of a
 line bundle over a curve), and indeed
Treibich showed that if $X$ with $g>1$ and $\phi :(\Gamma,p)\rightarrow
(X,q)$ are in tangential position, then $\Gamma$ and $X$ have to be
hyperelliptic with $q$ a Weierstrass point while $p$ is not
[T1]. Instead, in [DP] we ask the following question: are there
families of KP ``abelian solitons,'' namely Jacobians Jac$(\Gamma )$
containing an abelian subvariety $P$ of dimension $k>1$ in such a way
that the first $k$ KP flows (which span a hyperosculating space to the
Abel image of $\Gamma$ at the image of $\infty$) are tangent to $P$? We
found only a very special case, namely $P$ is ``coelliptic'' with
Jac$(\Gamma )/P$ isomorphic to a fixed elliptic curve $E$, but we were
able to show that they form a completely integrable system:


\definition{Definition} Fix any two integers $n\ge g\ge 2$ and an elliptic
curve $E$. We also fix the origin $q$  of $E$, and define the
$(g-1)$-dimensional family of coelliptic solitons to be the curves
$\Gamma$ with a map $\pi :\Gamma\rightarrow E$ of degree $n$,
generically ramified simply over $g-1$ points, with a branch point of
multiplicity $(g-1)$ at $q$.
\enddefinition

\proclaim{Theorem} {\rm [DP, Th. 1]} The family of degree $n$, genus
$g$ coelliptic solitons constructed above is birationally equivalent
to an algebraically completely integrable system with
$(g-1)$-dimensional Prym varieties $P$ as Liouville tori.
\endproclaim

Note that by construction, the first $(g-1)$ KP flows are indeed
tangent to the Prym variety $P=$ Prym$(\pi )$, namely the connected
component through the origin of the kernel of the norm map: Jac
$\Gamma\rightarrow E$. Our integrable system is obtained by symplectic
reduction from the Hitchin-Markman system of Higgs pairs $\hbox{\rm Higgs}_E
(n,d,\omega (q+\max \{ 0,n-g\} q^\prime ))$ (for any $q\not=
q^\prime\in E)$ over a subvariety of spectral curves cut out by
hamiltonians. Therefore, the hamiltonian flows are compatible with the
KP flows [DM \S 6].

\medskip
In view of this result, for $g=3$ e.g., one might still be able to
write a Lax pair for the integrable systems, describing the motion of
a divisor on $P$, a 2-dimensional abelian variety (``ultraelliptic''
case) and corresponding to genus-3 KP flows and their Baker
functions. To that end, in [EEP3, EP] we investigate the hyperelliptic
Kleinian functions and the differential equations that they satisfy. 
These analogs of the $\wp$ function were
developed in [BEL] and in genus 2 also introduced by D. Grant [G]
along with an analog of Jacobi's derivative formula,
and extended to genus 3 by Y. \^Onishi [O].
We plan to
%extend the work of Frobenius-Stickelberger; 
devise a proper Ansatz for the Baker function of
an ultraelliptic soliton, and follow Krichever [Kr] in showing that
the Lax pair for the system can be derived from a Frobenius-Stickelberger
formula. 



\head 3. Lam\'e \endhead
We begin with a different point of view, which in truth is closely
related to the theory of elliptic solitons. The classical spectral
problem for differential operators with elliptic coefficients:
$$L=\partial^n +u_{n-2}(x)\ \partial^{n-2}+\ldots +u_0(x),
$$
where $\partial ={d\over {{dx}}}$ and $u_j(x)\in\bbC (\wp,\wp^\prime
)=\bbC (E)$, with the normalization $u_{n-1}(x)=0$, can be viewed in
the context of isospectral deformations of commutative algebras of
differential operators, namely the KP flows:

\bigskip
\remark{Remark} If the ring of differential operators that commute with
$L$ is of (arithmetic) genus $>0$, which is to say that $L$ is
finite-gap, then the corresponding KP solution is an elliptic soliton.
\endremark
Indeed, the ring of operators that commute with $L$ is given by the
differential-operator part of the ring of formal pseudo-differential
operators:\hfil\break $\{ \sum_{j\le N}c_j (L^{1/n})^j,\
 c_j\in\bbC \}$ so all the
coefficients are periodic in $x$.

The question of when $L$ is finite-gap is surprisingly subtle:
\proclaim {Theorem} \hbox{\rm (cf. [I])}
$L=\partial^2 +c\wp (x)$ is finite gap if and only if the constant
$c=-n(n+1)$, where $n$ is a natural number.
\endproclaim
In fact, the first significant example of elliptic solition cited in
section 1 arises this way for $n=2$. 
A powerful and effective method was introduced
recently by F. Gesztesy and his collaborators
(cf. [GW] for a survey) by extending Picard's theorem 
to link the finite-gap property with the qualitative behavior 
of the eigenfunctions.
One is then faced with the
question of computing the equation for the isospectral curve $\Gamma$
whose Jacobian splits along $E$. There are several recursive
approaches, [Al], [DGU], [GRT],
to finding a differential operator $B$ of order $n$ that
commutes with $L$, so that the polynomial equation
$B^2=L^{2n+1}$ +(lower-order terms) that they satisfy gives the
(possibly singular) curve, but a more classical one can be found in
[He], namely an Ansatz for the expression of the 
eigenfunctions at the branched values of the spectral parameter.
{}For example when $n$ is odd and the branchpoint is of 
one of two possible types
(the expressions for the other cases are analogous):
%Baker function, which
%will have $n$ poles on the curve $E$ (cf. [TV2]) given by a Weierstrass
%equation: ${1\over 4}\mu^2 =(\lambda -e_1)(\lambda -e_2)(\lambda
%-e_3),$
$$\psi (x)=((\wp (x)-e_1)^{1/2}(\wp (x)-e_2))^{1/2} 
\sum_{j=0}^{(n-3)/2}b_j(\wp
(x)-e_3)^{n/2-j-1}
$$
(notice how this differs from the more general Ansatz for the KP
elliptic solitons used by Krichever:
$$\psi =\sum_{j=1}^n a_i(t,k,\alpha )\phi (x-x_i,\alpha )e^{xk+tk^2}
$$
where
$$\phi (x,\alpha )=-{{\sigma (x-\alpha )}\over {{\sigma (\alpha
)\sigma (x)}}}e^{\zeta (\alpha )x}
$$
and $\sigma,\zeta$ are the classical Weierstrass functions).

By expanding the Lam\'e equation (using MAPLE, e.g.) and solving
algebraic conditions of compatibility for the $b_j$, one can give 
examples of Lam\'e curves:

\bigskip
\noindent $n=2$\ \ \ \ \ \ \ $w^2=( z^2-3g_2)\prod_{i=1}^3 (z+3e_i)$

\bigskip
\noindent $n=3$\ \ \ \ \ \ \ $w^2=z \prod_{i=1}^3 (z^2-6ze_i+45e_i^2
-15g_2)${\hfil\break}
and
the general-$n$ case can be calculated recursively. There are several questions
to be addressed:

\medskip
$\bullet$ Which Lam\'e Jacobians are isogenous to a product of elliptic
curves (or even isomorphic to products?)

\medskip
$\bullet$ When that happens, 
what is an explicit expression for a KdV solution in
terms of elliptic functions?

\medskip
$\bullet$ In view of recent work on elliptic solitons, more general
elliptic potentials are known to be finite-gap, as reviewed in the
next proposition: the two questions above should be asked for these
too.

\proclaim {Proposition} \hbox{\rm ([T1,2])}. For any $n\ge 1$ and
$(a_i)\in\bbN^4$ such that $\sum_{i=0}^3 a_i(a_i+1)=2n$, there is a
unique hyperelliptic tangential cover $\Gamma\rightarrow E$ and a half
period $\omega\in \{ \omega_0=q,\omega_1,\omega_2,\omega_3\}$ such that the
potential 
$$u(x)=\sum_{i=0}^3 a_i(a_i+1)\wp (x-\omega_i)
$$
is the KdV initial value associated to the divisor $\pi^\ast (\omega
-q)$, and the other normalizing data. For any other point $\rho$ on
the elliptic curve, the potential
$$u(x)=2\wp (x-\rho )+2\wp (x+\rho )+\sum_{i=0}^3 a_i(a_i+1)\wp
(x-\omega_i)
$$
is finite-gap if and only if $\sum_{i=0}^3 (2a_i+1)^2\wp^\prime (\rho
-\omega_i)=0$. 
\endproclaim
 Recall that a tangential cover is called primitive if it does not
 factor nontrivially through another one, and exceptional if the
 (geometric) genus of its canonical image on the surface $S$ (defined
in section 2) is 0.


In [T1] Treibich proves some characterizing properties that
allow him to compute the tangential polynomials, that is to say the
equations as covers of the curve $E$, for the primitive exceptional
hyperelliptic tangential
covers of degree $n(=deg\ \pi)$ up to 6, for example when:

\medskip
$n=3$,\ \ \ \ \ $S_3(T)=T^3-3\wp T+\wp^\prime,\ u(x)=6\wp (x),\ \ g=2$

\bigskip
$n=4$,\ \ \ \ \ $S_4(T)=T^4+3(e_i-2\wp )T^2+4\wp^\prime
T-3\prod_{j\not= i}(\wp -e_j)$,

\medskip
$u(x)=2(3\wp (x)+\wp (x-\omega_i)),\ g=2$ (with $i=1,2,3$)

\bigskip
$n=5$\ \ \ \ \ \  also corresponds to genus 2;

\bigskip
$n=6$,\ \ \ \ \ $S_6(T)=T^6-15\wp T^4+20\wp^\prime T^3-{9\over
4}(20\wp^2 -3g_2)T^2+12\wp\wp^\prime T-{5\over 4}{\wp^\prime}^2$,

\medskip
$u(x)=12\wp (x),\ \ g=3$.

\bigskip
I posed these questions above rather than in the next section because
this is work in progress, in collaboration with J.C. Eilbeck and V.Z.
Enolskii [EEP2].  


These are some (very) preliminary observations which we are pursuing
in order to give a theory of Lam\'e tangential covers. The
hyperelliptic curve $\Gamma_n$  associated to the potential
$-n(n+1)\wp$, for some $n$ and some special elliptic curves will be
singular, though the arithmetic genus is always $n$. In coordinates 
$(z,w)$ as above, the ``KP basis'' of holomorphic differentials is $\{
\omega_j: z^j {{dz}\over w}\},\ 0\le j\le g-1$, in the sense that the KP
flows on Jac$(\Gamma_n)$ are parallel to the directions
$(0,\ldots,\int \omega_j,\ldots 0)$.

\remark{Observation 1} Aside from the cover that defines the elliptic
soliton, corresponding to the differential $z^{g-1} {{dz}\over w}$
(cf. [BC]), the time direction $t_3$ also is elliptic for $n=2,3$ and,
in the ``equianharmonic'' case (i.e. when $\wp$ is associated to the
curve with an automorphism of order 3) also for $n=4$ and 5.
\endremark


\remark{Observation 2} Jac $(\Gamma_3)$ splits completely in the
equianharmonic case. For $n=2$, Jac$(\Gamma_2)$ splits into two
non-isomorphic elliptic curves; for $n=3$, the curves along the
$x=t_1,t_5$ direction are both equianharmonic, while the $t=t_3$ curve
is not. 
\endremark

\medskip
The next insight into differential equations with periodic
coefficients was provided by Halphen. He generalized Ince's result and
proved [H]:

The operator $L=\partial^3 -(n^2-1)\wp\partial -
{{(n^2-1)}\over {{2}}}\wp^\prime =0$ is finite gap.


\medskip
The resulting elliptic solitons will be  solutions of the Boussinesq equation,
because the third KP time evolution corresponds to the equation

$$\partial_{t_3}L=[(L^{3/3})_{+}, L]=0.
$$

In this situation, to find the equation of the spectral curve one uses
the Halphen Ansatz for the eigenfunction:
$$\psi (x;\alpha )=e^{kx} \sum_{j=0}^{n-1} a_j(x,\mu, k) {{d^j}\over
{{dx^j}}}\phi (x;\alpha )
$$
where $\alpha$ is a complex number viewed as a point of the elliptic
curve $\nu^2=4\mu^3-g_2\mu -g_3$; by expanding the eigenvalue problems
$L\psi =z\psi$ and $B\psi =w\psi$, where $B$ is the lowest-order
differential operator that commutes with $L$ (not a polynomial
in $L$) and imposing compatibility, one obtains the equation
$F(w,z)=0$ of a (trigonal) ($3,N$) curve.

Halphen also pointed out that 
the equation $\partial^3 -{4\over 3}n^2\wp\partial +
{{2n(n-3)(4n+3)}\over {{27}}}\wp^\prime =0$ has solutions that
can be expressed in terms of elliptic functions, so it would be 
interesting to check whether the operator
$L=\partial^3 -{4\over 3}n^2\wp\partial +
{{2n(n-3)(4n+3)}\over {{27}}}\wp^\prime $ is finite-gap.  A Gr\"obner-basis
approach together with new examples of third-order finite-gap operators
is worked out in [Br].
\medskip
In [EEP1], we showed that the $(3,4)$ Halphen curve, which has genus 3,
in the case of the  equianharmonic $\wp$-function splits into three
elliptic curves, and what's more, again the three elliptic curves are
tangent to the three KP flows. In this case, all three curves are
isomorphic.

\medskip
The other task is to express explicitly the KP solution in terms of
elliptic functions, which is achieved by the general formula once the
evolution vectors $U_j$ are computed. In [EEP1] we worked out the solution
to the Krichever system, from which the KP flows are deduced. The
method we used involves knowledge of the ``Kleinian functions''
$\wp_{ij} (\underline{z})=-{{\partial^2}\over {{\partial z_i\partial
z_j}}}\log \sigma (\underline{z})$, and their use in giving an
algebraic solution of Jacobi's inversion problem. Our final result is
the following:
\proclaim {Theorem} The 5-particle Calogero-Moser-Krichever system for
the{\hfil\break} equianharmonic elliptic curve, with trigonal spectral curve
$$w^3=\left( z^2+{{25}\over 4}g_3\right)\left( z^2-{{135}\over
4}g_3\right)
$$
has evolution divisor $\sum_{i=1}^3 (z_i,w_i)$ depending on the
particles $x_i(t_1,t_2,t_4),\ 1\le i\le 5$ according to the equations
$$\left( {{df}\over {{dy}}}\right)^2 -f^6-4\left(
\widetilde{\wp}\left( {2\over 3}y\right)+2g(y)\right)f^2=0
$$
where
$$f={{dx_i (y)}\over {{dy}}},\ \ g(y)={{\partial x_i (t,y)}\over
{{\partial t}}}\biggl|_{t=0},
$$
denoting $y=t_1, \ t=t_2$, $\widetilde{\wp}^{\prime
2}=4\widetilde{\wp}^3 +(40g_3)^2$ (the curve tangent to the $t_4$
direction). In turn, the algebraic inversion problem is solved in
terms of $f,g$.
\endproclaim

\head 4. Problems\endhead
Kowalevski also gave the separation of variables for a
dynamical problem, which  is now known as the
``Kowalevski top'' [Ko2]. In this part, we looked at some links between
her work on reduction and the theory of integrable systems; we have
seen that it is quite exceptional that `reduction persist' through an
integrable system, and that this exceptional situation is related to
the motion of the poles of rational/elliptic KP solutions. One area of
KP theory which has so far eluded explicit integration is that of the
higher-rank solutions, which will be parametrized by rank-$r$ vector
bundles over a curve as well as $(r-1)$ arbitrary functions, and I
pose a problem in that area even if I don't have concrete suggestions,
mainly because I believe Veselov's little-advertised example deserves
further investigation.

\medskip
{\bf 4.1} As part of his thesis, Veselov [V] produced a hamiltonian system
  for the poles of certain solutions of the KP equation which turned
  out to be strictly of higher rank [La,LP]. Can one use the poles of
  the elliptic KP solutions of higher rank [KN, PW2] to construct an
  integrable system? Also, Veselov's system does not have a
  spectral-curve or aci model yet, to the best of my knowledge; the data
  however may be truly infinite-dimensional, because of the presence
  of the arbitrary function [PW1].

\medskip
{\bf 4.2} As for the abelian solitons of section 2,
  any matter of explicit expression (hamiltonians, Lax pairs, reduced
  KP solution) is open, and would give an application of the Kleinian
  $(\sigma )$-functions [BEL].

\medskip
{\bf 4.3} The two problems, reduction on one hand, and aci systems on the
  other, have come in contact in a fairly mysterious way, through
  elliptic/abelian solitons mainly. I do not know of an example of
  elliptic soliton that can be proved to be non-reducible to elliptic
  functions, although probably this is the generic situation.
 More significantly, at least two questions could be asked for
  the moduli space of curves: firstly, from
  experimental evidence [EEP2], as mentioned in section 3, when the
  elliptic soliton has lattice with extra symmetries there seem to be
  a better chance at reduction.
 Secondly, the question of finding equations for
  the subvarieties of ${\Cal M}_g$ where reduction occurs has been asked
  already [BE1], and the answer in genus 2 degree $N$, is a Humbert
  surface $H_{N^2}\subset {\Cal M}_2$, but one more nontrivial
  condition to detect would be whether the second KP flow is always
  tangent to the `second elliptic curve' in the splitting of Jac
  $(\Gamma )$. Again, this is unlikely, but the KP equation has a way
  of surprising us. Also, periodicity in $t$ has been calculated for
  2-dimensional elliptic solitons [S] but it might make sense to ask
  for periodicity of one of the higher flows $t_k,\ k>1$, for any
  solution, and for a moduli space of curves with this property.

\medskip
I would close by saying that the classical theory of elliptic
operators with regular-singular points still needs to be brought out
in the deeper context of algebraic geometry; one of Verdier's ideas
was indeed to embed the elliptic solitons in a moduli space of local
systems; this somewhat has been furthered by the theory of
Higgs fields, and Hitchin systems, where finally a dictionary between
completely integrable equations and monodromy problems may be
found. Sophie Kowalevski's thesis was one early place where the two
theories appeared together.



\refstyle{A}
\widestnumber\key{AMcKM}
\Refs


\ref\key{A1}
\by R.D.M. Accola
\paper 
On the number of automorphisms of a closed Riemann surface
\jour Trans. Amer. Math. Soc.
\vol 131
\yr 1968
\pages 398--408
\endref

%\ref\key{A2}
%\by R.D.M. Accola
%\paper
%Two theorems on Riemann surfaces with noncyclic automorphism groups.
%\jour Proc. Amer. Math. Soc.
%\vol 25
%\yr 1970
%\pages 598--602
%\endref

\ref\key{A2}
\by R.D.M. Accola
\paper
Plane models for Riemann surfaces 
admitting certain half-canonical linear series, I 
\inbook Riemann
surfaces and related topics:
Proceedings of the 1978 Stony Brook Conference 
\publ Ann. of Math. Stud.
\publaddr Princeton, N.J.
\yr 1981
\vol 97
\pages 7--20
\endref

\ref\key{AMcKM}
\by H. Airault, H.P. McKean and J. Moser
\paper
 Rational and elliptic solutions of the Korteweg-de Vries equation
 and a related many-body problem
\jour Comm. Pure Appl. Math.
\vol 30
\yr 1977
\pages 95--148
\endref

\ref\key{Al}
\by S.I. Al'ber
\paper
Investigation of equations of Korteweg - de Vries type by the method of 
recurrence relations
\jour J. London Math. Soc.
\vol 19 (2)
\yr 1979
\pages 467--480
\endref



\ref\key{BBM1} 
\by M.V. Babich, A.I. Bobenko and V.B. Matveev
\paper
Reductions of Riemann theta functions of genus $g$ to theta functions of
lesser genus, and symmetries of algebraic curves
\jour Dokl. Akad. Nauk SSSR
\vol 272 
\yr 1983
\pages 13--17
\endref

\ref\key{BBM2}
\by M.V.Babich, A.I. Bobenko and V.B. Matveev
\paper
Solution of nonlinear equations, integrable by the inverse problem method,
in Jacobi theta-functions and the symmetry of algebraic
curves
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\vol 49
\yr 1985
\pages 511--529
\endref

\ref\key{B}
\by H.F. Baker
\book An introduction to the theory of multiply periodic functions
\publ Cambridge Univ. Press
\publaddr Cambridge
\yr 1907
\endref


\ref\key{BM}
\by W. Barth and J. Michel
\paper
Modular curves and Poncelet polygons
\jour Math. Ann.
\vol 295
\yr 1993
\pages 25--49
\endref

\ref\key{BBME}
\by E. D. Belokolos, A.I. Bobenko, V.B. Matveev and V.Z. \`Enol'skii
\paper  Algebro-geometric
 principles of superposition of
finite-zone solutions of integrable nonlinear equations
\jour  Uspekhi Mat. Nauk.
\vol 41
\yr 1986
\pages 3--42
\endref


\ref\key{BE1}
\by E.D. Belokolos and V.Z. \`Enol'skii
\paper Algebraically integrable nonlinear equations and Humbert surfaces
\inbook 
Plasma
theory and nonlinear and turbulent processes in physics, Vol. 1,
2 (Kiev, 1987)
\publ World Sci. Publishing
\publaddr Singapore
\yr 1988
\pages 20--57
\endref 

\ref\key{BE2}
\by E.D. Belokolos and V.Z. \`Enol'skii
\paper
Verdier's elliptic solitons and the Weierstrass reduction
theory
\jour Funktsional. Anal. i Prilozhen
\vol 23
\yr 1989
\pages 57--58
\endref

\ref\key{BE3}
\by E. D. Belokolos and V.Z. \`Enol'skii
\paper  Reduction of theta functions and elliptic finite-gap
potentials
\jour  Acta Appl. Math.
\yr 1994
\vol 36
\pages 87--117
\endref



\ref\key{BT}
\by K. Berry and M. Tretkoff
\paper  The period matrix of Macbeath's curve of genus seven
\inbook
 Curves, Jacobians, and abelian Varieties (Amherst, MA, 1990),
Contemp. Math.
\publ  Amer. Math. Soc.
\publaddr Providence, RI
\yr 1992
\vol 136
\pages 31--40
\endref

\ref\key{Br}
\by Yu.V. Brezhnev
\paper  Elliptic solitons and Gr\"obner bases
\jour  preprint nlin.SI/0007028
\yr 
\vol
\pages 
\endref

\ref\key{BEL}
\by V.M.  Buchstaber, V.Z. Enol'skii and D.V. Leykin
\paper  Kleinian functions, hyperelliptic Jacobians and
applications
\inbook
  S. P. Novikov and I. M. Krichever, editors, { Reviews in
  Mathematics and Mathematical Physics}
\publ Gordon and Breach 
\publaddr  London
\vol 10:2
\yr 1997
\pages 1--125
\endref

\ref\key{BC}
\by J.L. Burchnall and  T.W. Chaundy
\paper  Commutative ordinary differential operators
\jour  Proc. Roy. Soc. London Ser. A
\year 1928
\vol 118
\pages 557--583
\endref

\ref\key{C}
\by A.B. Coble
\book
 Algebraic Geometry and Theta Functions.  Reprint of the 1929 edition
\publ American Mathematical Society Colloquium Publications
\publaddr Providence, RI
\yr 1982
\endref

\ref\key{DGU}
\by R. Dickson,  F. Gesztesy and K. Unterkofler
\paper  Algebro-geometric solutions of the Boussinesq hierarchy
\jour Rev. Math. Phys.
\yr 1999
\vol 11
\pages 823--879
\endref

\ref\key{D}
\by R. Donagi
\paper  Spectral covers
\inbook Current topics in complex algebraic geometry (Berkeley,
CA, 1992/93), Math. Sci. Res. Inst. Publ.
\publ Cambridge Univ. Press
\publaddr Cambridge
\vol 28
\yr 1995
\pages 65--86
\endref

\ref\key{DM}
\by R. Donagi and E. Markman
\paper  Spectral covers, algebraically completely integrable
Hamiltonian systems, and moduli of bundles
\inbook   Integrable systems and
quantum groups (Montecatini Terme, 1993),
Lecture Notes in Math.
\publ Springer
\publaddr Berlin
\vol 1620
\yr 1996
\pages 1--119
\endref

\ref\key{DP} 
\by R.Y. Donagi and E. Previato
\paper  Abelian solitons
\jour  Nonlinear waves: computation and theory
(Athens, GA, 1999) Math. Comput. Simulation 
\yr 2001
\vol 55
\pages 407--418
\endref

\ref\key{DKN} 
\by B.A. Dubrovin, I.M. Krichever and S.P. Novikov
\paper  Integrable systems. I
\inbook Current problems in
mathematics. Fundamental directions
\publ Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i
Tekhn. Inform.
\publaddr Moscow
\vol 4
\yr 1985
\pages 179--284
\endref 

\ref\key{DN} 
\by B.A. Dubrovin and S.P. Novikov
\paper  Periodic and conditionally periodic analogs of the
many-soliton solutions of the Korteweg-de Vries equation
\jour  Soviet Physics JETP
\yr 1974
\vol 40
\pages 1058--1063
\endref 

\ref\key{E1}
\by C.J. Earle
\paper Some Jacobian varieties which split
\inbook Complex Analysis Joensuu 1978  Lecture
Notes in Math.
\publ Springer
\publaddr Berlin
\vol 747
\yr 1979
\pages 101--107
\endref

\ref\key{E2} 
\by C.J. Earle
\paper
H. E. Rauch, function theorist
\inbook  Differential geometry and complex analysis
\pages 15--31 
\publ Springer
\publaddr Berlin
\yr 1985
\endref

\ref\key{E3}
\by C.J. Earle
\paper  Some Riemann surfaces whose Jacobians have strange product
structures
\inbook Curves, Jacobians, and abelian varieties
(Amherst, MA, 1990), Contemp. Math., vol. 136
\publ Amer. Math. Soc.
\publaddr Providence, RI
%\vol 136
\yr 1992
\pages 127--137
\endref 

\ref\key{EEP1}
\by J.C. Eilbeck, V. Z. Enolskii and  E. Previato
\paper  Varieties of elliptic solitons
\jour  J. Phys. A: Math. Gen.
\yr 2001
\vol 34
\pages 2215--2227
\endref

\ref\key{EEP2}
\by J.C. Eilbeck, V. Z. Enolskii and  E. Previato
\paper  Splittable Ince and Halphen potentials
\jour in preparation
\endref

\ref\key{EEP3}
\by J.C. Eilbeck, V.Z. Enolskii and  E. Previato
\paper  Frobenius-Stickelberger addition formula for $\sigma$
functions of higher genera
\jour in preparation
\endref


\ref\key{EP}
\by V.Z. Enolskii and E. Previato
\paper Varieties of 
ultraelliptic solitons
\jour in preparation
\endref

\ref\key{GRT}
\by F. Gesztesy, R. Ratnaseelan and G. Teschl
\paper  The KdV hierarchy and associated trace
formulas
\inbook  Recent Developments in Operator Theory and Its
Applications  (Winnipeg, MB, 1994) Oper. Theory. Adv. Appl.
\publ Birkh\"auser
\publaddr Basel
\vol 87
\yr 1996
\pages 125--163
\endref 

\ref\key{GW}
\by F. Gesztesy and R. Weikard
\paper  Elliptic algebro-geometric solutions of the KdV and AKNS
hierarchies--an analytic approach
\jour  Bull. Amer. Math. Soc. (N.S.)
\yr 1998
\vol 35
\pages 271--317
\endref

\ref\key{G}
\by D. Grant
\paper  
A generalization of Jacobi's derivative formula to dimension two
\jour  J. Reine Angew. Math. 
\vol 392
\yr 1988
\pages 125--136
\endref

\ref\key{Gr}
\by L. Greenberg
\paper  Maximal groups and signatures
\inbook Discontinuous groups and Riemann surfaces
(Proc. Conf. Univ. Maryland, College Park, MD, 1973)  Ann. of Math. Studies
\publ Princeton Univ. Press 
\publaddr Princeton, NJ
\vol 79
\yr 1974
\pages 207--226
\endref


\ref\key{H}
\by M. Halphen
\book Trait\'e des fonctions elliptiques et des leurs applications
\publ Gauthier-Villars
\publaddr Paris
\yr 1888
\endref

\ref\key{He}
\by C. Hermite
\book Oeuvres de Charles Hermite, III
\publ Gauthier-Villars
\publaddr Paris
\yr 1912
\endref

\ref\key{Hu1}
\by A. Hurwitz
\paper \"Uber einige besondere homogene lineare Differentialgleichungen
\jour Math. Ann.
\vol 26
\yr 1886
\pages 117-126
\endref

\ref\key{Hu2}
\by A. Hurwitz
\paper  \"Uber algebraische Gebilde mit eindeutingen
Transformationen in sich
\jour  Math. Ann.
\yr 1893
\vol 41
\pages 403--442
\endref

\ref\key{I}
\by E.L. Ince
\book
 Ordinary Differential Equations
\publ Dover Publications
\publaddr New York
\yr 1944
\endref

\ref\key{K}
\by V. Kanev
\paper  Spectral curves, simple Lie algebras, and Prym-Tjurin
varieties
\inbook   Theta functions (Brunswick, ME, 1987), Proc. Sympos. Pure Math.
\publ Amer. Math. Soc.
\publaddr Providence, RI
\vol 49, Part I
\yr 1989
\pages 627--645
\endref

\ref\key{Ke}
\by D. H. Kennedy
\book
 Little sparrow: a portrait of Sophia Kovalevsky
\publ Ohio University Press
\publaddr Athens, OH
\yr 1983
\endref

\ref\key{Ko1}
\by S. Kowalevski
\paper \"Uber die Reduction 
einer
bestimmten Klasse abel'scher Integrale dritten Ranges auf elliptische 
Integrale
\jour Acta Math.
\vol 4
\yr 1884 
\pages 393--414
\endref


\ref\key{Ko2}
\by S. Kowalevski
\paper 
M\'emoire sur un cas particulier de la
rotation d'un corps solide autour d'un point fixe
\jour Acta Math.
\vol 12
\yr 1889
\pages 177--232
\endref

%\ref\key{Kra}
%\by A. Krazer
%\book Lehrbuch der Thetafunktionen
%\publ Teubner
%\publaddr Leipzig
%\yr 1903
%\endref

\ref\key{Kr}
\by I.M. Krichever
\paper  Elliptic solutions of the Kadomcev-Petvia{\v s}vili
equations, and integrable systems of particles 
\jour  Funktsional. Anal. i Prilozhen
\yr 1980
\vol 14
\pages 45--54
\endref

%\ref\key{Kr2}
%\by I.M. Krichever
%\paper  Vector bundles and Lax equations on algebraic curves
%\jour Boston University Seminar (04/06/01)
%\endref

\ref\key{KN}
\by I.M. Krichever and  S.P. Novikov
\paper  Holomorphic vector bundles over Riemann surfaces and the
Kadomcev-Petvia\v svili equation, I
\jour  Funktsional. Anal. i Prilozhen
\yr 1978
\vol 12
\pages 41--52
\endref

\ref\key{Kuh}
\by R. Kuhn 
\paper  Curves of genus $2$ with split Jacobian
\jour  Trans. Amer. Math. Soc.
\yr 1988
\vol 307
\pages 41--49
\endref


\ref\key{Kuw}
\by M. Kuwata
\paper Morphisms from curves of genus 2 to elliptic curves and
Poncelet's closure theorem 
\finalinfo preprint 1999
\endref



\ref\key{L1}
\by H. Lange
\paper  \"Uber die Modulvariet\"at der Kurven vom Geschlecht $2$
\jour  J. Reine Angew. Math.
\yr 1976
\vol 281
\pages 80--96
\endref 

\ref\key{L2}
\by H. Lange
\paper  Abelian varieties with several principal polarizations
\jour  Duke Math. J.
\yr 1987
\vol 55
\pages 617--628
\endref

\ref\key{LB}
\by H. Lange and C. Birkenhake
\paper  Complex abelian varieties
\inbook  Grundlehren der Mathematischen Wissenschaften
\publ Springer-Verlag 
\publaddr Berlin
\vol 302
\yr 1992
\endref

\ref\key{LR}
\by H. Lange and S. Recillas
\paper  Abelian varieties with group action
\jour 
math.AG/0106055
\finalinfo preprint
\endref

\ref\key{La}
\by G.A. Latham
\paper Solutions of the KP equation associated to rank-three
commuting differential operator over a singular elliptic curve
\jour  Phys. D
\yr 1990
\vol 41
\pages 55--66
\endref

\ref\key{LP}
\by G.A. Latham and E. Previato
\paper  KP solutions generated from KdV by ``rank $2$''
transference
\jour  Phys. D
\yr 1996
\vol 94
\pages 95--102
\endref

\ref\key{Le}
\by M.H. Lee
\paper  Solutions of the Korteweg-de Vries equation associated to
hyperelliptic curves
\inbook  Hadronic Mechanics and Nonpotential Interactions,  Part 1
{(Cedar Falls, IA, 1990)}
\publ Nova Sci. Publ.
\publaddr Commack, NY
\pages 205--216
\yr 1992
\endref

\ref\key{M}
\by A.M. Macbeath
\paper   Hurwitz groups and surfaces
\inbook The Eightfold way, Math. Sci. Res. Inst. Publ.
\publ Cambridge Univ. Press 
\publaddr Cambridge
\yr 1999
\vol 35
\pages 103--113
\endref

\ref\key{Ma}
\by C. Maclachlan
\paper  A bound for the number of automorphisms of a compact
Riemann surface
\jour  J. London Math. Soc.
\yr 1969
\vol 44
\pages 265--272
\endref

\ref\key{Mar1}
\by H.H. Martens
\paper  Riemann matrices with  many polarizations
\inbook  Complex Analysis and its Applications  (Lectures,
Internat. Sem., Trieste, 1975)
\publ Internat. Atomic Energy Agency
\publaddr Vienna
\yr 1976
\vol III
\pages 35--48
\endref

\ref\key{Mar2}
\by H.H. Martens
\paper  On the reduction of Abelian integrals and a problem of
H. Hopf
\inbook Curves, 
 Jacobians and abelian varieties (Amherst, MA, 1990), Contemp. Math.
\publ Amer. Math. Soc.
\publaddr Providence, RI
\pages 287--296
\yr 1992
\vol 136
\endref

\ref\key{MS}
\by V.B. Matveev and A.O. Smirnov
\paper  On the Riemann theta function of a trigonal curve and
solutions of the Boussinesq and KP equations
\jour  Lett. Math. Phys. 
\yr 1987
\vol 14
\pages 25--31
\endref

\ref\key{Mu}
\by D. Mumford
\paper  Tata lectures on theta
\inbook Progress in Mathematics
\publ Birkh\"auser
\publaddr Boston, MA
\pages 287--296
\vol 28
\yr 1983
\endref

\ref\key{O}
\by Y. {\^O}nishi
\paper Complex multiplication formulae for hyperelliptic curves of genus three
\jour
Tokyo J. Math. 
\vol 21 
\yr 1998
\pages 381--431 
\endref


\ref\key{P}
\by E. Previato
\paper Poncelet's theorem in space
\jour Proc. Amer. Math. Soc. 
\yr 1999
\vol 127
\pages 2547--2556
\endref


\ref\key{PW1}
\by E. Previato and G. Wilson
\paper  Vector bundles over curves and solutions of the KP
equations
\inbook  Proc. Sympos. Pure Math.
\yr 1989
\vol 49
\pages 553--569
\endref

\ref\key{PW2}
\by E. Previato and G. Wilson
\paper  Differential operators and rank two bundles over elliptic
curves
\jour  Compositio Math.
\yr  1992
\vol 81
\pages 107--119
\endref

\ref\key{RL}
\by H.E. Rauch and J. Lewittes
\paper  The Riemann surface of Klein with 168
automorphisms
\inbook  Problems in Analysis (papers dedicated to Salomon
Bochner, 1969)
\publ Princeton Univ. Press
\publaddr Princeton, NJ
\pages 297--308
\yr 1970
\endref

\ref\key{R}
\by J. F. X.  Ries
\paper  Splittable Jacobi varieties
\inbook Curves, Jacobians, and
abelian varieties (Amherst, MA 1990), Contemp. Math.
\publ Amer. Math. Soc.
\publaddr Providence, RI
\pages 305-326
\vol 136
\yr 1992
\endref

%\ref\key{SW}
%\by G. Segal and G. Wilson
%\paper  Loop groups and equations of KdV type
%\jour  Inst. Hautes \'Etudes Sci. Publ. Math.
%\yr 1985
%\vol 61
%\pages 5--65
%\endref

\ref\key{SU}
\by M.F. Singer and F. Ulmer 
\paper  On a third order differential equation whose differential
Galois group is the simple group of $168$ elements
\inbook  Applied algebra, algebraic algorithms and error-correcting
codes (San Juan, PR, 1993),
Lecture Notes in Comput. Sci.
\publ Springer
\publaddr Berlin
\pages 316--324
\vol 673
\yr 1993
\endref

\ref\key{S}
\by A.O. Smirnov 
\paper  Finite-gap elliptic solutions of the KdV equation
\jour  Acta Appl. Math.
\yr 1994
\vol 36
\pages 125--166
\endref

\ref\key{T1}
\by A. Treibich
\paper  Tangential polynomials and elliptic solitons
\jour  Duke Math. J.
\yr 1989
\vol 59
\pages 611-627
\endref

\ref\key{T2}
\by A. Treibich
\paper  New elliptic potentials
\jour  Acta Appl. Math.
\yr 1994
\vol 36
\pages 27--48
\endref

\ref\key{TT}
\by C.L. Tretkoff and M.D. Tretkoff
\paper Combinatorial group theory, Riemann surfaces and
differential equations
\inbook Contributions to group theory,  Contemp. Math.
\publ  Amer. Math. Soc.
\publaddr Providence, RI
\pages 467--519
\vol 33
\yr 1984
\endref

\ref\key{TV1}
\by A. Treibich, J.-L. Verdier
\paper  Solitons elliptiques
\inbook The Grothendieck Festschrift, Vol. III, Progr. Math.
\publ Birkh\"auser
\publaddr Boston
\pages 437--480
\vol 88
\yr 1990
\endref

\ref\key{TV2}
\by A. Treibich, J.-L. Verdier
\paper  Vari\'et\'es de Kritchever des solitons elliptiques de
KP
\inbook Proceedings of the Indo-French Conference on Geometry (Bombay, 1989)
\publ Hindustan Book Agency
\publaddr Delhi
\pages 187--232
\yr 1993
\endref 

\ref\key{V}
\by A.P. Veselov
\paper  Rational solutions of the Kadomtsev Petviashvili equation
and Hamilton systems
\jour Uspekhi Mat. Nauk.
\yr 1980
\vol 35
\pages 195--196
\endref

%\ref\key{WW}
%\by E.T. Whittaker and G.N. Watson
%\paper  A course of modern analysis. An introduction to the
%general theory of infinite processes and of analytic functions; with
%an account of the principal transcendental functions
%\inbook Reprint of the
%fourth (1927) edition. Cambridge Mathematical Library 
%\publ Cambridge University Press
%\publaddr Cambridge
%\yr 1996
%\endref


\endRefs

\enddocument

Place where to find relation between the automorphisms of the curve
and splittability of Jacobian (with exapmle, Fermat, 
modular, Humbert, Drinfeld curve):
90h:14057 14K05 (14H35) 
Kani, E.(3-QEN); Rosen, M.(1-BRN) 
Idempotent relations and factors of Jacobians. 
Math. Ann. 284 (1989), no. 2, 307--327. 

Kamke, Erich Differentialgleichungen. (German) L\"osungsmethoden und 
L\"osungen. I: Gew\"ohnliche Differentialgleichungen. 
Neunte Auflage. Mit einem Vorwort von
Detlef Kamke. B. G. Teubner, Stuttgart, 1977.


\enddocument




