The purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic. The applications include explicit class field theory and Langlands conjectures for function fields, effect of the geometry of the theta divisor on factorization of analogs of Gauss sums, special values of function field Ga...

Let (P,Ξ) be the naturally polarized model of the Prym variety associated to the étale double cover π : C̃ → C of smooth connected curves, where Ξ ⊂ P ⊂ Pic2g−2(C̃), and g(C) = g. If L is any “non exceptional” singularity of Ξ, i.e. a point L on Ξ ⊂ Pic2g−2(C̃) such that h0(C̃, L) ≥ 4, but which cannot be expressed as π∗(M)(B) for any line bundle M on C with h0(C,M) ≥ 2 and effective divisor B ≥ 0 ...

Abstract. In this paper we prove a conjecture of Hershel Farkas [8] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infini...

We investigate the symplectic geometric and also differential aspects of moduli space connections on a compact connected Riemann surface X . Fix theta characteristic K 1 / 2 ; it defines divisor M stable vector bundles rank r degree zero. Given bundle E ? lying outside divisor, we construct natural holomorphic connection that depends holomorphically Using this connection, canonical isomorphism ...

Over a perfect field k of characteristic p > 0, we construct a “Witt vector cohomology with compact supports” for separated k-schemes of finite type, extending (after tensorisation with Q) the classical theory for proper k-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identifi...

It follows from [14, Section 17, p. 163] that 3Θ is very ample. In the singular case, D’Souza has constructed a natural compactification J̄0 for the Jacobian J0 of a complete, integral curve over an algebraically closed field [5]. The scheme J̄0 parametrizes torsion-free, rank 1 sheaves of Euler characteristic 0 on X . A natural question in this context is whether there is a canonical Cartier div...

Abstract. In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor) — denote θnull the locus of such ppavs. We describe the locus θ null ⊂ θnull where this singularity is not an ordinary double point. By using theta function methods we first show θ n...