The Schottky problem asks which principally polarized abelian varieties are Jacobians of algebraic curves, that is, what characterizes the image of the Torelli map from the moduli of genus-g curves into the moduli of abelian varieties. The integrable-systems approach answers it through the Kadomtsev-Petviashvili equation: a curve together with a marked point produces a Baker-Akhiezer function whose associated potential solves the KP equation, and geometrically this is visible as special secant lines of the Kummer variety, the image of the abelian variety under the linear system of second-order theta functions. Weil reducibility and the Riemann theta singularity theorem give the Fay-Gunning trisecant identity for Jacobians; Welters conjectured the converse, that a single trisecant forces the variety to be a Jacobian, and Krichever proved the most degenerate case in which the trisecant becomes a flex line, by constructing periodic wave solutions and a commutative ring of ordinary differential operators whose spectral curve is the sought-after curve.
Lecture notes from a BIMSA course (June 2024) presenting the integrable-systems solution of the Schottky problem. A compact complex algebraic curve C of genus g determines its Jacobian, the quotient …