Naive level structures on an elliptic curve #
Let E be an elliptic curve over a scheme S and A a group. A homomorphism φ : A →* E(S) to
the group of S-points of E is injective on every fibre
(EllipticCurveGeom.IsFiberwiseInjective) when for every a ≠ 1 in A the section φ a misses
the zero section at every point of S. Equivalently, for every point s of S, the composite of
φ with restriction to the fibre of E over the residue field κ(s) is injective
(EllipticCurveGeom.isFiberwiseInjective_iff_forall_residueField), and then so is its composite
with restriction to the fibre over any field-valued point of S, in particular over every geometric
point (EllipticCurveGeom.IsFiberwiseInjective.injective_pointsPullback).
For A = ℤ/Nℤ with N ≠ 0 such a homomorphism is an S-point P of E[N] that has exact order
N in every geometric fibre: none of P, 2P, …, (N - 1)P meets the zero section
(EllipticCurveGeom.isFiberwiseInjective_zmod_iff). For A = (ℤ/Nℤ)² it is a pair (P, Q) of
S-points of E[N] none of whose combinations aP + bQ with (a, b) ≠ 0 meets the zero section
(EllipticCurveGeom.isFiberwiseInjective_zmod_prod_iff), that is, a basis of the N-torsion in
every geometric fibre in which N is invertible: over a separably closed field in which N is
invertible, the injective homomorphisms (ℤ/Nℤ)² →* E(K) are exactly those with image E[N](K)
(EllipticCurveGeom.isFiberwiseInjective_iff_range_eq_ker), every elliptic curve there has
one (EllipticCurveGeom.exists_isFiberwiseInjective), and any two differ by a unique matrix in
GL₂(ℤ/Nℤ) acting on (ℤ/Nℤ)² (EllipticCurveGeom.IsFiberwiseInjective.existsUnique_eq_comp).
Over a base on which N is invertible these are the naive [Γ₁(N)]- and [Γ(N)]-structures,
which are compared with Drinfeld level structures in Katz–Mazur §3.7.
The condition is pointwise, so it is stable under arbitrary base change
(EllipticCurveGeom.IsFiberwiseInjective.pointsPullback), and it descends along surjective base
changes (EllipticCurveGeom.isFiberwiseInjective_pointsPullback_iff). The key input is that a base
changed point meets the zero section exactly over the points where the original point does
(EllipticCurveGeom.pointsPullback_left_apply_eq_zero_iff), and over a field a point is zero
exactly when it meets the zero section (EllipticCurveGeom.eq_one_iff_left_apply_eq_zero).
For a finite group A and a homomorphism u : A →* E(H) to the points of E with values in a
scheme H over S, the points of H at which u is injective on the fibre form an open subset,
the naive locus of u (EllipticCurveGeom.naiveLocus), since the zero section has closed
image. When u is universal among homomorphisms from A, as the universal pair of points of
E[N] is for A = (ℤ/Nℤ)², the naive locus represents the naive A-structures. If N is
invertible on S and every point u a is killed by N, the naive locus is also closed
(EllipticCurveGeom.isClopen_naiveLocus), because two points killed by N agree on an open and
closed set; so it is finite over S when H is (EllipticCurveGeom.isFinite_naiveLocusOver_hom).
The group law on points of E is written multiplicatively, following EllipticCurveGeom.mulBy.
Main definitions #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective E φ: the homomorphismφ : A →* E(S)is injective on every fibre ofEoverS.TauCeti.AlgebraicGeometry.EllipticCurveGeom.naiveLocus u: the points ofHat whichu : A →* E(H)is injective on the fibre;naiveLocusOpens uandnaiveLocusOver uare the corresponding open subscheme ofHand scheme overS, for finiteA.
Main results #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_iff_injective: over a field, injectivity on fibres is injectivity.TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.pointsPullback: injectivity on fibres is stable under base change.TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.eq_of_comp_left_eq: points in the image of a homomorphism injective on every fibre are distinguished by every field-valued point of the base.TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.left_apply_injectiveandTauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.disjoint_range: distinct indices give distinct points in each fibre and disjoint section images.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_iff_forall_residueField: injectivity on fibres is injectivity on the fibre over every residue field.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isOpen_naiveLocus: for finiteA, the naive locus is open, andrange_left_subset_naiveLocus_iffcharacterises the points landing in it.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isClopen_naiveLocusandTauCeti.AlgebraicGeometry.EllipticCurveGeom.isFinite_naiveLocusOver_hom: whereNis invertible andutakes values killed byN, the naive locus is open and closed, and finite overSwhenHis.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_zmod_iff: the naive[Γ₁(N)]condition, askPmissing the zero section for0 < k < N.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_zmod_prod_iff: the naive[Γ(N)]condition, asaP + bQmissing the zero section for(a, b) ≠ 0.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_zmod_three_prod_iff: the naive[Γ(3)]condition, asPandQmissing the zero section andQdiffering from±Pat every point.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isFiberwiseInjective_iff_range_eq_ker: over a separably closed field in whichNis invertible, the naive[Γ(N)]-structures are the bases ofE[N](K).TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.existsUnique_eq_comp: there,GL₂(ℤ/Nℤ)acts simply transitively on the naive[Γ(N)]-structures.
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §3.7 (naive level structures
where
Nis invertible) and 2.3.1.
Homomorphisms injective on every fibre #
A homomorphism φ : A →* E(S) from a group A to the group of S-points of an elliptic curve
E over S is injective on every fibre when, for every a ≠ 1, the section φ a of E
misses the zero section at every point of S: equivalently, φ followed by restriction to the
fibre of E over every field-valued point of S is injective
(isFiberwiseInjective_iff_forall_residueField, IsFiberwiseInjective.injective_pointsPullback).
For A = ℤ/Nℤ and A = (ℤ/Nℤ)², over a base on which N is invertible, these are the naive
[Γ₁(N)]- and [Γ(N)]-structures on E.
Instances For
A homomorphism is injective on every fibre exactly when the image of every a ≠ 1 misses the
zero section everywhere.
Injectivity on fibres is stable under base change.
Injectivity on fibres descends along surjective base changes.
Restricting a homomorphism injective on every fibre along an injective homomorphism gives a homomorphism injective on every fibre.
Over a nonempty base, a homomorphism injective on every fibre is injective.
The naive locus #
The naive locus of a family of homomorphisms. For an elliptic curve E over a scheme S, a
scheme H over S and a homomorphism u : A →* E(H) to the group of points of E with values in
H, the set of points h of H at which u is injective on the fibre: for every a ≠ 1, the
point u a misses the zero section of E at h.
Instances For
The defining condition of the naive locus.
For a finite group A, the naive locus is open: it is the complement of the finitely many
closed sets where some u a with a ≠ 1 meets the closed image of the zero section.
A point x of H with values in a scheme T over S lands in the naive locus of u exactly
when, for every a ≠ 1, the point x ≫ u a of E misses the zero section at every point
of T.
The naive locus of u, for a finite group A, as an open subscheme of H.
Instances For
The naive locus of u, for a finite group A, as a scheme over S.
Instances For
The inclusion of the naive locus of u into H, an open immersion.
Instances For
The naive locus of u has, over S, every property of morphisms of schemes which is stable
under composition, holds for open immersions and holds for H.
The naive locus is closed where N is invertible. If the natural number N is invertible
in every residue field of S and every point u a is killed by N, then the naive locus of u is
open and closed in H, for a finite group A. Each point u a with a ≠ 1 meets the zero section
on the set where it agrees with the zero point, and two points killed by N agree on an open and
closed set (EllipticCurveGeom.isClopen_range_equalizer_ι_left_of_mulBy_eq_one).
Where N is invertible and every point u a is killed by N, the inclusion of the naive locus
of u into H is a closed immersion as well as an open immersion.
The naive locus is finite where N is invertible. If H is finite over S, the natural
number N is invertible in every residue field of S and every point u a is killed by N, then
the naive locus of u is finite over S, for a finite group A: it is a closed subscheme of
H.
Over a field #
Over a field, injectivity on fibres is injectivity.
Injectivity on the fibre over a field-valued point. If φ : A →* E(S) is injective on
every fibre, then for every point t of S with values in a field K, in particular every
geometric point, φ followed by restriction to the fibre of E over t is injective.
Points of a fibrewise injective homomorphism are distinguished on field points. If
φ : A →* E(S) is injective on every fibre and φ c₁, φ c₂ agree after composition with a
point t : Spec K ⟶ S with values in a field, then c₁ = c₂.
The underlying points of a fibrewise injective family of sections are distinct over each point of the base.
Distinct sections of a fibrewise injective family have disjoint images.
The divisor of a fibrewise injective finite family is the scheme-theoretic union of its disjoint sections.
The divisor of a fibrewise injective finite family killed by N is contained in E[N],
expressed as the reverse inclusion of ideal sheaves.
Points agree on the fibre over a point exactly when they agree there. Two S-points P
and Q of E have the same base change to the fibre E_{κ(s)} of E over the residue field
κ(s) of a point s of S exactly when P and Q take the same value at s.
Injectivity on fibres is checked on residue fields. A homomorphism φ : A →* E(S) is
injective on every fibre exactly when, for every point s of S, φ followed by restriction to
the fibre E_{κ(s)} of E over the residue field κ(s) is injective.
Naive [Γ₁(N)]-structures #
The naive [Γ₁(N)] condition. A homomorphism φ : ℤ/Nℤ →* E(S), given by the S-point
P = φ 1 of E[N], is injective on every fibre exactly when none of P, 2P, …, (N - 1)P meets
the zero section: P has exact order N in every fibre.
Naive [Γ(N)]-structures #
The naive [Γ(N)] condition. A homomorphism φ : (ℤ/Nℤ)² →* E(S), given by the pair of
S-points P = φ (1, 0) and Q = φ (0, 1) of E[N], is injective on every fibre exactly when no
combination aP + bQ with (a, b) ≠ 0 in (ℤ/Nℤ)² meets the zero section.
The naive [Γ(3)] condition. A homomorphism φ : (ℤ/3ℤ)² →* E(S), given by the pair of
S-points P = φ (1, 0) and Q = φ (0, 1) of E[3], is injective on every fibre exactly when
P and Q miss the zero section and Q is neither P nor -P over every point of S: every
nonzero element of (ℤ/3ℤ)² is, up to sign, one of (1, 0), (0, 1), (1, 1) and (-1, 1).
Naive [Γ(N)]-structures over a separably closed field #
Naive [Γ(N)]-structures over a separably closed field are bases of E[N]. Over a
separably closed field K in which N is invertible, a homomorphism φ : (ℤ/Nℤ)² →* E(K) is
injective on fibres exactly when its image is the N-torsion E[N](K), that is, when it
identifies (ℤ/Nℤ)² with E[N](K).
Every elliptic curve over a separably closed field has a naive [Γ(N)]-structure when
N is invertible in the field: there is a homomorphism (ℤ/Nℤ)² →* E(K) injective on fibres.
GL₂(ℤ/Nℤ) acts simply transitively on the naive [Γ(N)]-structures over a separably
closed field. If N is invertible in K, then any two homomorphisms
φ ψ : (ℤ/Nℤ)² →* E(K) injective on fibres differ by a unique invertible matrix g acting on
(ℤ/Nℤ)²: ψ = φ ∘ g. Both are isomorphisms onto E[N](K), and g is the matrix of
φ⁻¹ ∘ ψ.