Schottky Problem Research Articles

Quasi-periodic waves to the defocusing nonlinear Schrödinger equation

A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the (1+1) -dimensional and (2+1) -dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann–Schottky problem for hyperelliptic curves with 2 infinite points.

Hirota varieties and rational nodal curves

The Hirota variety parameterizes solutions to the KP equation arising from a degenerate Riemann theta function. In this work, we study in detail the Hirota variety arising from a rational nodal curve. Of particular interest is the irreducible subvariety defined as the image of a parameterization map, we call this the main component. Proving that this is an irreducible component of the Hirota variety corresponds to solving a weak Schottky problem for rational nodal curves. We solve this problem up to genus nine using computational tools.

Characteristic cycles and the microlocal geometry of the Gauss map, I

We show that for the reductive Tannaka groups of semisimple holonomic $\mathscr{D}$-modules on abelian varieties, every Weyl group orbit of weights of their universal cover is realized by a conic Lagrangian cycle on the cotangent bundle. Applications include a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties and a criterion for the simplicity of the arising Lie algebras.

Computing theta functions with Julia

We present a new package Theta.jl for computing with the Riemann theta function. It is implemented in Julia and offers accurate numerical evaluation of theta functions with characteristics and their derivatives of arbitrary order. Our package is optimized for multiple evaluations of theta functions for the same Riemann matrix, in small dimensions. As an application, we report on experimental approaches to the Schottky problem in genus five.

Semicontinuity of Gauss maps and the Schottky problem

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti–Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav’s defined by the topological type of the theta divisor.

An explicit solution to the weak Schottky problem

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.

Characteristic cycles and the microlocal geometry of the Gauss map, II

Abstract We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

On the Schottky problem for genus five Jacobians with a vanishing theta null

We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.

Metric distortion in the geometric Schottky problem

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus \(J(\mathcal{M}_g)\) in the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties as an algebraic subvariety. By viewing \(\mathcal{A}_g\) as a noncompact metric space coming from its structure as a locally symmetric space and \(J(\mathcal{M}_g)\) as a metric subspace, we compare the subspace metric d and the induced length metric l on \(J(\mathcal{M}_g)\). Consequently, we clarify the nature of the metric distortion of the subspace \(J(\mathcal{M}_g)\) and hence settle a problem posed by Farb (2006) on the metric distortion of \(J(\mathcal{M}_g)\) inside \(\mathcal{A}_g\) in a certain sense (see Theorem 1.5 and Corollary 1.6).

Schottky algorithms: Classical meets tropical

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.

Standard conjecture of Künneth type with torsion coefficients

A. Venkatesh raised the following question, in the context of torsion automorphic forms: can the mod p analogue of Grothendieck’s standard conjecture of Kunneth type be true (especially for compact Shimura varieties)? In the first theorem of this article, by using a topological obstruction involving Bockstein, we show that the answer is in the negative and exhibit various counterexamples, including compact Shimura varieties. It remains an open geometric question whether the conjecture can fail for varieties with torsion-free integral cohomology. Turning to the case of abelian varieties, we give upper bounds (in p) for possible failures, using endomorphisms, the Hodge–Lefschetz operators, and invariant theory. The Schottky problem enters into consideration, and we find that, for the Jacobians of curves, the question of Venkatesh has an affirmative answer for every prime number p.

The degree of the Gauss map of the theta divisor

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppav's by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.

Hyperelliptic Schottky problem and stable modular forms

It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.

Cubic threefolds, Fano surfaces and the monodromy of the Gauss map

The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.

Theta divisors with curve summands and the Schottky problem

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

Corrigendum to: “Andreotti-Mayer loci and the Schottky problem” [cf. Documenta Math. 13 (2008) 398–440

The main result of [2] (C. Ciliberto, G. van der Geer: Andreotti-Mayer Loci and the Schottky problem. Documenta Math. 13 (2008), 389–440) is based on a proposition whose proof is incorrect. We therefore give an alternative proof of the main result of [2].

Algebraic and rigid geometry on the Schottky problem

Abstract We study the Schottky problem by giving the KP characterization of Jacobian varieties among abelian varieties in terms of their algebraic or nonarchimedean theta functions.

Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

On the Teichmüller geodesic generated by the L-shaped translation surface tiled by three squares

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmuller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperelliptic automorphism comes from the 3-square-tiled translation surface itself. This makes the computation of an exact expression for period matrices of other Riemann surfaces in that family by the classical method impossible. We nevertheless give the solution to the Schottky problem for that family: we exhibit explicit necessary and sufficient conditions for a Riemann matrix to be a period matrix of a Riemann surface in the family, involving the vanishing of a genus 3 theta characteristic on a family of double covers.

On the second successive minimum of the Jacobian of a Riemann surface

To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show that in the case of a hyperelliptic surface this geometric invariant is bounded from above by a constant and that for any surface of genus g the square of the second successive minimum is equally of order log(g). We obtain improved bounds for the kth successive minimum of the Jacobian, if the surface contains small simple closed geodesics.