In mathematics, a Picard modular group, studied by Picard (1881), is a group of the form SU(J,L), where L is a 3-dimensional 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 act on the unit sphere in C2 and the quotient is called a Picard modular surface.
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In mathematics, a Picardmodulargroup, studied by Picard (1881), is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers...
a Picardmodular surface, studied by Picard (1881), is a complex surface constructed as a quotient of the unit ball in C2 by a Picardmodulargroup. Picard...
{\displaystyle \mathbb {C} } . An important example of this type of group is the Picardmodulargroup SU ( 2 , 1 ; Z [ i ] ) {\displaystyle \operatorname {SU}...
performed with unitary groups of hermitian forms, a well-known example is the Picardmodulargroup. When G {\displaystyle G} is a Lie group one can define an...
of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over...
of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picardgroup PSL(2,Z[i]) is...
class Serre's multiplicity conjectures Albanese variety PicardgroupModular form Moduli space Modular equation J-invariant Algebraic function Algebraic form...
of this surface into a projective space. Hilbert modular form Picardmodular surface Siegel modular variety Ihara, Yasutaka; Nakamura, Hiroaki (1997)...
groups of Lie type; with Michael Rapoport, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application to modular forms...
discontinuous (discrete group) theory was built up by Klein, Lie, Henri Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy...
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...
continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré...
the Picard scheme (at any point) is equal to h 0 , 1 {\displaystyle h^{0,1}} . In characteristic 0 a result of Pierre Cartier showed that all groups schemes...
ISBN 0-387-90108-6, MR 0396773 Kolchin, E. R. (1948), "Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations"...
(the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily...
stack of elliptic curves Moduli spaces of K-stable Fano varieties Modular curve Picard functor Moduli of semistable sheaves on a curve Kontsevich moduli...
{\displaystyle {\mbox{Div}}^{0}(E').} Alternatively, we can use the smaller Picardgroup P i c 0 {\displaystyle {\mathrm {Pic} }^{0}} , a quotient of Div 0 ....
points on a Shimura variety modulo a prime of good reduction in The zeta functions of Picardmodular surfaces, Publications du CRM, 1992, pp. 151-253....
the quotient of Teichmüller space by the mapping class group. In this case it is the modular curve. In the remaining cases X {\displaystyle X} is a hyperbolic...
is. The ideal class group is generally denoted Cl K, Cl O, or Pic O (with the last notation identifying it with the Picardgroup in algebraic geometry)...
Pic ( C ) {\displaystyle \operatorname {Pic} (C)} the Picardgroup of it; i.e., the group of isomorphism classes of line bundles on C. Since C is smooth...
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