Algebraic variety that is a moduli space for principally polarized abelian varieties
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 elliptic 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]
^ abCite error: The named reference survey was invoked but never defined (see the help page).
^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. Vol. 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. 65 (1). The Johns Hopkins University Press: 1–86. doi:10.2307/2371774. JSTOR 2371774.
^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. Vol. 4. American mathematical Society and Clay Mathematics Institute. pp. 265–378. ISBN 978-0-8218-3844-0.
^Cite error: The named reference entropy was invoked but never defined (see the help page).
and 23 Related for: Siegel modular variety information
mathematics, a Siegelmodularvariety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension...
theta-functions. The Siegelmodularvarieties, which describe Siegelmodular forms, are recognised as part of the moduli theory of abelian varieties. In all this...
projective variety associated to the graded ring formed by modular forms (in the Siegel case, Siegelmodular forms; see also Siegelmodularvariety). The non-uniqueness...
this surface into a projective space. Hilbert modular form Picard modular surface Siegelmodularvariety Ihara, Yasutaka; Nakamura, Hiroaki (1997). "Some...
Hilbert modular surfaces and Siegelmodularvarieties are among the best known classes of Shimura varieties. Special instances of Shimura varieties were...
of abelian varieties, such as the Siegelmodularvariety. This is the problem underlying Siegelmodular form theory. See also Shimura variety. Using techniques...
Arithmetic of abelian varieties Birch and Swinnerton-Dyer conjecture Moduli of algebraic curves SiegelmodularvarietySiegel's theorem on integral points...
intersection theory of Deligne–Mumford stacks Schottky problem Siegelmodularvariety Moduli stack of elliptic curves Moduli of algebraic curves Hilbert...
a test case for the general theory of Shimura varieties. Hilbert modular surface Siegelmodularvariety Langlands, Robert P.; Ramakrishnan, Dinakar, eds...
totally real number field. Siegelmodular forms are associated to larger symplectic groups in the same way in which classical modular forms are associated to...
compactification of the Siegelmodularvariety A2(3). Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of SiegelModularVarieties". Advanced Studies in...
powers of the modular curve. Work by Pila and Jacob Tsimerman, demonstrated the André–Oort conjecture in the case of the Siegelmodularvariety. Pila has...
theorem Moduli of algebraic curves Moduli space Hilbert modular surface Siegelmodularvariety Artin, M. (2015-12-31), "Algebraization of formal moduli:...
compactification of the Siegelmodularvariety A2(2). Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of SiegelModularVarieties". Advanced Studies in...
conjecture for Siegelmodularvarieties. Later, he completed the proof of the full André-Oort conjecture for all moduli spaces of abelian varieties by reducing...
Niemeier lattices of dimension 24. Freitag also demonstrated that the Siegelmodularvariety Ag is of general type when g = 8. with Rolf Busam: Funktionentheorie...
conjecture which was later shown to imply the André–Oort conjecture for Siegelmodularvarieties by Jacob Tsimerman. (with Tong Zhang) "Effective Bound of Linear...
of abelian varieties via what is known as Parshin's trick, which gives an embedding of an algebraic curve into the Siegelmodularvariety. Shafarevich...
Stern–Brocot tree Dedekind sum Egyptian fraction Montgomery reduction Modular exponentiation Linear congruence theorem Method of successive substitution...
biographical drama film directed by John Lee Hancock and written by Robert Siegel. Starring Michael Keaton as businessman Ray Kroc, the film depicts the story...
Algebraic Geometry and Theta Functions 1939 Siegelmodular forms c. 1940 André Weil defines "abelian variety" 1952 Weil defines an intermediate Jacobian...
awards and has appeared in over 50 publications. Robert Siegel Architects has designed a variety of building types, including major civic commissions such...