Drinfeld bases of the N-torsion of an elliptic curve #
Let E be an elliptic curve over a scheme S and N a positive integer. Following Drinfeld and
Katz–Mazur, a Drinfeld basis of E[N], or [Γ(N)]-structure on E, is a homomorphism
φ : (ℤ/Nℤ)² →* E(S) to the group of S-points of E such that the effective Cartier divisor
D = ∑_{a ∈ (ℤ/Nℤ)²} [φ a]
on E is the closed subscheme E[N] (EllipticCurveGeom.IsDrinfeldBasis). In the language of
Katz and Mazur, φ is a (ℤ/Nℤ)²-generator of the subgroup scheme E[N] of the curve E; they
show that this says that the N² points φ a, counted with multiplicity, form a full set of
sections of the finite locally free S-scheme E[N]. That comparison is not made here.
The definition makes sense over every base, including bases on which N is not invertible, where
the points φ a need not be distinct: for a supersingular elliptic curve over a field of
characteristic p, the subgroup scheme E[p] is the divisor p²[0], so the homomorphism all of
whose points are the zero section is a Drinfeld basis of E[p]. It is the integral counterpart
of the naive [Γ(N)]-structures (EllipticCurveGeom.IsFiberwiseInjective), which ask instead
that the φ a be distinct in every fibre and are the right notion only where N is invertible.
Every naive full level structure is a Drinfeld basis over an arbitrary base, and the converse holds
where N is invertible; both directions are proved in DrinfeldBasis.Naive.
The divisor ∑_a [φ a] is the divisor of a finite family of S-points of E
(EllipticCurveGeom.pointsDivisor), the product of the ideal sheaves of the sections φ a; it is
a relative effective Cartier divisor on E over S. The N-torsion E[N] is the inverse
image [N]⁻¹[0] of the zero-section divisor (EllipticCurveGeom.ker_torsionι_left), so a Drinfeld
basis is a homomorphism with ∑_a [φ a] = [N]⁻¹[0] (EllipticCurveGeom.isDrinfeldBasis_iff).
Both ∑_a [φ a] and E[N] are compatible with base change
(EllipticCurveGeom.pointsDivisor_pointsPullback,
EllipticCurveGeom.ker_torsionι_left_of_isPullback), so Drinfeld bases are stable under arbitrary
base change (EllipticCurveGeom.IsDrinfeldBasis.pointsPullback). The divisor does not depend on
how its points are indexed, so precomposing a Drinfeld basis with an automorphism of (ℤ/Nℤ)², in
particular with an element of GL₂(ℤ/Nℤ), gives a Drinfeld basis
(EllipticCurveGeom.isDrinfeldBasis_comp_mulEquiv_iff). For N = 1 every homomorphism is a
Drinfeld basis (EllipticCurveGeom.isDrinfeldBasis_one).
Following EllipticCurveGeom.mulBy, the group law on points of E is written multiplicatively.
Main definitions #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis E N φ: the homomorphismφ : (ℤ/Nℤ)² →* E(S)is a Drinfeld basis ofE[N].
Main results #
TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff: a Drinfeld basis is a homomorphism with∑_a [φ a] = [N]⁻¹[0].TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.pointsPullback: Drinfeld bases are stable under base change.TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_comp_mulEquiv_iff: Drinfeld bases are stable under automorphisms of(ℤ/Nℤ)².TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_one: every homomorphism is a Drinfeld basis ofE[1].TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.range_eq_ker: over any field, the image of a Drinfeld basis is exactly the group of rationalN-torsion points.
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies
108, Princeton University Press, 1985, Chapter 1 (generators of a subgroup scheme of a smooth
curve, and full sets of sections) and §3.1 (
[Γ(N)]-structures). - V. G. Drinfeld, Elliptic modules, Mat. Sb. 94 (136) (1974), 594–627.
Drinfeld bases of E[N] (Drinfeld; Katz–Mazur §3.1). A homomorphism
φ : (ℤ/Nℤ)² →* E(S) to the group of S-points of an elliptic curve E over S is a Drinfeld
basis of the N-torsion E[N], or [Γ(N)]-structure on E, when the divisor
∑_{a ∈ (ℤ/Nℤ)²} [φ a] on E is the closed subscheme E[N]: their ideal sheaves agree.
Equivalently (isDrinfeldBasis_iff), ∑_a [φ a] is the inverse image [N]⁻¹[0] of the zero
section. The points φ a are counted with multiplicity and need not be distinct.
Instances For
The divisor ∑_a [φ a] of a Drinfeld basis is the closed subscheme E[N].
Over a field, the image of a Drinfeld basis of E[N] is E[N](K). The points φ a are
exactly the K-points of E killed by N. No hypothesis on the characteristic of K is
needed.
A Drinfeld basis is a homomorphism with ∑_a [φ a] = [N]⁻¹[0].
Drinfeld bases are stable under automorphisms of (ℤ/Nℤ)². Precomposing with an
automorphism σ, for instance the right action of an element of GL₂(ℤ/Nℤ), reindexes the
divisor ∑_a [φ a] without changing it.
Every homomorphism is a Drinfeld basis of E[1]: both [0] and E[1] are the zero
section.
Drinfeld bases are stable under base change (Katz–Mazur §3.1): if φ : (ℤ/Nℤ)² →* E(S) is
a Drinfeld basis of E[N], then so is its composite with base change of points,
(ℤ/Nℤ)² →* E'(S').