The n-torsion of an elliptic curve over a scheme #
Let E be an elliptic curve over a scheme S, with its group law (EllipticCurveGeom.grpObj),
and let n be an integer. The n-torsion E[n] of E is the scheme-theoretic kernel of
multiplication by n, [n] : E ⟶ E (EllipticCurveGeom.mulBy): the kernel of the homomorphism
Grp.ofHom (E.mulBy n) of group schemes over S, constructed in TauCeti.GroupScheme. It is a
group scheme over S, with a homomorphism E[n] ⟶ E of group schemes over S
(EllipticCurveGeom.torsionι). The construction specializes TauCeti.GroupScheme.torsion to
E.toCommGrp.
On underlying schemes, E[n] is the fibre of [n] over the zero section, so its inclusion into E
is a closed immersion. Its points with values in any Z over S are the points of E with values
in Z killed by [n], as a group. Its formation commutes with arbitrary base change: a pointed
morphism of elliptic curves E' ⟶ E forming a pullback square over a morphism of bases
S' ⟶ S induces a morphism E'[n] ⟶ E[n] forming a pullback square over S' ⟶ S.
Multiplication transitions E[mn] ⟶ E[n] and inclusions E[m] ⟶ E[mn] specialize the
commutative group-scheme API. Their identity and composition laws include transports between
equal indices, supplying the coherence of torsion towers.
Nothing here asserts that E[n] is finite or flat over S: for n ≠ 0 it is finite locally
free of rank n², which rests on the finiteness and flatness of [n].
Main definitions #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsion E n: then-torsionE[n], a group object ofOver S.TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionι E n: its inclusionE[n] ⟶ E, a homomorphism of group schemes overS.TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionLift: the point ofE[n]given by a point ofEkilled by[n].TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionPointsMulEquiv: the points ofE[n]with values inZare the kernel of[n]on the points ofEwith values inZ.TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionPointsEquiv: the pointsxofEwith values inZwithx ^ n = 1are the points ofE[n]with values inZ.TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionMap: the morphismE'[n] ⟶ E[n]induced by a base change square of elliptic curves.TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsionPairHom E N: the universal pair of points ofE[N], the homomorphism(ℤ/Nℤ)² →* E(E[N] ×_S E[N])sending(a, b)toaP + bQ.
Main results #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.torsion_eq_kernel:E[n]is the kernel of the integernof the endomorphism ring ofE.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isPullback_torsion:E[n]is the fibre of[n]over the zero section.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isCommMonObj_torsion:E[n]is a commutative group scheme overS.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isClosedImmersion_torsionι_left:E[n] ⟶ Eis a closed immersion; consequentlyE[n]is proper overS(TauCeti.AlgebraicGeometry.EllipticCurveGeom.isProper_torsion_hom).TauCeti.AlgebraicGeometry.EllipticCurveGeom.ker_torsionι_left: as a closed subscheme ofE,E[n]is the inverse image[n]⁻¹[0]of the zero section.TauCeti.AlgebraicGeometry.EllipticCurveGeom.locallyOfFinitePresentation_torsion_hom:E[n]is locally of finite presentation overS.TauCeti.AlgebraicGeometry.EllipticCurveGeom.subschemeι_ker_torsionι_left_structureMap: the closed subscheme of the ideal sheaf ofE[n]isE[n]overS, so it is locally of finite presentation overSas well (TauCeti.AlgebraicGeometry.EllipticCurveGeom.locallyOfFinitePresentation_ker_torsionι_left).TauCeti.AlgebraicGeometry.EllipticCurveGeom.isPullback_torsionMap: then-torsion commutes with base change.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isPullback_torsionMap_torsionιandTauCeti.AlgebraicGeometry.EllipticCurveGeom.ker_torsionι_left_of_isPullback:E'[n]is the inverse image ofE[n]underE' ⟶ E, as a closed subscheme.TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_comp_torsionPairHom: forN ≠ 0,E[N] ×_S E[N]represents homomorphisms(ℤ/Nℤ)² →* E(Z).
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, 2.3.1.
The n-torsion as a kernel #
The n-torsion E[n] of an elliptic curve E over a scheme S: the scheme-theoretic
kernel of multiplication by n, as a group object of Over S.
Instances For
The n-torsion is the kernel of the endomorphism n. The n-torsion of E is the kernel
of the image in Grp (Over S) of the integer n of the endomorphism ring End E.toCommGrp.
The inclusion E[n] ⟶ E of the n-torsion of E, a morphism of Over S.
Instances For
The inclusion of the n-torsion is a homomorphism of group schemes over S.
The inclusion of the n-torsion is a monomorphism.
The n-torsion is commutative. Its group law is the restriction of that of E.
Multiplication by n kills the n-torsion.
Multiplication by n kills the n-torsion.
The inclusion of the n-torsion, as a point of E with values in E[n], is killed by n.
Every point of E with values in Z that factors through the n-torsion is killed by
n.
The point of the n-torsion E[n] given by a point x of E with values in Z which is
killed by multiplication by n.
Instances For
The point torsionLift x hx of E[n] maps to x in E.
The point torsionLift x hx of E[n] maps to x in E.
Two points of the n-torsion E[n] agree if their images in E do.
A point of E with values in Z lies in the n-torsion E[n] exactly when it is killed by
multiplication by n.
A homomorphism φ : (ℤ/Nℤ)² →* E(Z) takes values in the N-torsion of E(Z).
The points of E[n]. The points of the n-torsion E[n] with values in Z form the
kernel of multiplication by n on the group of points of E with values in Z. This specializes
TauCeti.GroupScheme.torsionPointsMulEquiv, identifying multiplication by n with n-th powers.
Instances For
A point of E[n] corresponds to its image in E.
The point of E[n] corresponding to a point x of E killed by [n] maps to x in E.
The points x of E with values in Z with x ^ n = 1 are the points of E[n] with values
in Z. This is torsionPointsMulEquiv, with the kernel of [n] written as a subtype.
Instances For
The point of E[n] corresponding to a point x of E with x ^ n = 1 maps to x in E.
A point of E[n] corresponds to its image in E.
Underlying schemes #
E[n] is the fibre of [n] over the zero section. On underlying schemes, the square
formed by the inclusion E[n] ⟶ E, the structure morphism of E[n], multiplication by n and the
zero section of E is a pullback square.
The inclusion E[n] ⟶ E lies over S.
The inclusion E[n] ⟶ E lies over S.
The n-torsion is a closed subscheme. The inclusion E[n] ⟶ E is a closed immersion,
being the base change of the zero section of the separated morphism E ⟶ S.
E[n] is the inverse image of the zero section under [n]: the ideal sheaf of the closed
subscheme E[n] of E is the inverse image along [n] of the ideal sheaf of the zero section.
The n-torsion of an elliptic curve is proper over the base.
The n-torsion is locally of finite presentation over S: E[n] is the base change of
[n] along the zero section.
The closed subscheme of the ideal sheaf of E[n] is E[n] over S.
The closed subscheme of the ideal sheaf of E[n] is locally of finite presentation over
S.
Base change #
The morphism E'[n] ⟶ E[n] induced by a morphism g : E' ⟶ E of elliptic curves lying over
f : S' ⟶ S, forming a pullback square with the structure morphisms and carrying the zero section
of E' to that of E. It is the restriction of g (torsionMap_torsionι).
Instances For
The morphism E'[n] ⟶ E[n] is the restriction of g.
The morphism E'[n] ⟶ E[n] is the restriction of g.
The morphism E'[n] ⟶ E[n] lies over f.
The morphism E'[n] ⟶ E[n] lies over f.
The n-torsion commutes with base change. For a morphism g : E' ⟶ E of elliptic curves
lying over f : S' ⟶ S, forming a pullback square with the structure morphisms and carrying the
zero section of E' to that of E, the square formed by the induced morphism E'[n] ⟶ E[n], the
structure morphisms of E'[n] and E[n] and f is a pullback square: E'[n] is the base change
of E[n] along f.
E'[n] is the inverse image of E[n]. For a morphism g : E' ⟶ E of elliptic curves
forming a pullback square over f : S' ⟶ S and carrying the zero section of E' to that of E,
the square formed by the induced morphism E'[n] ⟶ E[n], the two inclusions and g is a pullback
square.
The ideal sheaf of E[n] commutes with base change: for a morphism g : E' ⟶ E of
elliptic curves forming a pullback square over f : S' ⟶ S and carrying the zero section of E'
to that of E, the ideal sheaf of E'[n] is the inverse image along g of that of E[n].
The universal point of E[N] #
The universal point of E[N]. The homomorphism ℤ/Nℤ →* E(E[N]) sending a to aP,
where P is the inclusion of E[N] into E, viewed as a point of E with values in E[N].
Every homomorphism ℤ/Nℤ →* E(Z), for Z over S, factors uniquely through it
(existsUnique_comp_torsionPointHom).
Instances For
The universal point of E[N] sends a to aP.
The universal point of E[N] sends 1 to the inclusion P of E[N] into E.
E[N] represents homomorphisms from ℤ/Nℤ. Every homomorphism φ : ℤ/Nℤ →* E(Z), for
Z over S, is a ↦ x ≫ torsionPointHom E N a for a unique point x of E[N] with values in
Z, namely the point of E[N] given by φ 1.
Pairs of points of E[N] #
The universal pair of points of E[N]. The homomorphism (ℤ/Nℤ)² →* E(E[N] ×_S E[N])
sending (a, b) to aP + bQ, where P and Q are the two projections
E[N] ×_S E[N] ⟶ E[N] followed by the inclusion of E[N] into E. Every homomorphism
(ℤ/Nℤ)² →* E(Z), for Z over S, factors uniquely through it
(existsUnique_comp_torsionPairHom).
Instances For
The universal pair of points of E[N] sends (a, b) to aP + bQ.
The universal pair of points of E[N] sends (1, 0) to the first projection P.
The universal pair of points of E[N] sends (0, 1) to the second projection Q.
E[N] ×_S E[N] represents homomorphisms from (ℤ/Nℤ)². Every homomorphism
φ : (ℤ/Nℤ)² →* E(Z), for Z over S, is a ↦ x ≫ torsionPairHom E N a for a unique point x
of E[N] ×_S E[N] with values in Z, namely the pair of points of E[N] given by φ (1, 0) and
φ (0, 1).
Transitions between torsion levels #
Multiplication by m restricted to E[mn], with values in E[n].
Instances For
The transition E[mn] ⟶ E[n] sends a torsion point to its multiple by m.
The transition E[mn] ⟶ E[n] sends a torsion point to its multiple by m.
On schemes, the torsion transition is the restriction of multiplication by m.
On schemes, the torsion transition is the restriction of multiplication by m.
The torsion transition is a homomorphism of group schemes.
Multiplication by one on torsion is the identity, after identifying 1 * n with n.
Successive torsion transitions multiply their indices, with the associativity transport.
Successive torsion transitions multiply their indices, with the associativity transport.
The inclusion of E[m] in E[mn].
Instances For
The inclusion between torsion levels preserves the underlying point of E.
The inclusion between torsion levels preserves the underlying point of E.
On schemes, inclusions between torsion levels preserve the point of E.
On schemes, inclusions between torsion levels preserve the point of E.
The inclusion between torsion levels is a monomorphism.
The inclusion between torsion levels is a homomorphism of group schemes.
The inclusion between torsion levels is a closed immersion.
Inclusion into the same torsion level is the identity, after identifying m * 1 with m.
Successive torsion inclusions multiply their indices, with the associativity transport.
Successive torsion inclusions multiply their indices, with the associativity transport.
The transition kills the included m-torsion.
The transition kills the included m-torsion.
The torsion transition is the base change of multiplication by m on E.
The torsion transition is locally of finite presentation, even when m = 0.
The kernel of E[mn] ⟶ E[n] is precisely the included E[m], scheme-theoretically.
Inclusions between torsion levels commute with arbitrary base change.
Inclusions between torsion levels commute with arbitrary base change.
Torsion transitions commute with arbitrary base change.
Torsion transitions commute with arbitrary base change.