# Siegel modular variety

Last updated 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,  a 20th century German mathematician who specialized in number theory. He introduced  Siegel modular varieties in a 1943 paper. 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. 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.  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.  They also have applications to black hole entropy and conformal field theory. 

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. 

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. 

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. 

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.   Furthermore, it was shown by Yung-Sheng Tai, Eberhard Freitag, and David Mumford that Ag is of general type when g  7. Eberhard Freitag is a German mathematician, specializing in complex analysis and especially modular forms. 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.  In particular, a compactification of A2(2) is birationally equivalent to the Segre cubic which is in fact rational.  Similarly, a compactification of A2(3) is birationally equivalent to the Burkhardt quartic which is also rational.  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. 

## Applications

Siegel modular forms arise as vector-valued differential forms on Siegel modular varieties.  Siegel modular varieties have been used in conformal field theory via the theory of Siegel modular forms.  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. 

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.   In 1983 and 1984, Gerd Faltings completed the proof of the Mordell conjecture by proving the Shafarevich finiteness conjecture.    The main idea of Faltings' proof is the comparison of Faltings heights and naive heights via Siegel modular varieties. 

## Related Research Articles In number theory, 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.

In mathematics, 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. John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at the University of Texas at Austin. He was awarded the Abel Prize in 2010. Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.

In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.

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. Yuri Ivanovitch Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book "Computable and Uncomputable".

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. In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.

This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

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.

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space, refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

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

In mathematics, the André–Oort conjecture is an open problem in number theory that generalises the Manin–Mumford conjecture. 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. Aleksei Nikolaevich Parshin, sometimes romanized as Alexey Nikolaevich Paršin, is a Russian mathematician, specializing in number theory and algebraic geometry. Dan Abramovich, born March 12, 1963 in Haifa, is a mathematician working in the fields of algebraic geometry and arithmetic geometry. As of 2019, he holds the title of L. Herbert Ballou University Professor at Brown University, and he is an Elected Fellow of the American Mathematical Society.

1. Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties". Advanced Studies in Pure Mathematics. 35: 89–156. arXiv:. doi:10.2969/aspm/03510089.
2. Oda, Takayuki (2014). "Intersections of Two Walls of the Gottschling Fundamental Domain of the Siegel Modular Group of Genus Two". In Heim, Bernhard; Al-Baali, Mehiddin; Rupp, Florian (eds.). Automorphic Forms, Research in Number Theory from Oman. Springer Proceedings in Mathematics & Statistics. 115. Springer. pp. 193–221. doi:10.1007/978-3-319-11352-4_15. ISBN   978-3-319-11352-4.
3. Siegel, Carl Ludwig (1943). "Symplectic Geometry". American Journal of Mathematics. The Johns Hopkins University Press. 65 (1): 1–86. doi:10.2307/2371774.
4. Milne, James S. (2005). "Introduction to Shimura Varieties". In Arthur, James; Ellwood, David; Kottwitz, Robert (eds.). Harmonic Analysis, the Trace Formula, and Shimura Varieties (PDF). Clay Mathematics Proceedings. 4. American mathematical Society and Clay Mathematics Institute. pp. 265–378. ISBN   978-0-8218-3844-0.
5. Belin, Alexandre; Castro, Alejandra; Gomes, João; Keller, Christoph A. (11 April 2017). "Siegel modular forms and black hole entropy" (PDF). Journal of High Energy Physics. 2017 (4). arXiv:. doi:10.1007/JHEP04(2017)057.See Section 1 of the paper.
6. van der Geer, Gerard (2013). "The cohomology of the moduli space of Abelian varieties". In Farkas, Gavril; Morrison, Ian (eds.). The Handbook of Moduli, Volume 1. 24. Somerville, Mass.: International Press. arXiv:. ISBN   9781571462572.
7. Tai, Yung-Sheng (1982). "On the Kodaira dimension of the moduli space of abelian varieties". Inventiones Mathematicae. 68: 425–439. doi:10.1007/BF01389411.
8. Freitag, Eberhard (1983). Siegelsche Modulfunktionen (in German). Springer-Verlag. doi:10.1007/978-3-642-68649-8.
9. Mumford, David (1983). "On the Kodaira dimension of the Siegel modular variety". In Ciliberto, C.; Ghione, F.; Orecchia, F. (eds.). Algebraic Geometry - Open Problems, Proceedings of the Conference held in Ravello, May 31 - June 5, 1982. Lecture Notes in Mathematics. 997. Springer. pp. 348–375. doi:10.1007/BFb0061652.
10. Belin, Alexandre; Castro, Alejandra; Gomes, João; Keller, Christoph A. (7 November 2018). "Siegel paramodular forms and sparseness in AdS3/CFT2". Journal of High Energy Physics. 2018 (11). arXiv:. doi:10.1007/JHEP11(2018)037.
11. Parshin, A. N. (1968). "Algebraic curves over function fields I" (PDF). Izv. Akad. Nauk. SSSR Ser. Math. 32: 1191–1219.
12. Cornell, Gary; Silverman, Joseph H., eds. (1986). Arithmetic geometry. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. doi:10.1007/978-1-4613-8655-1. ISBN   0-387-96311-1. MR   0861969.
13. Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. doi:10.1007/BF01388432. MR   0718935.
14. Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German). 75 (2): 381. doi:10.1007/BF01388572. MR   0732554.
15. "Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof." Bloch, Spencer (1984). "The Proof of the Mordell Conjecture" (PDF). The Mathematical Intelligencer. 6 (2): 44.