The naive moduli problems [Γ₁(N)] and [Γ(N)] #
Let A be a group. A naive A-structure on an elliptic curve E over a scheme S is a
homomorphism φ : A →* E(S) to the group of S-points of E which is injective on every fibre
of E over S (EllipticCurveGeom.IsFiberwiseInjective): for a ≠ 1 the section φ a misses
the zero section everywhere. The condition is pointwise, so it is stable under base change
(EllipticCurveGeom.IsFiberwiseInjective.pointsPullback), and the naive A-structures form a
moduli problem over Ell/B (ModuliProblem.naiveLevel), an instance of ModuliProblem.homLevel.
For A = ℤ/Nℤ this is the naive moduli problem [Γ₁(N)] (ModuliProblem.gammaOneNaive): for
N ≠ 0, a naive [Γ₁(N)]-structure is an S-point P of E[N] none of whose multiples
P, …, (N - 1)P meets the zero section (ModuliProblem.gammaOneNaiveEquiv).
For A = (ℤ/Nℤ)² it is the naive moduli problem [Γ(N)] (ModuliProblem.gammaNaive): a pair
(P, Q) of S-points of E[N] none of whose combinations aP + bQ with (a, b) ≠ 0 meets the
zero section (ModuliProblem.gammaNaiveEquiv), which, when N is invertible on S, is a basis of
the N-torsion in every geometric fibre
(EllipticCurveGeom.isFiberwiseInjective_iff_range_eq_ker). Restriction along an injective
homomorphism A' →* A is a morphism of moduli problems (ModuliProblem.naiveLevelMap); keeping
the first component P of a pair (P, Q), the first basis vector when N is invertible, gives
[Γ(N)] ⟶ [Γ₁(N)] (ModuliProblem.gammaNaiveToGammaOneNaive).
The automorphisms of A act on the right on naive A-structures, φ ↦ φ ∘ σ
(ModuliProblem.naiveLevelAction), and the action is free over a nonempty base because naive
structures are injective. Through GL₂(ℤ/Nℤ) ≃* Aut((ℤ/Nℤ)²) (TauCeti.zmodProdMulAutEquiv)
this is the right action of GL₂(ℤ/Nℤ) on [Γ(N)] (ModuliProblem.gammaNaiveAction) in the
row-vector convention of Katz and Mazur: g = !![a, b; c, d] sends (P, Q) to
(aP + cQ, bP + dQ). The morphism [Γ(N)] ⟶ [Γ₁(N)] is invariant under the matrices with first
column (1, 0).
Naive [Γ(N)]-structures rigidify for N ≥ 3 (Katz–Mazur 2.7.2): an automorphism of an elliptic
curve E/S fixing a naive [Γ(N)]-structure is the identity (ModuliProblem.isRigid_gammaNaive),
with no hypothesis on S; such a structure can only exist where N is invertible. More generally
the naive A-structures are rigid as soon as A contains elements a and b with a² ≠ 1,
b ≠ 1 and b ≠ a^{±1} (ModuliProblem.isRigid_naiveLevel), since an automorphism fixing φ a
and φ b is trivial (EllipticCurveGeom.IsFiberwiseInjective.eq_id_of_isPullback). The bound
N ≥ 3 is sharp: for N ≤ 2, the automorphism [-1] fixes every point killed by N. Taking
a = 1 and b = 2 in ℤ/Nℤ shows that [Γ₁(N)] is rigid for N ≥ 4
(ModuliProblem.isRigid_gammaOneNaive).
The definitions make sense over every base B, but they are the naive level structures of Katz and
Mazur only over a base on which N is invertible, where E[N] is finite étale. Over a general base
the level structures are Drinfeld's (ModuliProblem.gammaOne for [Γ₁(N)], ModuliProblem.gamma
for [Γ(N)]).
Relative representability. For a finite group A, suppose that the homomorphisms
A →* E(T), for schemes T over S, are represented by a scheme H over S with a universal
homomorphism u : A →* E(H). Then the naive A-structures on the base changes E_T are
represented by the open subscheme of H on which u is injective on fibres, the naive locus of
u (ModuliProblem.naiveLevelRepresentableBy): a T-point of E_T is the same as a point of E
with values in T (EllObj.baseChangePointsEquiv), and such a point meets the zero section of
E_T exactly where its image meets the zero section of E. For A = ℤ/Nℤ, N ≠ 0, the
N-torsion E[N] with its universal point is such an H
(EllipticCurveGeom.existsUnique_comp_torsionPointHom), so the naive problem [Γ₁(N)] is
relatively representable by the open subscheme of E[N] of points of exact order N in every
fibre (ModuliProblem.gammaOneNaiveRelRepData); when N is invertible on B, E[N] is étale,
so [Γ₁(N)] is étale over Ell/B (ModuliProblem.hasProperty_etale_gammaOneNaiveRelRepData).
For A = (ℤ/Nℤ)², N ≠ 0, the fibre product E[N] ×_S E[N] with the universal pair of points
of E[N] is such an H (EllipticCurveGeom.existsUnique_comp_torsionPairHom), so the naive
problem [Γ(N)] is relatively representable over every base (ModuliProblem.gammaNaiveRelRepData).
Its representing schemes are open subschemes of E[N] ×_S E[N], so they inherit from E[N] every
property stable under base change and composition which open immersions have
(ModuliProblem.hasProperty_gammaNaiveRelRepData), for instance local finite presentation, and
étaleness when N is invertible on B (ModuliProblem.hasProperty_etale_gammaNaiveRelRepData).
Finiteness and surjectivity where N is invertible. When N is invertible on B, the
representing schemes of [Γ₁(N)] and [Γ(N)] are also closed in E[N] and E[N] ×_S E[N], so
both problems are finite over Ell/B (ModuliProblem.hasProperty_isFinite_gammaOneNaiveRelRepData,
ModuliProblem.hasProperty_isFinite_gammaNaiveRelRepData). They are also surjective over Ell/B
(ModuliProblem.hasProperty_surjective_gammaOneNaiveRelRepData,
ModuliProblem.hasProperty_surjective_gammaNaiveRelRepData): over an algebraic closure of a residue
field of the base, the N-torsion of an elliptic curve is (ℤ/Nℤ)², so every group embedding in
(ℤ/Nℤ)² gives a naive structure there (ModuliProblem.hasProperty_surjective_naiveLevel).
Main definitions #
TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevel B A: the moduli problem of naiveA-structures overEll/B.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelEquiv: the naiveA-structures on an object(E/S)ofEll/Bare the homomorphismsA →* E(S)injective on every fibre.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelMap: restriction of naive structures along an injective homomorphism of groups.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelAction: the right action ofAut Aon naiveA-structures.TauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveAction: the right action ofGL₂(ℤ/Nℤ)on the naive moduli problem[Γ(N)].TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaive B NandTauCeti.AlgebraicGeometry.ModuliProblem.gammaNaive B N: the naive moduli problems[Γ₁(N)]and[Γ(N)].TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaiveEquivandTauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveEquiv: forN ≠ 0, the naive[Γ₁(N)]-structures are theS-points ofE[N]of exact orderNin every fibre, and the naive[Γ(N)]-structures are the pairs ofS-points ofE[N]independent in every fibre.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLocusEquiv: for a universal homomorphismu, the points of the naive locus ofuwith values inTare the naive structures onE_T.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelRepresentableBy: the naive locus represents the fibre of the problem of naiveA-structures.TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaiveRelRepDataandTauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveRelRepData: relative-representability data for the naive problems[Γ₁(N)]and[Γ(N)].
Main results #
TauCeti.AlgebraicGeometry.EllObj.Hom.isFiberwiseInjective_pointsPullback: naive structures pull back along arrows ofEll/B.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelEquiv_map: the pull-back of a naive structure along an arrow ofEll/Bis its composite with base change of points.TauCeti.AlgebraicGeometry.ModuliProblem.naiveLevelEquiv_gammaNaiveToGammaOneNaive_app_apply: the morphism[Γ(N)] ⟶ [Γ₁(N)]restricts a naive[Γ(N)]-structure alongz ↦ (z, 0).TauCeti.AlgebraicGeometry.ModuliProblem.eq_one_of_naiveLevelAction_hom_app_eq_selfandTauCeti.AlgebraicGeometry.ModuliProblem.eq_one_of_gammaNaiveAction_hom_app_eq_self: the actions are free over a nonempty base.TauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveEquiv_gammaNaiveAction_hom_app_fst_torsionιand itssndanalogue:(P, Q) g = (aP + cQ, bP + dQ)forg = !![a, b; c, d].TauCeti.AlgebraicGeometry.ModuliProblem.isRigid_naiveLevel,TauCeti.AlgebraicGeometry.ModuliProblem.isRigid_gammaNaiveandTauCeti.AlgebraicGeometry.ModuliProblem.isRigid_gammaOneNaive: naiveA-structures are rigid whenAcontains a suitable pair of elements, the naive moduli problem[Γ(N)]is rigid forN ≥ 3, and[Γ₁(N)]is rigid forN ≥ 4.TauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveAction_hom_comp_gammaNaiveToGammaOneNaive: keeping the first point is invariant under the matrices with first column(1, 0).TauCeti.AlgebraicGeometry.ModuliProblem.gammaOneNaiveEquiv_map_torsionιandTauCeti.AlgebraicGeometry.ModuliProblem.gammaNaiveEquiv_map_fst_torsionι: the points ofE[N]of a pulled-back naive structure are the base changes of the original points.TauCeti.AlgebraicGeometry.ModuliProblem.isFiberwiseInjective_baseChange_iff: injectivity on the fibres of a base changeE_T, read on the points ofEwith values inT.TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_gammaOneNaiveRelRepDataandTauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_gammaNaiveRelRepData: the representing morphisms of[Γ₁(N)]and[Γ(N)]inherit the properties ofE[N]stable under base change and composition and holding for open immersions.TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_etale_gammaOneNaiveRelRepDataandTauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_etale_gammaNaiveRelRepData: ifNis invertible onB, the naive problems[Γ₁(N)]and[Γ(N)]are étale overEll/B.TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_isFinite_gammaOneNaiveRelRepDataandTauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_isFinite_gammaNaiveRelRepData: ifNis invertible onB, the naive problems[Γ₁(N)]and[Γ(N)]are finite overEll/B.TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_surjective_naiveLevel,TauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_surjective_gammaOneNaiveRelRepDataandTauCeti.AlgebraicGeometry.ModuliProblem.hasProperty_surjective_gammaNaiveRelRepData: ifNis invertible onB, naive structures by a group embedding in(ℤ/Nℤ)², in particular the naive problems[Γ₁(N)]and[Γ(N)], are surjective overEll/B.
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §3.7 (naive level structures
where
Nis invertible), 3.7.1 (relative representability of naive full levelN), Chapter 4, and §7.1 (the action ofGL₂(ℤ/Nℤ)on[Γ(N)]). - D. Loeffler, Modular curves, lecture notes, Proposition 3.8.2 (naive level structures are
relatively representable and finite étale over
Ell/ℤ[1/N]).
Naive A-structures pull back along arrows of Ell/B: an arrow f : X ⟶ Y exhibits X
as a base change of Y, and a homomorphism A →* E(S) injective on every fibre stays so after
composing with base change of points along f.
The moduli problem of naive A-structures over Ell/B. A naive A-structure on an
object (E/S) of Ell/B is a homomorphism A →* E(S) injective on every fibre of E over S
(naiveLevelEquiv), and it pulls back along an arrow of Ell/B by base change of points
(naiveLevelEquiv_map). The group A lies in Type, as the groups (ℤ/Nℤ)ʳ of level structures
do, so that the naive structures lie in the universe Type (u + 1) of moduli problems.
Instances For
The naive A-structures on an object (E/S) of Ell/B are the homomorphisms A →* E(S)
injective on every fibre of E over S.
Instances For
The pull-back of a naive A-structure along an arrow φ of Ell/B is its composite with
base change of points along φ.
Restriction of naive structures. An injective homomorphism ψ : A' →* A restricts naive
A-structures to naive A'-structures, φ ↦ φ ∘ ψ.
Instances For
Restriction along ψ composes a naive structure with ψ.
Two morphisms of moduli problems into the naive A-structures agree once they assign the same
homomorphism A →* E(S) to every structure.
The action of the automorphisms of A #
The right action of the automorphisms of A on naive A-structures. An automorphism σ
of A sends a naive A-structure φ : A →* E(S) to φ ∘ σ
(naiveLevelEquiv_naiveLevelAction_hom_app). Since φ ∘ (σ ∘ τ) = (φ ∘ σ) ∘ τ, this is a right
action, that is, a homomorphism from the opposite group (Aut A)ᵐᵒᵖ to the automorphisms of the
moduli problem.
Instances For
An automorphism σ of A sends a naive A-structure φ to φ ∘ σ.
The inverse of the automorphism of the moduli problem given by σ sends a naive
A-structure φ to φ ∘ σ⁻¹.
The action of the automorphisms of A on naive A-structures is free. Over a nonempty
base, an automorphism of A fixing a naive A-structure is the identity, because a naive
structure is injective. Over the empty base every automorphism fixes the unique structure.
Rigidity #
Naive A-structures are rigid when A contains a full level pair. If A contains a and
b with a² ≠ 1, b ≠ 1, b ≠ a and b ≠ a⁻¹, then an automorphism of an elliptic curve
E/S over the identity of S fixing a naive A-structure is the identity. No hypothesis is made
on the base.
The naive moduli problem [Γ₁(N)] over Ell/B: a naive [Γ₁(N)]-structure on (E/S) is
a homomorphism ℤ/Nℤ →* E(S) injective on every fibre, that is, for N ≠ 0, an S-point P of
E[N] none of whose multiples P, …, (N - 1)P meets the zero section
(EllipticCurveGeom.isFiberwiseInjective_zmod_iff). It is the naive level structure of Katz and
Mazur over a base on which N is invertible.
Instances For
The naive moduli problem [Γ(N)] over Ell/B: a naive [Γ(N)]-structure on (E/S) is a
homomorphism (ℤ/Nℤ)² →* E(S) injective on every fibre. Over a base on which N is invertible,
it is a pair of S-points of E[N] forming a basis of the N-torsion in every geometric fibre
(EllipticCurveGeom.isFiberwiseInjective_iff_range_eq_ker).
Instances For
Full level structures rigidify for N ≥ 3 (Katz–Mazur 2.7.2). An automorphism of an
elliptic curve E/S over the identity of S fixing a naive [Γ(N)]-structure (P, Q) is the
identity, because it fixes P, Q and -P, which are distinct in every fibre. No hypothesis is
made on the base, which may be nonreduced or non-affine.
Naive [Γ₁(N)]-structures rigidify for N ≥ 4. An automorphism of an elliptic curve
E/S over the identity of S fixing a naive [Γ₁(N)]-structure P is the identity, because it
fixes P, 2P and -P, which are distinct in every fibre. No hypothesis is made on the base. The
bound is sharp: on Y² + Y = X³ over a field containing a primitive cube root of unity ζ, the
automorphism (X, Y) ↦ (ζX, Y) fixes the point (0, 0) of order 3.
The morphism [Γ(N)] ⟶ [Γ₁(N)] of naive moduli problems which keeps the first component P
of a naive [Γ(N)]-structure (P, Q), that is, restricts it along z ↦ (z, 0). When N is
invertible, P is the first vector of the basis (P, Q) of the N-torsion.
Instances For
The morphism [Γ(N)] ⟶ [Γ₁(N)] restricts a naive [Γ(N)]-structure along z ↦ (z, 0).
Naive level structures as points of E[N] #
Naive [Γ₁(N)]-structures are points of E[N]. For N ≠ 0, a naive
[Γ₁(N)]-structure φ on (E/S) is determined by the S-point P = φ 1 of E[N], and the
S-points of E[N] arising this way are those of exact order N in every fibre: none of
P, 2P, …, (N - 1)P meets the zero section (EllipticCurveGeom.isFiberwiseInjective_zmod_iff).
Instances For
The S-point of E[N] of a naive [Γ₁(N)]-structure φ is φ 1.
The naive [Γ₁(N)]-structure of an S-point P of E[N] sends a to aP.
The S-point of E[N] of the pull-back of a naive [Γ₁(N)]-structure along an arrow φ of
Ell/B is the base change along φ of its S-point of E[N].
Naive [Γ(N)]-structures are pairs of points of E[N]. For N ≠ 0, a naive
[Γ(N)]-structure φ on (E/S) is determined by the pair (P, Q) = (φ (1, 0), φ (0, 1)) of
S-points of E[N], and the pairs arising this way are those none of whose combinations aP + bQ
with (a, b) ≠ 0 meets the zero section (EllipticCurveGeom.isFiberwiseInjective_zmod_prod_iff).
When N is invertible on S, these are the pairs forming a basis of the N-torsion in every
geometric fibre.
Instances For
The first S-point P of E[N] of a naive [Γ(N)]-structure φ is φ (1, 0).
The second S-point Q of E[N] of a naive [Γ(N)]-structure φ is φ (0, 1).
The naive [Γ(N)]-structure of a pair (P, Q) of S-points of E[N] sends (a, b) to
aP + bQ.
The first S-point of E[N] of the pull-back of a naive [Γ(N)]-structure along an arrow φ
of Ell/B is the base change along φ of its first S-point of E[N].
The second S-point of E[N] of the pull-back of a naive [Γ(N)]-structure along an arrow
φ of Ell/B is the base change along φ of its second S-point of E[N].
The action of GL₂(ℤ/Nℤ) on naive [Γ(N)]-structures #
The right action of GL₂(ℤ/Nℤ) on the naive moduli problem [Γ(N)]. A matrix
g = !![a, b; c, d] acts on (ℤ/Nℤ)² by multiplication of column vectors
(TauCeti.zmodProdMulAutEquiv), and so sends the naive [Γ(N)]-structure with points (P, Q) to
the one with points (P, Q) g = (aP + cQ, bP + dQ)
(gammaNaiveEquiv_gammaNaiveAction_hom_app_fst_torsionι,
gammaNaiveEquiv_gammaNaiveAction_hom_app_snd_torsionι).
Instances For
A matrix g sends a naive [Γ(N)]-structure φ : (ℤ/Nℤ)² →* E(S) to φ ∘ g.
The inverse of the automorphism of [Γ(N)] given by a matrix g sends a naive
[Γ(N)]-structure φ : (ℤ/Nℤ)² →* E(S) to φ ∘ g⁻¹.
The naive [Γ(N)]-structures are acted on freely by GL₂(ℤ/Nℤ) over a nonempty base.
Keeping the first point is invariant under the matrices fixing (1, 0). A matrix whose
first column is (1, 0) does not change the first point P of a naive [Γ(N)]-structure, so the
morphism [Γ(N)] ⟶ [Γ₁(N)] is invariant under the subgroup of such matrices.
The row-vector convention, first point. The first point of E[N] of the naive
[Γ(N)]-structure (P, Q) g is aP + cQ, where (a, c) is the first column of g.
The row-vector convention, second point. The second point of E[N] of the naive
[Γ(N)]-structure (P, Q) g is bP + dQ, where (b, d) is the second column of g.
Relative representability #
Injectivity on fibres of a base change, read on the original curve. For an object
X = (E/S) of Ell/B and a scheme T over S, a homomorphism φ : A →* E_T(T) is injective on
every fibre of the base change E_T exactly when, for every a ≠ 1, the point of E with values
in T given by φ a (EllObj.baseChangePointsEquiv) misses the zero section of E at every
point of T.
The naive structure of a point of the naive locus. For a point y of the naive locus of
u with values in a scheme T over S, the homomorphism A →* E_T(T) sending a to the
T-point of the base change E_T given by the point y ≫ u a of E with values in T. It is
injective on every fibre (isFiberwiseInjective_naiveLocusHom).
Instances For
The naive structure of a point of the naive locus is injective on every fibre.
The naive structures given by the naive locus. For a universal homomorphism u, the points
of the naive locus of u with values in a scheme T over S correspond to the naive
A-structures on the base change E_T, a point y corresponding to naiveLocusHom u y.
Instances For
The correspondence naiveLocusEquiv is natural in the scheme T over S.
Naive structures are represented by the naive locus. Let X = (E/S) be an object of
Ell/B, A a finite group, and u : A →* E(H) a universal homomorphism to the points of E with
values in a scheme H over S: every homomorphism φ : A →* E(T), for T over S, is
a ↦ x ≫ u a for a unique point x of H with values in T. Then the naive A-structures on the
base changes E_T are represented by the open subscheme of H on which u is injective on
fibres (naiveLocusEquiv).
Instances For
Surjectivity where N is invertible #
Naive structures exist over geometric points. Let A be a group embedding in (ℤ/Nℤ)²,
where N is invertible in every residue field of B. Then every relatively representable problem
of naive A-structures is surjective over Ell/B: an elliptic curve acquires a naive
A-structure over an algebraic closure of each residue field of its base, since there its
N-torsion is (ℤ/Nℤ)² (EllipticCurveGeom.exists_isFiberwiseInjective).
Relative representability of the naive problem [Γ₁(N)] #
The naive moduli problem [Γ₁(N)] is relatively representable. For an object
X = (E/S) of Ell/B, the naive [Γ₁(N)]-structures on the base changes E_T of E are
represented by the open subscheme of E[N] of points P none of whose multiples aP with
a ≠ 0 meets the zero section, the points of exact order N in every fibre: the naive locus of
the universal point of E[N] (EllipticCurveGeom.torsionPointHom).
Instances For
The representing morphisms of the naive problem [Γ₁(N)] have every property of morphisms of
schemes which is stable under composition, holds for open immersions, and holds for the
N-torsion of every elliptic curve: each representing scheme is an open subscheme of E[N].
The naive moduli problem [Γ₁(N)] is étale over Ell/B when N is invertible on B:
if N is invertible in every residue field of B, every representing scheme is étale over its
base, as an open subscheme of the étale N-torsion.
The naive moduli problem [Γ₁(N)] is finite over Ell/B when N is invertible on B:
if N is invertible in every residue field of B, every representing scheme is finite over its
base, as a closed subscheme of the finite N-torsion.
The naive moduli problem [Γ₁(N)] is surjective over Ell/B when N is invertible on
B: every elliptic curve acquires a point of exact order N in every fibre over an algebraic
closure of each residue field of its base.
Relative representability of the naive problem [Γ(N)] #
The naive moduli problem [Γ(N)] is relatively representable. For an object X = (E/S)
of Ell/B, the naive [Γ(N)]-structures on the base changes E_T of E are represented by the
open subscheme of E[N] ×_S E[N] of pairs (P, Q) none of whose combinations aP + bQ with
(a, b) ≠ 0 meets the zero section: the naive locus of the universal pair of points of E[N]
(EllipticCurveGeom.torsionPairHom). This holds over every base B; over a base on which N is
invertible these are the naive full level N structures of Katz and Mazur.
Instances For
The representing morphisms of the naive problem [Γ(N)] have every property of morphisms of
schemes which is stable under base change and composition, holds for open immersions, and holds
for the N-torsion of every elliptic curve: each representing scheme is an open subscheme of
E[N] ×_S E[N], whose structure morphism is the composite of that open immersion with the
morphism from E[N] ×_S E[N] to the base.
The naive problem [Γ(N)] is locally of finite presentation over Ell/B.
The naive moduli problem [Γ(N)] is étale over Ell/B when N is invertible on B
(Katz–Mazur, 3.7.1): if N is invertible in every residue field of B, each representing scheme
is étale over its base, as an open subscheme of E[N] ×_S E[N].
The naive moduli problem [Γ(N)] is finite over Ell/B when N is invertible on B:
if N is invertible in every residue field of B, every representing scheme is finite over its
base, as a closed subscheme of the finite scheme E[N] ×_S E[N].
The naive moduli problem [Γ(N)] is surjective over Ell/B when N is invertible on
B: every elliptic curve acquires a basis of its N-torsion over an algebraic closure of each
residue field of its base.