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- Abelian_variety abstract "In 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. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.".
- Abelian_variety wikiPageExternalLink art21.pdf.
- Abelian_variety wikiPageExternalLink av.html.
- Abelian_variety wikiPageID "247152".
- Abelian_variety wikiPageLength "17992".
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- Abelian_variety wikiPageRevisionID "674344813".
- Abelian_variety wikiPageWikiLink Abelian_group.
- Abelian_variety wikiPageWikiLink Abelian_integral.
- Abelian_variety wikiPageWikiLink Abstract_variety.
- Abelian_variety wikiPageWikiLink Albanese_variety.
- Abelian_variety wikiPageWikiLink Algebraic_Geometry.
- Abelian_variety wikiPageWikiLink Algebraic_curve.
- Abelian_variety wikiPageWikiLink Algebraic_geometry.
- Abelian_variety wikiPageWikiLink Algebraic_geometry_and_analytic_geometry.
- Abelian_variety wikiPageWikiLink Algebraic_group.
- Abelian_variety wikiPageWikiLink Algebraic_number_field.
- Abelian_variety wikiPageWikiLink Algebraic_number_fields.
- Abelian_variety wikiPageWikiLink Algebraic_torus.
- Abelian_variety wikiPageWikiLink Algebraic_variety.
- Abelian_variety wikiPageWikiLink Algebraically_closed_field.
- Abelian_variety wikiPageWikiLink American_Mathematical_Society.
- Abelian_variety wikiPageWikiLink André_Weil.
- Abelian_variety wikiPageWikiLink Arithmetic_of_abelian_varieties.
- Abelian_variety wikiPageWikiLink Automorphism.
- Abelian_variety wikiPageWikiLink Automorphism_group.
- Abelian_variety wikiPageWikiLink Bernhard_Riemann.
- Abelian_variety wikiPageWikiLink Birational_geometry.
- Abelian_variety wikiPageWikiLink Carl_Gustav_Jacob_Jacobi.
- Abelian_variety wikiPageWikiLink Carl_Gustav_Jakob_Jacobi.
- Abelian_variety wikiPageWikiLink Cartier_dual.
- Abelian_variety wikiPageWikiLink Category:Abelian_varieties.
- Abelian_variety wikiPageWikiLink Category:Algebraic_curves.
- Abelian_variety wikiPageWikiLink Category:Algebraic_surfaces.
- Abelian_variety wikiPageWikiLink Category:Geometry_of_divisors.
- Abelian_variety wikiPageWikiLink Category:Niels_Henrik_Abel.
- Abelian_variety wikiPageWikiLink Characteristic_(algebra).
- Abelian_variety wikiPageWikiLink Charles_Émile_Picard.
- Abelian_variety wikiPageWikiLink Commutative.
- Abelian_variety wikiPageWikiLink Commutative_property.
- Abelian_variety wikiPageWikiLink Complete_variety.
- Abelian_variety wikiPageWikiLink Complex_analysis.
- Abelian_variety wikiPageWikiLink Complex_manifold.
- Abelian_variety wikiPageWikiLink Complex_number.
- Abelian_variety wikiPageWikiLink Complex_numbers.
- Abelian_variety wikiPageWikiLink Complex_torus.
- Abelian_variety wikiPageWikiLink Connected_space.
- Abelian_variety wikiPageWikiLink Contravariant_functor.
- Abelian_variety wikiPageWikiLink Coprime.
- Abelian_variety wikiPageWikiLink Coprime_integers.
- Abelian_variety wikiPageWikiLink Cubic_function.
- Abelian_variety wikiPageWikiLink Cubic_polynomial.
- Abelian_variety wikiPageWikiLink Cyclic_group.
- Abelian_variety wikiPageWikiLink Dedekind_domain.
- Abelian_variety wikiPageWikiLink Definite_quadratic_form.
- Abelian_variety wikiPageWikiLink Differential_of_the_first_kind.
- Abelian_variety wikiPageWikiLink Dynamical_system.
- Abelian_variety wikiPageWikiLink Elliptic_curve.
- Abelian_variety wikiPageWikiLink Elliptic_function.
- Abelian_variety wikiPageWikiLink Elliptic_integral.
- Abelian_variety wikiPageWikiLink Endomorphism_ring.
- Abelian_variety wikiPageWikiLink Equivalence_relation.
- Abelian_variety wikiPageWikiLink Ferdinand_Georg_Frobenius.
- Abelian_variety wikiPageWikiLink Field_(mathematics).
- Abelian_variety wikiPageWikiLink Finite_field.
- Abelian_variety wikiPageWikiLink Finitely_generated_abelian_group.
- Abelian_variety wikiPageWikiLink Finitely_generated_group.
- Abelian_variety wikiPageWikiLink Free_abelian_group.
- Abelian_variety wikiPageWikiLink Function_field_of_an_algebraic_variety.
- Abelian_variety wikiPageWikiLink Functor.
- Abelian_variety wikiPageWikiLink Generalised_Riemann_hypothesis.
- Abelian_variety wikiPageWikiLink Generalized_Riemann_hypothesis.
- Abelian_variety wikiPageWikiLink Generating_set_of_a_group.
- Abelian_variety wikiPageWikiLink Genus_(mathematics).
- Abelian_variety wikiPageWikiLink Geometric_fiber.
- Abelian_variety wikiPageWikiLink Global_field.
- Abelian_variety wikiPageWikiLink Glossary_of_algebraic_geometry.
- Abelian_variety wikiPageWikiLink Grothendiecks_relative_point_of_view.
- Abelian_variety wikiPageWikiLink Group_(mathematics).
- Abelian_variety wikiPageWikiLink Group_law.
- Abelian_variety wikiPageWikiLink Group_scheme.
- Abelian_variety wikiPageWikiLink Hamiltonian_system.
- Abelian_variety wikiPageWikiLink Henri_Poincaré.
- Abelian_variety wikiPageWikiLink Hermitian_form.
- Abelian_variety wikiPageWikiLink Hyperelliptic_curve.
- Abelian_variety wikiPageWikiLink Hyperelliptic_integral.
- Abelian_variety wikiPageWikiLink Identity_element.
- Abelian_variety wikiPageWikiLink Imaginary_part.
- Abelian_variety wikiPageWikiLink Integer.
- Abelian_variety wikiPageWikiLink Isogeny.
- Abelian_variety wikiPageWikiLink Isomorphic.
- Abelian_variety wikiPageWikiLink Isomorphism.
- Abelian_variety wikiPageWikiLink Jacobian_variety.
- Abelian_variety wikiPageWikiLink Karl_Weierstrass.
- Abelian_variety wikiPageWikiLink Kodaira_dimension.
- Abelian_variety wikiPageWikiLink Lattice_(group).
- Abelian_variety wikiPageWikiLink Lefschetz_principle.
- Abelian_variety wikiPageWikiLink Line_bundle.
- Abelian_variety wikiPageWikiLink Local_field.
- Abelian_variety wikiPageWikiLink Localization_of_a_ring.