Jacobians of genus 2 curves

URI: https://gptkb.org/entity/Jacobians_of_genus_2_curves

GPTKB entity

Statements (48)

Predicate	Object
gptkbp:instanceOf	gptkb:algebraic_geometry principally polarized abelian surface
gptkbp:dimensions	2
gptkbp:hasConnection	true
gptkbp:hasEndomorphismRing	may be larger than integers
gptkbp:hasMordellWeilGroup	yes
gptkbp:hasRationalPoints	depends on base field
gptkbp:hasSpecialCase	gptkb:Jacobian_variety
gptkbp:hasThetaDivisor	true
gptkbp:hasTorsionPoints	yes
https://www.w3.org/2000/01/rdf-schema#label	Jacobians of genus 2 curves
gptkbp:isAbelianSurface	true
gptkbp:isAbelianVariety	true
gptkbp:isAlgebraicGroup	true
gptkbp:isAlgebraicGroupVariety	true
gptkbp:isCommutativeGroup	true
gptkbp:isComplexTorus	true
gptkbp:isFunctorial	true
gptkbp:isGroupVariety	true
gptkbp:isIrreducible	true
gptkbp:isModuliPoint	gptkb:Siegel_modular_variety_of_genus_2
gptkbp:isomorphicTo	Picard group of degree 0 divisors on genus 2 curve
gptkbp:isParameterSpaceFor	line bundles of degree 0
gptkbp:isPrincipallyPolarized	true
gptkbp:isProductOfEllipticCurves	sometimes
gptkbp:isProjective	true
gptkbp:isQuotientOf	complex vector space by lattice
gptkbp:isSimple	sometimes
gptkbp:isSmooth	true
gptkbp:isUniversal	for degree 0 line bundles
gptkbp:parameter	moduli space of genus 2 curves
gptkbp:polarization	principal polarization
gptkbp:relatedTo	gptkb:Torelli_theorem gptkb:Frobenius_endomorphism gptkb:Siegel_modular_forms gptkb:moduli_space_of_abelian_surfaces modular forms Weil pairing duality theory hyperelliptic curve endomorphism algebra isogeny genus 2 curve
gptkbp:usedIn	cryptography number theory arithmetic geometry
gptkbp:bfsParent	gptkb:abelian_surfaces
gptkbp:bfsLayer	7