# Siegel modular variety

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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, the 20th-century German number theorist who introduced the varieties in 1943. [2] [3]

## 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]

## 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]

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]

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

## 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]

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 the Mordell conjecture (now known as Faltings's theorem) would hold if the Shafarevich finiteness conjecture was true by introducing Parshin's trick. [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]

## Related Research Articles

In arithmetic geometry, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001.

In mathematics, in particular 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. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli.

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.

In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory.

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In algebraic geometry, the Kodaira dimensionκ(X) measures the size of the canonical model of a projective variety X.

In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

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This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.

In algebraic geometry, the Barth–Nieto quintic is a quintic 3-fold in 4 dimensional projective space studied by Wolf Barth and Isidro Nieto (1994) that is the Hessian of the Segre cubic.

In mathematics, the André–Oort conjecture is an open problem in Diophantine geometry, a branch of number theory, that builds on the ideas found in the Manin–Mumford conjecture, which is now a theorem. A prototypical version of the conjecture was stated by Yves André in 1989 and a more general version was conjectured by Frans Oort in 1995. The modern version is a natural generalisation of these two conjectures.

Yujiro Kawamata is a Japanese mathematician working in algebraic geometry.

Aleksei Nikolaevich Parshin, sometimes romanized as Alexey Nikolaevich Paršin, is a Russian mathematician, specializing in arithmetic geometry.

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## References

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