In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Picard modular group, studied by , is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
of the form SU(''J'',''L''), where ''L'' is a 3-dimensional
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
over the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
of an
imaginary quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...
and ''J'' is a
hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
on ''L'' of signature (2, 1). Picard modular groups
act on the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
in C
2 and the quotient is called a
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 ...
.
See also
*
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...
*
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
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 , volume=10 , pages=305–322
Group theory
Automorphic forms