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Picard Modular Surface
In mathematics, a Picard modular surface, studied by , is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group. Picard modular surfaces are some of the simplest examples of Shimura varieties and are sometimes used as a test case for the general theory of Shimura varieties. See also *Hilbert modular surface *Siegel modular variety 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 pola ... References * *{{Citation , last1=Picard , first1=Émile , authorlink=Émile Picard, title= Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques , url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 , year=1881 , journal=Annales Scientifiques de l'École Normale Supérieure , series=Série 2 , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complex Surface
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico (rapper), Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Picard Modular Group
In mathematics, a Picard modular group, studied by , is a group (mathematics), group of the form SU(''J'',''L''), where ''L'' is a 3-dimensional lattice (group), lattice over the ring of integers of an imaginary quadratic field and ''J'' is a hermitian form on ''L'' of signature (2, 1). Picard modular groups Group action (mathematics), act on the unit sphere in C2 and the quotient is called a Picard modular surface. See also *Fuchsian group *Kleinian group References * * Group theory Automorphic forms {{group-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shimura Varieties
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. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties. Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. In 1979, Robert Langlands remarked that Shimura varieties form a natural realm of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hilbert Modular Surface
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before. Definitions If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |