Arithmetic variety
Encyclopedia
In mathematics, an arithmetic variety is the quotient space
of a Hermitian symmetric space
by an arithmetic subgroup of the associated algebraic Lie group
.
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of a Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...
by an arithmetic subgroup of the associated algebraic Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
.
Further reading
- "Introduction to modern number theory", By Yu I. Manin, Alekseĭ A. Panchishkin
- "On arithmetic varieties" by David Kazhdan, Israel J. Math. 44 (1983), no. 2, 139–159.
See also
- Arithmetic Chow groups
- Arithmetic of abelian varietiesArithmetic of abelian varietiesIn mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures...
- Abelian varietyAbelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...