Schottky problem

Schottky problem
Name	Schottky problem
Field	Algebraic geometry, Complex analysis, Number theory
Introduced	Early 20th century
Notable people	Friedrich Schottky, David Mumford, Igusa, Jun-ichi Igusa
Related	Torelli theorem, Jacobian variety, Siegel modular variety

Schottky problem The Schottky problem asks for a characterization of principally polarized abelian varieties that are Jacobians of algebraic curves, and it lies at the intersection of Riemann surface, Algebraic curve, Abelian variety, Theta function, and Siegel modular form theory. It connects classical work by Friedrich Schottky, modern advances by David Mumford, Igusa, and relations to the Torelli theorem, Hodge conjecture, Moduli space of curves, and computational approaches from Arakelov theory. The problem motivates research across Complex manifold, Number theory, Representation theory, Integrable system, and Mathematical physics communities.

Introduction

The question originates in the setting of period matrices for Riemann surfaces, asking which points in the Siegel upper half space correspond to period matrices of Jacobians of Algebraic curves, and it is tightly tied to the classical theory of Theta function, Riemann theta, Theta divisor, and Principally polarized abelian variety. Early formulations by Friedrich Schottky and later clarification by Igusa and Andreotti reframed the problem in terms of equations on the moduli of Principally polarized abelian varietys and loci inside the Siegel modular variety. This locus, commonly called the Jacobian locus, relates to fundamental results such as the Torelli theorem, the Schottky–Jung relations, and to techniques developed by David Mumford and collaborators.

Historical Background

Origins trace to late 19th- and early 20th-century work on period matrices by Friedrich Schottky and contemporaries studying Riemann surface uniformization, while subsequent milestones include contributions by Jun-ichi Igusa on theta constants, Andreotti and Mayer on singularities of theta divisors, and foundational geometric methods by David Mumford. The development of moduli theory by Alexander Grothendieck, Igor Dolgachev, and work on compactifications of Siegel modular varietys by Pierre Deligne and David Mumford influenced rigorous formulations, with connections to results by Griffiths on variations of Hodge structure and to later advances by Arbarello and Cornalba on the moduli space of curves. Analytic and arithmetic perspectives were advanced by André Weil, Igusa, and later researchers in Arithmetic geometry.

Formulation and Equivalent Statements

One standard formulation asks: which points of the Siegel modular variety correspond to Jacobians of genus g curves, equivalently which principally polarized abelian varieties satisfy conditions expressible by vanishing of suitable Theta function relations, and how to express these by modular forms such as Siegel modular forms of degree g studied by Igusa. Equivalences include characterizations via the geometry of the Theta divisor per work of Andreotti and Mayer, representation-theoretic descriptions involving Symplectic group (Sp) actions as in the theory of Siegel modular forms, and characterizations using the Torelli theorem and Schottky–Jung relations studied by Friedrich Schottky and later by Schottky and Jung commentators. Alternative formulations involve Prym variety criteria, integrable systems via the Kadomtsev–Petviashvili equation and connections developed by Novikov and Shiota, and Hodge-theoretic descriptions linked to the Hodge conjecture framework championed by Phillip Griffiths.

Methods of Attack and Major Results

Methods combine analytic techniques in Theta function theory from Riemann and Weierstrass, algebraic geometry via the work of David Mumford, Alexander Grothendieck, and Arbarello, arithmetic approaches from Igusa and André Weil, and integrable-systems methods by Sato, Novikov, and Shiota. Major results include Igusa's characterization in low genera by explicit theta-null relations, the Andreotti–Mayer codimension bounds for the singular locus of the theta divisor, Shiota's solution of the characterization of Jacobians via the KP equation following ideas of Novikov and Dubrovin, and Mumford's contributions to moduli compactification. Further progress involves characterization of the Prym locus by work of Beauville and Mumford, and explicit modular form descriptions in small genus by Igusa and computational advances by researchers associated with Institut des Hautes Études Scientifiques and Max Planck Institute for Mathematics.

Applications and Connections

The problem has ramifications for the Torelli theorem and reconstruction of Algebraic curves from period data as used in Enumerative geometry and Topological quantum field theory approaches, applications in Cryptography when using Jacobians in discrete-log settings studied by Vladimir S. Miller and Peter L. Montgomery, links to Integrable system soliton theory via the KP equation and work by Dubrovin and Krichever, and implications for arithmetic geometry surrounding rational points inspired by Faltings and André Oort conjectures. It intersects computational algebraic geometry developed by David Cox and Bernd Sturmfels and informs research at institutions like Clay Mathematics Institute when considering moduli and period mapping problems.

Open Problems and Conjectures

Significant open questions include explicit characterization of the Jacobian locus in higher genera via modular form equations sought by practitioners at IHÉS and MPIM, precise determination of defining ideals for the Jacobian locus as conjectured in various forms by Mumford and respondents, effective descriptions of Prym and intermediate Jacobian loci studied by Beauville and Clemens, and arithmetic refinements related to the André–Oort conjecture and the distribution of rational points pursued by Yves André and Richard Pink. Other active directions involve extensions to Tropical geometry by researchers influenced by Bernd Sturmfels and Grigory Mikhalkin, advances in computational certification by teams at Max Planck Institute for Mathematics and Microsoft Research, and further Hodge-theoretic bridges to the Hodge conjecture community.

Category:Algebraic geometry