Characterization of Jacobians of curves with involution

A problem of characterization of Jacobians among indecomposable principally polarized abelian varieties is the famous Riemann-Schottky problem. The first effective solution of the Riemann-Schottky problem was obtained by T. Shiota, who proved the famous Novikov conjecture: the Jacobians of smooth algebraic curves are exactly those indecomposable principally polarized Abelian varieties (ppavs) whose theta functions define solutions of the Kadomtsev-Petviashvili (KP) equation. Subsequently, the author proved Welters' triple secant conjecture, which is by far the most powerful characterization of Jacobians. The talk will deal with the characterization of the Jacobians of curves with involution. Such curves arise when constructing solutions of soliton equations with symmetries.