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To a complete non-singular [[Algebraic curve|algebraic curve]]  $  C $
of genus  $  g $
one can associate its Jacobian (cf. [[Jacobi variety|Jacobi variety]]). This is an [[Abelian variety|Abelian variety]]  $  J( C) $
of dimension  $  g $
together with a principal polarization  $  \Theta $(
cf. [[Polarized algebraic variety|Polarized algebraic variety]]). B. Riemann showed in 1857 that algebraic curves of genus  $  g $
depend on  $  3g- 3 $
parameters (for  $  g > 1 $). 
But principally-polarized Abelian varieties of dimension  $  g $
depend on  $  {g( g+ 1)/2 } $
parameters. Since for  $  g \geq  4 $
one has  $  g( g+ 1)/2 > 3g- 3 $, 
the question arises which principally-polarized Abelian varieties are Jacobians. This question was posed by Riemann, but is known as the Schottky problem. More precisely, if  $  {\mathcal M} _ {g} $
is the moduli space of curves of genus  $  g $(
i.e. the parameter space of isomorphism classes of such curves of genus  $  g $, 
cf. [[Moduli theory|Moduli theory]]) and if  $  {\mathcal A} _ {g} $
is the moduli space of principally-polarized Abelian varieties of dimension  $  g $, 
there is a mapping  $  {\mathcal M} _ {g} \rightarrow {\mathcal A} _ {g} $, 
and the problem is to characterize the closure  $  J _ {g} $
of its image. For  $  g \leq  3 $
one has  $  J _ {g} = {\mathcal A} _ {g} $.

Here only the case where the curve is a complex curve or Riemann surface is considered. If one chooses a symplectic basis  $  \alpha _ {1} \dots \alpha _ {g} $, 
$  \beta _ {1} \dots \beta _ {g} $
of the homology  $  H _ {1} ( C, \mathbf Z ) $
and a basis of the space of holomorphic  $  1 $-
forms  $  \omega _ {1} \dots \omega _ {g} $
on  $  C $
such that  $  \int _ {\alpha _ {i}  } \omega _ {j} = \delta _ {ij }  $(
Kronecker delta), one obtains the period matrix  $  \tau = ( \tau _ {ij }  )= ( \int _ {\beta _ {i}  } \omega _ {j} ) $. 
This matrix lies in the Siegel upper half-space  $  {\mathcal H} _ {g} $, 
the set of all complex symmetric  $  ( g \times g ) $-
matrices whose imaginary part is positive definite.

The Jacobian  $  J( C) $
is given by the complex torus  $  \mathbf C  ^ {g} / \Lambda _  \tau  $, 
where  $  \Lambda _  \tau  = \mathbf Z  ^ {g} + \tau \mathbf Z  ^ {g} $, 
and  $  \Theta $
is then the divisor of Riemann's [[Theta-function|theta-function]]

$$ 
\theta ( \tau , z)  =  \sum _ {m \in \mathbf Z  ^ {g} }
e ^ {\pi i (  ^ {t} m \tau m + 2  ^ {t} mz) } .
$$

The moduli space  $  {\mathcal A} _ {g} $
can be obtained as  $  {\mathcal H} _ {g} /  \mathop{\rm Sp} ( 2g, \mathbf Z ) $, 
where  $   \mathop{\rm Sp} ( 2g, \mathbf Z ) $
acts naturally on  $  {\mathcal H} _ {g} $. 
Coordinates on (a covering of)  $  {\mathcal A} _ {g} $
are provided by the  "theta constanttheta constants" , which are the values at  $  z= 0 $
of the theta-functions

$$ 
\theta \left [

for  $  \epsilon  ^  \prime  , \epsilon ^ {\prime\prime } \in ( 1 / 2) \mathbf Z  ^ {g} / \mathbf Z  ^ {g} $.

The first result is due to W. Schottky for  $  g = 4 $. 
He showed in 1888 that a certain polynomial of degree 16 in the theta constants vanished on  $  J _ {4} $, 
but not everywhere on  $  {\mathcal A} _ {4} $. 
J.-I. Igusa showed much later that its [[Zero divisor|zero divisor]] equals  $  J _ {4} $.

The next step was made by Schottky and H.W.E. Jung in 1909, who constructed expressions in the theta constants that vanish on  $  J _ {g} $
by means of double unramified coverings of curves. (Today this is called the theory of Prym varieties.) These expressions define a certain locus  $  S _ {g} $(
called the Schottky locus) which contains  $  J _ {g} $. 
It is conjectured that  $  S _ {g} = J _ {g} $, 
and this is the Schottky problem in restricted sense. B. van Geemen showed in 1983 that  $  J _ {g} $
is an irreducible component of  $  S _ {g} $.

Since the Schottky problem asks for a characterization, different approaches might lead to different answers. One approach uses the fact that the theta divisor of a Jacobian is singular (for  $  g \geq  4 $): 
the dimension of its singular locus is  $  \geq  g- 4 $. 
It is therefore natural to consider the locus  $  N _ {g}  ^ {m} \subset  {\mathcal A} _ {g} $
of principally-polarized Abelian varieties  $  ( A, \Theta ) $
of dimension  $  g $
with  $   \mathop{\rm dim} (  \mathop{\rm Sing} ( \Theta )) \geq  m $. 
Then  $  J _ {g} \subset  N _ {g} ^ {g- 4 } $, 
and  $  J _ {g} $
is indeed an irreducible component, as A. Andreotti and A. Mayer showed. However,  $  N _ {g} ^ {g- 4 } $
has other components, so this is not strong enough.

Another way to try to distinguish Jacobians and general principally-polarized Abelian varieties uses trisecants. Below, only principally-polarized Abelian varieties that are indecomposable, i.e. not products, are considered. For such an Abelian variety  $  ( X= \mathbf C  ^ {g} / \Lambda _  \tau  , \Theta ) $
there is a mapping to projective space  $  \Phi = \Phi _ {2 \Theta }  :  X \rightarrow \mathbf P  ^ {N} $
with  $  N= 2  ^ {g} - 1 $
given by the theta-functions  $  \theta [ \begin{array}{c}
\epsilon  ^  \prime  \\
 0 
\end{array}
 ]( 2 \tau , 2z) $
and its image is the Kummer variety  $  K( X, \Theta ) $(
cf. [[Kummer surface|Kummer surface]]). If  $  X = J( C) $
and  $  a, b, c, d \in C $
and if  $  r \in X $
is such that  $  2r= a+ b- c- d $, 
then the three points  $  \Phi ( r) $, 
$  \Phi ( r- b+ c) $
and  $  \Phi ( r- b+ d) $
in  $  \mathbf P  ^ {N} $
are collinear, i.e. define a trisecant. (Translated in terms of theta-functions this is Fay's trisecant identity.) It is conjectured that if  $  ( X, \Theta ) $
is an indecomposable principally-polarized Abelian variety whose Kummer variety admits a trisecant, then it is a Jacobian. (It was proved by A. Beauville and O. Debarre that the existence of a trisecant implies  $   \mathop{\rm dim}  {\textrm{ Sing  } } ( \Theta ) \geq  g- 4 $.) 
A weakened version of this is proved by R.C. Gunning and G. Welters. A simplified form reads: If the Kummer variety admits a continuous family of trisecants and  $   \mathop{\rm dim}  {\textrm{ Sing  } } ( \Theta ) = g- 4 $, 
then the Abelian variety is a Jacobian.

One can consider an infinitesimal version of this. Let  $  \theta _ {2} [ \sigma ]( \tau , z) = \theta [ \begin{array}{c}
\sigma \\
 0 
\end{array}
 ]( 2 \tau , 2z) $. 
Then the criterion is: The Abelian variety is a Jacobian if there exist constant vector fields  $  D _ {1} $, 
$  D _ {2} $, 
$  D _ {3} $
on  $  X $
and  $  d \in \mathbf C $
such that

$$ \tag{* }
\left ( \left ( D _ {1}  ^ {4} - D _ {1} D _ {3+} {
\frac{3}{4}
 } D _ {2}  ^ {2} + d \right
) \theta _ {2} [ \sigma ] \right ) ( \tau , 0) = 0 \  {\textrm{ 
for  all   } } \sigma .
$$

This is known as Novikov's conjecture and was proved by M. Shiota in 1986. Another (more geometric) proof was given by E. Arbarello and C. De Concini. The equation (*) is called the K–P-equation (after Kadomtsev–Petviashvili) and generalizes the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]. It is the first of a whole hierarchy of equations.

By the Riemann identity

$$ 
\theta ( x+ y) \theta ( x- y)  =  \sum _  \sigma 
\theta _ {2} [ \sigma ]( x) \theta _ {2} [ \sigma ]( y) ,
$$

this translates into the fact that  $  u= D _ {1}  ^ {2}  \mathop{\rm log}  \theta $
satisfies the partial differential equation

$$ 
D _ {1} ( D _ {1}  ^ {3} u + uD _ {1} u+ D _ {2} u)- D _ {3}  ^ {2} u  =  0.
$$

Yet another approach (due to van Geemen and G. van der Geer) and gives a connection to the approach using the K–P-equation. The theta-functions  $  \theta _ {2} [ \sigma ] ( \tau , z) $
map an Abelian variety to  $  \mathbf P  ^ {N} $. 
They also define a mapping from the moduli space  $  {\mathcal A} _ {g} ( 2, 4) $(
a covering of  $  {\mathcal A} _ {g} $) 
to this projective space, where the image of the class of  $  ( X, \Theta ) $
is the image of the origin of  $  X $
under  $  \Phi $. 
It is conjectured that the intersection of the image of the moduli space and the Kummer variety of an indecomposable Abelian variety has dimension  $  \leq  2 $
and equals  $  2 $
exactly when  $  ( X, \Theta ) $
is a Jacobian. Similarly for the intersection of the Kummer variety and the tangent space to the moduli space at the point defined by  $  ( X, \Theta ) $. 
By the heat equation for the theta-function this last conjecture is equivalent to a statement about the set of common zeros  $  F _ {X} \subset  X $
of the space  $  \Gamma _ {00 }  \subset  \Gamma ( X, O( 2 \Theta )) $
of theta-functions  $  \sum _  \sigma  a _  \sigma  \theta _ {2} [ \sigma ]( \tau , z) $(
for fixed  $  \tau $) 
which vanish with multiplicity  $  \geq  4 $
at the origin of  $  X= \mathbf C  ^ {g} / \Lambda _  \tau  $. 
For a Jacobian  $  X= J( C) $
this set  $  F _ {X} $
consists of the image  $  \{ {x- y } : {x, y \in C } \} \subset  J( C) $
of  $  C- C $
in the Jacobian (with a slight exception for  $  g= 4 $), 
as was proved by Welters. Conjecturally, Jacobians are now characterized by  $  F _ {X} \neq \{ 0 \} $.

These conjectures were refined by R. Donagi in [[#References|[a2]]] to a much stronger conjecture which describes the Schottky locus  $  {\mathcal R} {\mathcal S} _ {g} $
in  $  {\mathcal R} {\mathcal A} _ {g} $, 
the moduli space of principally-polarized Abelian varieties  $  ( A, \Theta , p) $
with a non-zero point  $  p $
of order  $  2 $
on  $  A $. 
This conjecture is very strong: it implies a strengthened version of the Novikov conjecture and all the conjectures of van Geemen and van der Geer. These last ones are obtained by intersecting  $  {\mathcal R} {\mathcal S} _ {g} $
with the boundary of the compactified moduli space  $  \overline{ {{\mathcal R} {\mathcal A} }}\; _ {g} $, 
while the Novikov conjecture follows by infinitesimalizing this. Donagi proved his conjecture for  $  g \leq  5 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beauville,   "Le problème de Schottky et la conjecture de Novikov"  ''Astérisque'' , '''152–153'''  (1988)  pp. 101–112  (Sém Bourbaki, Exp. 675)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Donagi,   "The Schottky problem"  E. Sernesi (ed.) , ''Theory of Moduli'' , ''Lect. notes in math.'' , '''1337''' , Springer  (1988)  pp. 84–137</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. van der Geer,   "The Schottky problem"  F. Hirzebruch (ed.)  J. Schwermer (ed.)  S. Suter (ed.) , ''Arbeitstagung Bonn 1984'' , ''Lect. notes in math.'' , '''1111''' , Springer  (1985)  pp. 385–406</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Mumford,   "Curves and their Jacobians" , Univ. Michigan Press  (1975)</TD></TR></table>