**Integrable linear
equations and Riemann-Schottky type problems
**

**Igor Krichever**

A connection discovered by Mumford of the celebrated Fay trisecant formula with a theory of soliton
equations eventually had led Welters to his remarkable conjecture: an
indecomposable principally polarized abelian variety *X* is the Jacobian
of a curve if and only if there exists a trisecant of its Kummer variety *K*(*X*).
It was motivated by Gunning's theorem and by another famous conjecture: the
Jacobians of curves are exactly the indecomposable principally polarized
abelian varieties whose theta-functions provide explicit solutions of the
so-called KP equation. The latter was proposed earlier by Novikov and was
unsettled at the time of the Welter's work. It was proved later by
T. Shiota and until recently has remained the most effective solution of
the classical Riemann-Schottky problem.

The characterization of the Jacobians proposed by the trisecant conjecture is
much stronger. The proof of this conjecture based on a notion of integrable *linear
equations* and a new type of cubic identities for the theta-functions valid
for the case of

Jacobians on the theta-divisor will be presented. We will also discuss applications
of integrable equations of the soliton theory for the characterization problem
of Prym varieties.