The fine moduli scheme `Y₁(N)` for `N ≥ 4` #

Let B be a scheme. The universal Tate normal form over B (EllObj.universalTate B) is the object of Ell/B given by the projective model of Y² + AXY + BY = X³ + BX² over B × Spec ℤ[A, B][Δ⁻¹], with its point (0, 0) (EllObj.universalTatePoint B). It classifies elliptic curves with a section P such that none of P, 2P and 3P meets the zero section: for such a pair (E/S, P) there is exactly one arrow (E/S) ⟶ EllObj.universalTate B of Ell/B pulling the point (0, 0) back to P (EllObj.existsUnique_hom_universalTate). This is the universal property EllipticCurveGeom.existsUnique_tateNormalForm, read in Ell/B.

Now let N ≥ 4. A naive [Γ₁(N)]-structure on E/S is a point P of exact order N in every fibre, so P, 2P and 3P miss the zero section, and (E, P) is classified by an arrow to the universal Tate normal form. That arrow is a naive [Γ₁(N)]-structure exactly when its morphism of bases factors through the locus where (0, 0) has naive exact order N, the subscheme Y₁(N) of B × Spec ℤ[A, B][Δ⁻¹] on which N(0, 0) = 0 and k(0, 0) misses the zero section for 0 < k < N (EllipticCurveGeom.naiveExactOrderPreimage). Hence the naive moduli problem [Γ₁(N)] is represented by the restriction EllObj.yOne B N of the universal Tate normal form to Y₁(N) (ModuliProblem.gammaOneNaiveRepresentableBy): Y₁(N) is a fine moduli scheme. When N is invertible on B, it is a closed subscheme of B × Spec ℤ[A, B][Δ⁻¹] (EllObj.isClosedImmersion_yOneι_base), so it is affine over B (EllObj.isAffineHom_yOne_structureMap), and affine when B is, for example for B = Spec ℤ[1/N]. It is then also smooth over B (EllObj.smooth_yOne_structureMap): the point (0, 0) embeds Y₁(N) as a retract, over B, of the scheme of points of naive exact order N on the universal Tate normal form, the representing scheme of the naive problem [Γ₁(N)] (ModuliProblem.gammaOneNaiveRelRepData), which is finite étale over the smooth B-scheme B × Spec ℤ[A, B][Δ⁻¹]; the retraction classifies the tautological point.

The hypothesis N ≥ 4 is what makes the Tate normal form apply: for N ≤ 3, a point P with NP = 0 has one of P, 2P and 3P equal to the zero section. For N ≤ 2 the automorphism [-1] fixes every naive [Γ₁(N)]-structure, so [Γ₁(N)] is not rigid.

Main definitions #

TauCeti.AlgebraicGeometry.EllObj.universalTate B: the universal Tate normal form over B, as an object of Ell/B.
TauCeti.AlgebraicGeometry.EllObj.universalTatePoint B: its point (0, 0).
TauCeti.AlgebraicGeometry.EllObj.yOne B N: the restriction of the universal Tate normal form to the locus Y₁(N) where (0, 0) has naive exact order N.
TauCeti.AlgebraicGeometry.EllObj.yOnePoint: the point (0, 0) of the universal elliptic curve over Y₁(N), which has naive exact order N (EllObj.isOfNaiveExactOrder_yOnePoint).
TauCeti.AlgebraicGeometry.EllObj.naiveExactOrderPoint N X: the tautological point of E_Z, for the scheme Z of points of naive exact order N of an object X = (E/S) of Ell/B, the naive locus of the universal point of E[N]; it has naive exact order N (EllObj.isOfNaiveExactOrder_naiveExactOrderPoint).

Main results #

TauCeti.AlgebraicGeometry.EllObj.existsUnique_hom_universalTate: the universal property of the universal Tate normal form in Ell/B.
TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaiveRepresentableBy: for N ≥ 4, the naive moduli problem [Γ₁(N)] over Ell/B is represented by EllObj.yOne B N.
TauCeti.AlgebraicGeometry.EllObj.isAffineHom_yOne_structureMap: Y₁(N) is affine over B when N is invertible on B.
TauCeti.AlgebraicGeometry.EllObj.existsUnique_hom_yOne: for N ≥ 4, a point of naive exact order N on an object of Ell/B is the pullback of (0, 0) along exactly one arrow to EllObj.yOne B N.
TauCeti.AlgebraicGeometry.EllObj.smooth_yOne_structureMap: for N ≥ 4 invertible on B, Y₁(N) is smooth over B.

References #

N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, 2.2.3 (Tate normal form) and Chapter 4.
D. Loeffler, Modular Curves, lecture notes, 3.3.4 and §3.8.
D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), 193–237.

The modular curve `Y₁(N)` from the universal Tate normal form #

Let T = Spec ℤ[A, B][Δ⁻¹] be the base of the universal Tate normal form Y² + AXY + BY = X³ + BX², with its point (0, 0). An elliptic curve E over a scheme S with an S-point P such that none of P, 2P, 3P meets the zero section is the pullback of the universal Tate normal form with its point (0, 0) along a unique morphism S ⟶ T (TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_tateNormalForm). This file cuts the modular curve Y₁(N) out of T:

the N-torsion locus TauCeti.AlgebraicGeometry.UniversalTate.torsionLocus N is the closed subscheme of T on which N • (0, 0) = 0, the equalizer of N • (0, 0) and the zero section. A morphism t : S ⟶ T factors through it exactly when the pullback of N • (0, 0) along t is the pullback of the zero section, an equality of morphisms rather than a condition at points (TauCeti.AlgebraicGeometry.UniversalTate.exists_comp_torsionLocusι_eq_iff);
TauCeti.AlgebraicGeometry.UniversalTate.yOne N is the open subscheme of the N-torsion locus at whose points none of (0, 0), 2 • (0, 0), …, (N - 1) • (0, 0) meets the zero section (TauCeti.AlgebraicGeometry.UniversalTate.exists_comp_yOneι_eq_iff).

The main result is that, for N ≥ 4, Y₁(N) is the fine moduli scheme of elliptic curves with an S-point P of exact order N in every fibre: N • P = 0 and none of P, 2P, …, (N - 1)P meets the zero section. Such a pair (E, P) is the pullback of the universal Tate normal form with its point (0, 0) along a unique morphism S ⟶ Y₁(N), by a unique cartesian morphism carrying the zero section to the zero section and P to (0, 0) (TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_yOne); conversely, every such pullback along a morphism S ⟶ Y₁(N) has a point of exact order N in every fibre (TauCeti.AlgebraicGeometry.EllipticCurveGeom.exists_yOne_iff). The bound N ≥ 4 makes P, 2P and 3P miss the zero section, so that the Tate normal form applies. A point of exact order N in every fibre is the same as a homomorphism ℤ/Nℤ →* E(S) injective on every fibre, a naive [Γ₁(N)]-structure in the sense of Katz and Mazur (TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_yOne_of_isFiberwiseInjective).

No hypothesis on S or on N beyond N ≥ 4 is needed: S may be nonreduced, and N need not be invertible on S. The condition N • P = 0 is scheme-theoretic, while the exact-order condition is a condition at the points of S.

Main definitions #

TauCeti.AlgebraicGeometry.UniversalTate.torsionLocus n, TauCeti.AlgebraicGeometry.UniversalTate.torsionLocusι n: the closed subscheme of Spec ℤ[A, B][Δ⁻¹] on which n • (0, 0) = 0, with its inclusion.
TauCeti.AlgebraicGeometry.UniversalTate.exactOrderOpens N: the open subset of Spec ℤ[A, B][Δ⁻¹] at whose points none of the multiples k • (0, 0), 0 < k < N, meets the zero section.
TauCeti.AlgebraicGeometry.UniversalTate.yOne N, TauCeti.AlgebraicGeometry.UniversalTate.yOneι N: the modular curve Y₁(N), the open subscheme yOneOpens N of the N-torsion locus lying over exactOrderOpens N, with its immersion into Spec ℤ[A, B][Δ⁻¹].

Main results #

TauCeti.AlgebraicGeometry.UniversalTate.exists_comp_torsionLocusι_eq_iff: the morphisms to the base factoring through the n-torsion locus are those along which n • (0, 0) vanishes.
TauCeti.AlgebraicGeometry.UniversalTate.exists_comp_yOneι_eq_iff: the morphisms to the base factoring through Y₁(N) are those along which N • (0, 0) vanishes and no k • (0, 0), 0 < k < N, meets the zero section.
TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_yOne: for N ≥ 4, Y₁(N) is a fine moduli scheme for points of exact order N in every fibre.
TauCeti.AlgebraicGeometry.EllipticCurveGeom.exists_yOne_iff: the pullback of (0, 0) along a morphism to Y₁(N) has exact order N in every fibre, and these are all such points.
TauCeti.AlgebraicGeometry.EllipticCurveGeom.existsUnique_yOne_of_isFiberwiseInjective: the same for naive [Γ₁(N)]-structures.

References #

N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, §2.2 and Chapter 4.
D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), 193–237.
D. Loeffler, Modular curves, lecture notes, §3.3.

The fine moduli scheme Y₁(N) for N ≥ 4 #

Main definitions #

Main results #

References #

The modular curve Y₁(N) from the universal Tate normal form #

Main definitions #

Main results #

References #

The torsion locus #

Y₁(N) #

Classifying pairs #

The moduli property of Y₁(N) #

The universal Tate normal form in Ell/B #

The locus Y₁(N) #

Smoothness of Y₁(N) #

The fine moduli scheme `Y₁(N)` for `N ≥ 4` #

The modular curve `Y₁(N)` from the universal Tate normal form #

`Y₁(N)` #

The moduli property of `Y₁(N)` #

The universal Tate normal form in `Ell/B` #

The locus `Y₁(N)` #

Smoothness of `Y₁(N)` #