Integrable linear equations and the Riemann-Schottky problem

Abstract

We prove that an indecomposable principally polarized abelian variety X is the Jacobain of a curve if and only if there exist vectors U = 0, V such that the roots x i (y) of the theta-functional equation θ(U x + V y + Z) = 0 satisfy the equations of motion of the formal infinite-dimensional Calogero-Moser system.

A characterization of Prym varieties

“…The proof is identical to that in the part (b) of Lemma 12 in [3] (compare with the proof of the corollary in [7]). …”

“…The proof of the lemma is identical to the proof of lemma 3.5 in [7]. The function ψ is first defined for Z / ∈ Σ − as the inverse image ψ = j * ψ BA of the Baker-Akhiezer function, which is known to be globally defined on Pic(Γ).…”

“…The goal of this work is to solve the characterization problem of the Prym varieties using the new approach proposed in the author's previous work [7], where it was shown that KP equation contains excessive information and the Jacobian locus can be characterized in terms only of one of its auxiliary linear equations. where p, E are constants.…”

“…Note that equations (1.11) are analogues of the equations derived in [8] and called in [7] the formal Calogero-Moser system. In a similar way, if we represent an entire function τ as a product…”

“…At the local level the main problem is to find the translational invariant normalization of ξ ± s which defines wave solutions uniquely up to a (x, t)-independent factor. Following the ideas of [11] and [7] we fix such a normalization using extensions of ξ In the last section we show that for each Z / ∈ Σ ± a local λ-periodic wave solution is the common eigenfunction of a commutative ring A Z ± of ordinary differential operators. The coefficients of these operators are independent of ambiguities in the construction of ψ.…”