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] }

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] }

The Siegel modular variety *A*_{g}, 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 *A*_{g}(*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] }

The Siegel modular variety *A*_{g} has dimension *g*(*g* + 1)/2.^{ [1] }^{ [6] } Furthermore, it was shown by Yung-Sheng Tai, Eberhard Freitag, and David Mumford that *A _{g}* is of general type when

Siegel modular varieties can be compactified to obtain projective varieties.^{ [1] } In particular, a compactification of *A*_{2}(2) is birationally equivalent to the Segre cubic which is in fact rational.^{ [1] } Similarly, a compactification of *A*_{2}(3) is birationally equivalent to the Burkhardt quartic which is also rational.^{ [1] } Another Siegel modular variety, denoted *A*_{1,3}(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] }

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] }

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.

**Yuri Ivanovich 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 dimension***κ*(*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.

A **height function** is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties to the real numbers.

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.

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.

**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.

**Walter Lewis Baily, Jr.** was an American mathematician.

- 1 2 3 4 5 6 7 8 9 10 11 Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties".
*Higher Dimensional Birational Geometry*. Advanced Studies in Pure Mathematics.**35**. pp. 89–156. arXiv: math/9810153 . doi:10.2969/aspm/03510089. ISBN 978-4-931469-85-3. - ↑ 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. - ↑ Siegel, Carl Ludwig (1943). "Symplectic Geometry".
*American Journal of Mathematics*. The Johns Hopkins University Press.**65**(1): 1–86. doi:10.2307/2371774. JSTOR 2371774. - 1 2 Milne, James S. (2005). "Introduction to Shimura Varieties" (PDF). In Arthur, James; Ellwood, David; Kottwitz, Robert (eds.).
*Harmonic Analysis, the Trace Formula, and Shimura Varieties*. Clay Mathematics Proceedings.**4**. American mathematical Society and Clay Mathematics Institute. pp. 265–378. ISBN 978-0-8218-3844-0. - 1 2 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): 57. arXiv: 1611.04588 . Bibcode:2017JHEP...04..057B. doi:10.1007/JHEP04(2017)057. See Section 1 of the paper. - ↑ 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: 1112.2294 . ISBN 9781571462572. - ↑ Tai, Yung-Sheng (1982). "On the Kodaira dimension of the moduli space of abelian varieties".
*Inventiones Mathematicae*.**68**(3): 425–439. Bibcode:1982InMat..68..425T. doi:10.1007/BF01389411. - ↑ Freitag, Eberhard (1983).
*Siegelsche Modulfunktionen*. Grundlehren der mathematischen Wissenschaften (in German).**254**. Springer-Verlag. doi:10.1007/978-3-642-68649-8. ISBN 978-3-642-68650-4. - ↑ 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. ISBN 978-3-540-12320-0. - ↑ 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): 37. arXiv: 1805.09336 . Bibcode:2018JHEP...11..037B. doi:10.1007/JHEP11(2018)037. - ↑ Parshin, A. N. (1968). "Algebraic curves over function fields I" (PDF).
*Izv. Akad. Nauk. SSSR Ser. Math.***32**(5): 1191–1219. Bibcode:1968IzMat...2.1145P. doi:10.1070/IM1968v002n05ABEH000723. - 1 2 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. - ↑ 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. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432. MR 0718935.CS1 maint: ref=harv (link) - ↑ Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
*Inventiones Mathematicae*(in German).**75**(2): 381. doi: 10.1007/BF01388572 . MR 0732554.CS1 maint: ref=harv (link) - ↑ "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. doi:10.1007/BF03024155.

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