Schottky problem
Encyclopedia
In mathematics
, the Schottky problem, named after Friedrich Schottky
, is a classical question of algebraic geometry
, asking for a characterisation of Jacobian varieties amongst abelian varieties.
s C of a given genus
g, and their Jacobians J. There is a moduli space
Mg of such curves, and a moduli space Ag of abelian varieties of dimension g, which are principally polarized. There is a morphism
which on points (geometric points, to be more accurate) takes C to J. The content of Torelli's theorem is that ι is injective (again, on points). The Schottky problem asks for a description of the image of ι.
It is discussed for g ≥ 4: the dimension of Mg is 3g − 3, for g ≥ 2, while the dimension of Ag is g(g + 1)/2. This means that the dimensions are the same (0, 1, 3, 6) for g = 0, 1, 2, 3. Therefore g = 4 is the first interesting case, and this was studied by F. Schottky in the 1880s. Schottky applied the theta constants, which are modular form
s for the Siegel upper half-space, to define the Schottky locus in Ag. A more precise form of the question is to determine whether the image of ι essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).
of a particular type, arising from a lattice
in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surface
s.
One of the major achievements of Bernhard Riemann
was his theory of complex tori and theta functions. Using the Riemann theta function, necessary and sufficient conditions on a lattice were written down by Riemann for a lattice in Cg to have the corresponding torus embed into complex projective space
. (The interpretation may have come later, with Solomon Lefschetz
, but Riemann's theory was definitive.) The data is what is now called a Riemann matrix. Therefore the complex Schottky problem becomes the question of characterising the period matrices of compact Riemann surfaces of genus g, formed by integrating a basis for the abelian integrals round a basis for the first homology group, amongst all Riemann matrices.
theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Schottky problem, named after Friedrich Schottky
Friedrich Schottky
Friedrich Hermann Schottky was a German mathematician who worked on elliptic, abelian, and theta functions and invented Schottky groups. He was born in Breslau, Germany and died in Berlin....
, is a classical question of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, asking for a characterisation of Jacobian varieties amongst abelian varieties.
Geometric formulation
More precisely, one should consider algebraic curveAlgebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s C of a given genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g, and their Jacobians J. There is a moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
Mg of such curves, and a moduli space Ag of abelian varieties of dimension g, which are principally polarized. There is a morphism
- ι: Mg → Ag
which on points (geometric points, to be more accurate) takes C to J. The content of Torelli's theorem is that ι is injective (again, on points). The Schottky problem asks for a description of the image of ι.
It is discussed for g ≥ 4: the dimension of Mg is 3g − 3, for g ≥ 2, while the dimension of Ag is g(g + 1)/2. This means that the dimensions are the same (0, 1, 3, 6) for g = 0, 1, 2, 3. Therefore g = 4 is the first interesting case, and this was studied by F. Schottky in the 1880s. Schottky applied the theta constants, which are modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s for the Siegel upper half-space, to define the Schottky locus in Ag. A more precise form of the question is to determine whether the image of ι essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).
Period lattice formulation
If one describes the moduli space Ag in intuitive terms, as the parameters on which an abelian variety depends, then the Schottky problem asks simply what condition on the parameters implies that the abelian variety comes from a curve's Jacobian. The classical case, over the complex number field, has received most of the attention, and then an abelian variety A is simply a complex torusComplex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense...
of a particular type, arising from a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surface
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
s.
Riemann's matrix formulation
NB a Riemann matrix is quite different from any Riemann tensorOne of the major achievements of Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
was his theory of complex tori and theta functions. Using the Riemann theta function, necessary and sufficient conditions on a lattice were written down by Riemann for a lattice in Cg to have the corresponding torus embed into complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
. (The interpretation may have come later, with Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
, but Riemann's theory was definitive.) The data is what is now called a Riemann matrix. Therefore the complex Schottky problem becomes the question of characterising the period matrices of compact Riemann surfaces of genus g, formed by integrating a basis for the abelian integrals round a basis for the first homology group, amongst all Riemann matrices.
Geometry of the problem
There are a number of geometric approaches, and the question has also been shown to implicate the Kadomtsev–Petviashvili equation, related to solitonSoliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
theory.