Abelian solutions of the soliton equations and Riemann–Schottky problems

The present article is an exposition of the author's talk at the conference dedicated to the 70th birthday of S. P. Novikov. The talk contained the proof of Welters' conjecture which proposes a solution of the classical Riemann–Schottky problem of characterizing the Jacobians of smooth algebraic curves in terms of the existence of a trisecant of the associated Kummer variety, and a solution of another classical problem of algebraic geometry, that of characterizing the Prym varieties of unramified covers.