Siegel modular variety

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A 2D slice of a Calabi-Yau quintic. One such quintic is birationally equivalent to the compactification of the Siegel modular variety A1,3(2). CalabiYau5.jpg
A 2D slice of a Calabi–Yau quintic. One such quintic is birationally equivalent to the compactification of the Siegel modular variety A1,3(2).

In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, [2] a 20th century German mathematician who specialized in number theory. He introduced [2] Siegel modular varieties in a 1943 paper. [3]

Algebraic variety object of study in algebraic geometry

Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

Abelian variety projective Abelian algebraic group

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.

Contents

Siegel modular varieties are the most basic examples of Shimura varieties. [4] Siegel modular varieties generalize moduli spaces of algebraic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions. [1] They also have applications to black hole entropy and conformal field theory. [5]

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.

In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.

Construction

The Siegel modular variety Ag, which parametrize principally polarized abelian varieties of dimension g, can be constructed as the complex analytic spaces constructed as the quotient of the Siegel upper half-space of degree g by the action of a symplectic group. Complex analytic spaces have naturally associated algebraic varieties by Serre's GAGA. [1]

In mathematics, a complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic spaces are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that

In mathematics, the Siegel upper half-space of degree g is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939).

The Siegel modular variety Ag(n), which parametrize principally polarized abelian varieties of dimension g with a level n-structure, arises as the quotient of the Siegel upper half-space by the action of the principal congruence subgroup of level n of a symplectic group. [1]

In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X.

A Siegel modular variety may also be constructed as a Shimura variety defined by the Shimura datum associated to a symplectic vector space. [4]

In mathematics, a symplectic vector space is a vector space V over a field F equipped with a symplectic bilinear form.

Properties

The Siegel modular variety Ag has dimension g(g + 1)/2. [1] [6] Furthermore, it was shown by Yung-Sheng Tai, Eberhard Freitag, and David Mumford that Ag is of general type when g  7. [1] [7] [8] [9]

Eberhard Freitag German mathematician

Eberhard Freitag is a German mathematician, specializing in complex analysis and especially modular forms.

David Mumford American mathematician

David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.

Siegel modular varieties can be compactified to obtain projective varieties. [1] In particular, a compactification of A2(2) is birationally equivalent to the Segre cubic which is in fact rational. [1] Similarly, a compactification of A2(3) is birationally equivalent to the Burkhardt quartic which is also rational. [1] Another Siegel modular variety, denoted A1,3(2), has a compactification that is birationally equivalent to the Barth–Nieto quintic which is birationally equivalent to a modular Calabi–Yau manifold with Kodaira dimension zero. [1]

Applications

Siegel modular forms arise as vector-valued differential forms on Siegel modular varieties. [1] Siegel modular varieties have been used in conformal field theory via the theory of Siegel modular forms. [10] In string theory, the function that naturally captures the microstates of black hole entropy in the D1D5P system of supersymmetric black holes is a Siegel modular form. [5]

In 1968, Aleksei Parshin showed that showed that the Mordell conjecture (now known as Faltings's theorem) would hold if the Shafarevich finiteness conjecture was true by introducing Parshin's trick, which gives an embedding of a algebraic curve into the Siegel modular variety. [11] [12] In 1983 and 1984, Gerd Faltings completed the proof of the Mordell conjecture by proving the Shafarevich finiteness conjecture. [13] [14] [12] The main idea of Faltings' proof is the comparison of Faltings heights and naive heights via Siegel modular varieties. [15]

See also

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References

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