THE PICARD GROUP OF ℳ_{1,1}

WILLIAM FULTON AND MARTIN OLSSON

Abstract. We compute the Picard group of the moduli stack of elliptic curves and its canonical compactification over general base schemes

1. Introduction

Let ℳ_{1,1} denote the moduli stack (over ℤ) classifying elliptic curves, and for a scheme S let ℳ_{1,1,S} denote the fiber product S ×_{Spec(ℤ)}ℳ_{1,1}. In his 1965 paper [7], Mumford computed the Picard group Pic(ℳ_{1,1,S}) when S is the spectrum of a field of characteristic not 2 or 3 and found it to be cyclic of order 12. Our aim in this paper is to compute the Picard group Pic(ℳ_{1,1,S}) for more general base schemes S, as well as to compute the Picard group Pic(\overline{ℳ} _{1,1,S}) for the standard compactification \overline{ℳ} _{1,1} of ℳ_{1,1}.

Recall that on ℳ_{1,1} there is the Hodge bundle λ. For any morphism t : T → ℳ_{1,1} corresponding to an elliptic curve f : E → T the pullback t^{∗}λ is the line bundle f_{∗}Ω_{E/T}^{1} . Equivalently, if f : ℰ → ℳ_{1,1} denotes the universal elliptic curve then λ = f_{∗}Ω_{ℰ /ℳ_{1,1}}^{1} . This bundle extends canonically to \overline{ℳ} _{1,1}. Namely, let f̄ : \overline{ℰ} → \overline{ℳ} _{1,1} denote the extension of ℰ provided by the Tate curve and let ω_{\overline{ℰ} /\overline{ℳ_{1,1}}} denote the relative dualizing sheaf. Then the sheaf f̄_{∗}ω_{\overline{ℰ} /\overline{ℳ_{1,1}}} is a line bundle on \overline{ℳ} _{1,1} extending λ. In what follows we will abuse notation and write also λ for this line bundle on \overline{ℳ} _{1,1}.

If Λ is a ring and t : Spec(Λ) → ℳ_{1,1} is a morphism corresponding to an elliptic curve E/Λ, then after replacing Λ by an étale extension the family E can be described by an equation

(1.0.1)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Define

(1.0.2) b_{2} = a_{1}^{2}+4a_{2}, b_{4} = a_{1}a_{3}+2a_{4}, b_{6} = a_{3}^{2}+4a_{6}, b_{8} = −a_{1}a_{3}a_{4}−a_{4}^{2}+a_{1}^{2}a_{6}+a_{2}a_{3}^{2}+4a_{2}a_{6},

and the discrimant

(1.0.3)  Δ = −b_{2}^{2}b_{8} − 8b_{4}^{3} − 27b_{6}^{2} + 9b_{2}b_{4}b_{6} ∈ Λ^{∗}.

With these chosen coordinates a basis for t^{∗}λ is given by the invariant differential

(1.0.4)  π = dx/(2y + a_{1}x + a_{3}).

Any two choices of coordinates 1.0.1 differ by a transformation

(1.0.5)  x′ = u^{2}x + r, y′ = u^{3}y + su^{2}x + t,

where u ∈ Λ^{∗} and r, s, t ∈ Λ. One can compute that the invariant differential π′ obtained from the coordinates (x′, y′) is equal to u^{−1}π, and that the discrimant Δ′ in the coordinates (x′, y′) is equal to u^{12}Δ. In particular, the element Δπ^{⊗12} ∈ t^{∗}λ^{⊗12} is independent of the choice of coordinates, and therefore defines a trivialization of λ^{⊗12} over ℳ_{1,1}.

Let p : ℳ_{1,1,S} → 𝔸_{S}^{1} be the map defined by the j-invariant

(1.0.6)  j = (b_{2}^{2} − 24b_{4})^{3}/Δ.

Theorem 1.1. Let S be a scheme. Then the map

(1.1.1)  ℤ/(12) × Pic(𝔸_{S}^{1}) → Pic(ℳ_{1,1,S}), (i, ℒ) ↦ λ^{⊗i} ⊗ p^{∗}ℒ

is an isomorphism if either of the following hold:

(i) S is a ℤ[1/2]-scheme.

(ii) S is reduced.

Remark 1.2. As we observe in 6.4 the theorem fails for nonreduced schemes in characteristic 2.

Theorem 1.3. The map

(1.3.1)  ℤ × Pic(S) → Pic(\overline{ℳ} _{1,1,S}) (n, M) ↦ λ^{n} ⊗_{𝒪_{S}} M

is an isomorphism for any scheme S.

Remark 1.4. By standard limit arguments it suffices to prove the above results in the case when S is noetherian. In what follows we will therefore restrict to the category of noetherian schemes unless otherwise stated.

1.5. Acknowledgements. Olsson partially supported by NSF grant DMS-0555827 and an Alfred P. Sloan fellowship.

2. When 6 is invertible on S

Though the case when 6 is invertible follows from the more technical work in subsequent sections, we include here a proof in the case of a ℤ[1/6]-scheme since it is much easier than the more general cases.

Let s̃_{4} : S → ℳ_{1,1,S} be the section corresponding to the elliptic curve with automorphism group μ_{4} (y^{2} = x^{3}+x with Δ = −64, j = 1728) and s̃_{6} : S → ℳ_{1,1,S} the section corresponding to the elliptic curve with automorphism group μ_{6} (y^{2} + y = x^{3} with Δ = −27, j = 0). These sections define closed immersions s_{4} : Bμ_{4,S} ↪ ℳ_{1,1,S} and s_{6} : Bμ_{6,S} ↪ ℳ_{1,1,S}. For any line bundle ℒ on ℳ_{1,1,S} the pullback s_{4}^{∗}ℒ (resp. s_{6}^{∗}ℒ) corresponds to a line bundle M_{4} (resp. M_{6}) on S with action of the group μ_{4} (resp. μ_{6}). We thus get maps

ρ_{4} : μ_{4} → \underline{Aut}(M_{4}) = 𝔾_{m}, ρ_{6} : μ_{6} → \underline{Aut}(M_{6}) ≃ 𝔾_{m}

defining characters χ_{4} ∈ ℤ/(4) and χ_{6} ∈ ℤ/(6).

Lemma 2.1. The pair (χ_{4}, χ_{6}) lies in ℤ/(12) ⊂ ℤ/(4) × ℤ/(6).

Proof. The construction of the pair (χ_{4}, χ_{6}) commutes with arbitrary base change on S, so it suffices to consider the case when S is the spectrum of an algebraically closed field, S = Spec(k). We have to show that ρ_{4}|_{μ_{2}} = ρ_{6}|_{μ_{2}}. Write k[[t]] for the completion of the local ring of 𝔸_{j}^{1} at j = 1728 and let k[[z]] be the completion of the local ring of ℳ_{1,1,S} at the point coresponding to the curve y^{2} = x^{3} + x. Then the map k[[t]] → k[[z]] sends t to z^{2} (with suitable choices of coordinates) and the action of μ_{4} is given by ζ ∗ z = ζ^{2} · z. Write ℒ |_{k[[z]]]} = k[[z]] · e for some basis e. Then ρ_{4} acts by ζ ∗ e = ζ^{χ_{4}}e. From this we see that ρ_{4}|_{μ_{2}} is equal to the character defined by the action of μ_{2} on the fiber of ℒ at the generic point of ℳ_{1,1,S}. Similarly, ρ_{6}|_{μ_{2}} is equal to the action on the generic fiber. □

We therefore obtain a map

(2.1.1)  Pic(ℳ_{1,1,S}) → ℤ/(12), ℒ ↦ (χ_{4}, χ_{6}),

and it follows from the construction that this map is a homomorphism. Let K denote the kernel.

Recall that a Deligne-Mumford stack 𝒳 is called tame if for every algebraically closed field Ω and point x̄ : Spec(Ω) → 𝒳 the order of the automorphism group of x̄ is relatively prime to the characteristic of Ω.

Lemma 2.2. Let 𝒳 be a tame Deligne–Mumford stack with coarse moduli space π : 𝒳 → X. Let ℒ be an invertible sheaf on 𝒳 such that for every geometric point x̄ → 𝒳 the action of the stabilizer group Gx̄ on ℒ (x̄) is trivial. Then π_{∗}ℒ is an invertible sheaf on X and π^{∗}π_{∗}ℒ → ℒ is an isomorphism.

Proof. It suffices to prove the lemma after passing to the strict henselization of X at a geometric point x̄. Let A = 𝒪_{X,x̄} and B = 𝒪_{𝒳 ,x̄} . Then as explained in [8, 2.12] if Γ denotes the stabilizer group of x̄ then there is a natural action of Γ on B such that 𝒳 = [Spec(B)/Γ]. Let M be the free B–module with Γ–action of rank 1 defining ℒ. Since Γ has order invertible in k(x̄) (since 𝒳 is tame) the representation category of Γ is semisimple. By our assumptions the reduction M ⊗ k(x̄) is generated by an invariant element and choosing a lifting to an invariant element of M we see that we can write M = B · e where Γ acts trivially on e. Then π_{∗}ℒ is just A · e and the lemma is immediate. □

Corollary 2.3. The homomorphism π^{∗} : Pic(𝔸_{S}^{1}) → K is an isomorphism.

Proof. We show that if ℒ is a line bundle with (χ_{4}, χ_{6}) = (0, 0), then π_{∗}ℒ is an invertible sheaf on 𝔸_{S}^{1} and π^{∗}π_{∗}ℒ → ℒ is an isomorphism. By 2.2 it suffices to show that for any geometric point x̄ → ℳ_{1,1,S} the action of the stabilizer group of x̄ on ℒ (x̄) is trivial. For this we may assume that S is the spectrum of an algebraically closed field. By our assumptions the actions ρ_{4} and ρ_{6} are trivial. By the argument used in the proof of 2.1 this implies that the action of the generic stabilizer is also trivial. From this it follows that the action is trivial at every point since over 𝔸^{1} − {0, 1728} the stack ℳ_{1,1,S} is a μ_{2}–gerbe. □

Lemma 2.4. The image of λ in ℤ/(12) is a generator. In particular 2.1.1 is surjective.

Proof. It suffices to consider the case when S is the spectrum of a field in which case the above shows that Pic(ℳ_{1,1,S}) injects into ℤ/(12). We can in fact compute directly the image of λ in ℤ/(4) × ℤ/(6). The image in ℤ/(4) corresponds to the representation of μ_{4} given by the action on the invariant differential dx/2y of the curve y^{2} = x^{3} + x. An element ζ ∈ μ_{4} acts by (x, y) ↦ (ζ^{2}x, ζy) and therefore the action on dx/2y is equal to multiplication by ζ. Therefore the image of λ in ℤ/(4) is equal to 1.

Similarly, the image of λ in ℤ/(6) corresponds to the character given by the invariant differential dx/(2y + 1) of the curve y^{2} + y = x^{3}. Write μ_{6} = μ_{2} × μ_{3}. Then (−1, 1) acts by (x, y) ↦ (x, −y − 1) and (1, ζ) acts by (x, y) ↦ (ζx, y). Therefore (−1, 1) acts on the invariant differential by multiplication by −1 and (1, ζ) acts by multiplication by ζ. It follows that λ maps to 1 in ℤ/(6) which implies that λ is a generator in ℤ/(12). □

Corollary 2.5. The map λ × π^{∗} : (ℤ/12) × Pic(𝔸_{S}^{1}) → Pic(ℳ_{1,1,S}) is an isomorphism.

3. The case of a normal affine scheme S

Write S = Spec(Λ) with Λ a normal ring. Let U be the scheme

(3.0.1)  U := Spec(Λ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}][1/Δ]),

where Δ is defined as in 1.0.3. The equation 1.0.1 defines a family of elliptic curves E → U. Let G denote the group scheme with underlying scheme Spec(Λ[u^{±}, r, s, t]) with group law defined by

(3.0.2)  (u′, r′, s′, t′) · (u, r, s, t) = (uu′, u^{2}r′ + r, us′ + s, u^{3}t′ + u^{2}r′s + t).

Then ℳ_{1,1,S} is isomorphic to the stack theoretic quotient [U/G].

Proposition 3.1. The pullback map

(3.1.1)  Pic(S) → Pic(U)

is an isomorphism.

Proof. The key point is the following result of Ischebeck [5, §4].

Lemma 3.2. Let Δ ∈ ℤ[t_{1}, …, t_{n}] be a polynomial satisfying

(i) The greatest common divisor of the coefficients of its nonconstant monomials is 1. (ii) For any field k the image of Δ in k[t_{1}, …, t_{n}] is irreducible.

Then for any noetherian normal ring Λ, the pullback homomorphism

(3.2.1)  Pic(Λ) → Pic(Λ[t_{1}, …, t_{n}][1/Δ])

is an isomorphism.

Proof. The assumptions are used as follows:

(1) Assumption (i) implies that the map Spec(ℤ[t_{1}, …, t_{n}][1/Δ]) → Spec(ℤ) is surjective and hence faithfully flat. It follows that the map

(3.2.2)  Spec(Λ[t_{1}, …, t_{n}][1/Δ]) → Spec(Λ)

is also faithfully flat.

(2) By the preceding observation the divisor V (Δ) ⊂ Spec(Λ[t_{1}, …, t_{n}]) does not contain any fibers, and its generic fiber is nonempty and irreducible. From this it follows that V (Δ) is irreducible.

It follows that there is an exact sequence of Weil divisor class groups [3, 1.8]

(3.2.3)  ℤ[V (Δ)] ⟶^{0} Cl(Λ[t_{1}, …, t_{n}]) ⟶ Cl(Λ[t_{1}, …, t_{n}][1/Δ]) ⟶ 0.

We conclude that

(3.2.4)  Cl(Λ) ≃ Cl(Λ[t_{1}, …, t_{n}]) ≃ Cl(Λ[t_{1}, …, t_{n}][1/Δ]).

The normality of Λ implies that the natural maps from the Picard groups to the Weil divisor class groups are injective. Thus it suffices to show that if D ∈ Cl(Λ) is a Weil divisor whose image in Cl(Λ[t_{1}, …, t_{n}][1/Δ]) is in the image of Pic(Λ[t_{1}, …, t_{n}][1/Δ]) then D is obtained from a line bundle on Spec(Λ). This follows from the observation that Λ → Λ[t_{1}, …, t_{n}][1/Δ] is faithfully flat [5, §4, Satz 6]. □

We apply the lemma to Δ ∈ ℤ[a_{1}, …, a_{6}]. Then (i) is immediate and (ii) follows from the calculations in [2, §3, §4] (note that though these sections concern characteristics prime to 6 the same calculations give the irreducibility of Δ over arbitrary fields). □

The isomorphism ℳ_{1,1,S} ≃ [U/G] defines a morphism σ : ℳ_{1,1,S} → BG. For a character χ : G → 𝔾_{m} defining a line bundle on BG let L_{χ} be the line bundle on ℳ_{1,1,S} obtained by pull back along σ.

Lemma 3.3. Let ℒ be a line bundle on ℳ_{1,1,S} such that the pullback L of ℒ to U is trivial. Then ℒ ≃ L_{χ} for some character χ : G → 𝔾_{m}.

Proof. Fix a basis e ∈ L.

Let F be the sheaf on the category of affine S-schemes (with the étale topology) which to any morphism of affine schemes S′ → S associates Γ(U_{S′}, 𝒪_{U_{S′}}^{∗}). There is an inclusion of sheaves 𝔾_{m} ⊂ F given by the inclusions Γ(S′, 𝒪_{S′}^{∗}) ⊂ Γ(U_{S′}, 𝒪_{U_{S′}}^{∗}). For any S′ → S and g ∈ G(S′), we get an element u_{g} ∈ F(S′) defined by the condition that g(e) = u_{g} · e ∈ L. This defines a map of sheaves (not necessarily a homomorphism)

(3.3.1)  f : G → F.

To prove the lemma it suffices to show that f has image contained in 𝔾_{m} ⊂ F (note that it is clear that if this holds then the map G → 𝔾_{m} is a homomorphism).

Since G is an affine scheme the map f is determined by a section u_{0} ∈ F(G). Since G is normal and connected, this section u_{0} ∈ Γ(U_{G}, 𝒪_{U_{G}}^{∗}) can be written uniquely as βΔ^{m}, where β ∈ Γ(G, 𝒪_{G}^{∗}) and m ∈ ℤ. We need to show that m = 0. For this note that the image of u_{0} under the map F(G) → F(S) defined by the identity section e : S → G is equal to 1. It follows that e^{∗}(β) · Δ^{m} is equal to 1 in Γ(U, 𝒪_{U}^{∗}) which implies that m = 0. □

By 3.1, if ℒ is a line bundle on ℳ_{1,1,S} then the pullback of ℒ to U is isomorphic to the pullback of a line bundle M on S. It follows that any line bundle on ℳ_{1,1,S} is isomorphic to M ⊗ L_{χ} for some character χ : G → 𝔾_{m}. More such a line bundle M ⊗ L_{χ} is trivial if and only if M is trivial and L_{χ} is trivial.

Lemma 3.4. Any homomorphism G → 𝔾_{m} factors through the projection

(3.4.1)  χ_{0} : G → 𝔾_{m}, (u, r, s, t) ↦ u.

Proof. There are three injective homomorphisms

(3.4.2)  j_{r}, j_{s}, j_{t} : 𝔾_{a} ↪ G

sending x ∈ 𝔾_{a} to (1, x, 0, 0), (1, 0, x, 0), and (1, 0, 0, x) respectively. The formula

(3.4.3)  (1, r, 0, 0)(1, 0, s, 0)(1, 0, 0, t − rs) = (1, r, s, t)

shows that the subgroup of G generated by the images of these three inclusions is equal to the kernel of χ_{0}. Since any homomorphism 𝔾_{a} → 𝔾_{m} is trivial, it follows that any homomorphism G → 𝔾_{m} has kernel containing Ker(χ_{0}). □

The line bundle λ is trivialized over U by the invariant differential π defined in 1.0.4 and as mentioned in the introduction the action of (u, r, s, t) ∈ G on π is through the character G → 𝔾_{m} sending (u, r, s, t) to u^{−1}. Putting all this together we find that 1.1.1 is surjective.

In fact, if χ : G → 𝔾_{m} is a character, a trivialization of L_{χ} is given by a unit θ ∈ Γ(U, 𝒪_{U}^{∗}) such that for any (u, r, s, t) ∈ G we have (u, r, s, t) ∗ θ = χ^{−1}(u, r, s, t)θ. Any unit θ on U can be written as βΔ^{m} for β ∈ Λ^{∗} and m ∈ ℤ. We have

(3.4.4)  (u, r, s, t) ∗ (βΔ^{m}) = βu^{12m}Δ^{m}.

It follows that L_{χ} is trivial if and only if χ = χ_{0}^{12m}, for some m.

This completes the proof of 1.1 in the case when S is affine and normal. □

A very similar argument can be used to prove 1.3 in the case when the base scheme S is affine and normal. Let c_{4} = b_{2}^{2} − 24b_{4}. Then one can show (see for example [10, III.1.4]) that 1.0.1 is nodal precisely when Δ = 0 and c_{4} ≠ 0. Let Ũ denote

(3.4.5)  Spec(Λ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}]) − V (Δ, c_{4}).

Again the group scheme G acts on Ũ and \overline{ℳ} _{1,1,S} ≃ [Ũ/G].

Lemma 3.5. (i) The map

(3.5.1)  Pic(Λ) → Pic(Ũ)

is an isomorphism.

(ii) The map Λ^{∗} → Γ(Ũ, 𝒪∗ ; Ũ) is an isomorphism.

Proof. Statement (ii) is immediate. Statement (i) follows from a very similar argument to the proof of 3.1. The only new ingredient is that the polynomial c_{4} is not irreducible over fields of characteristics 2 and 3 but it is a power of an irreducible polynomial (in characteristic 2 it is equal to a_{1}^{4} and in characteristic 3 is is equal to (a_{1} + a_{2})^{2}. □

Using this one sees as before that the map

(3.5.2)  Pic(Λ) × Pic(BG) → Pic(\overline{ℳ} _{1,1,S})

is an isomorphism with the character 3.4.1 mapping to λ^{−1}.

4. The case when S is reduced

4.1. If S is an arbitrary scheme, and ℒ a line bundle on ℳ_{1,1,S} then there is a unique function s ↦ l(s) ∈ ℤ/(12) which associates to a point s the unique power l(s) of λ such that ℒ_{s} ⊗ λ^{−l(s)} on ℳ_{1,1,k(s)} descends to 𝔸_{k(s)}^{1}.

Lemma 4.2. The function s ↦ l(s) is a locally constant function on S.

Proof. The assertion is local on S so we may assume that S is affine. Furthermore, the assertion can be verified on each irreducible component so we may assume that S is integral. Finally if S̃ → S is the normalization then it suffices to verify the assertion for S̃. In this case the result follows from section 3. □

4.3. In particular if S is connected we obtain a homomorphism

(4.3.1)  Pic(ℳ_{1,1,S}) → ℤ/(12)

sending λ to 1. Thus in general to prove 1.1 we need to show that the kernel of 4.3.1 is isomorphic to Pic(𝔸_{S}^{1}).

Lemma 4.4. For any locally noetherian scheme S, the map π : ℳ_{1,1,S} → 𝔸_{S}^{1} given by the j-invariant identifies 𝔸_{S}^{1} with the coarse moduli space of ℳ_{1,1,S}.

Proof. Let π̃ : ℳ_{1,1,S} → X be the coarse moduli space (which exists by [6]). By the universal property of the coarse moduli space, there exists a unique morphism f : X → 𝔸_{j}^{1} such that f ∘ π̃ = π. Since π is proper and quasi-finite, the morphism f is also proper and quasi-finite and therefore f is finite. Furthermore, by [6] we have π̃_{∗}𝒪_{ℳ_{1,1,S}} = 𝒪_{X}. It therefore suffices to show that the map 𝒪_{𝔸_{j}^{1}} → π_{∗}𝒪_{ℳ_{1,1,S}} is an isomorphism. It suffices to verify this locally in the flat topology on S, so we may further assume that S is the spectrum of a complete noetherian local ring A. In addition, since the morphism π is proper, the theorem on formal functions for stacks [9, §3] implies that it suffices to show the result over Spec(A/𝔪_{A}^{n}) for all n. This reduces the proof to the case when S is the spectrum of an artinian local ring A. Let k be the residue field of A, and let J ⊂ A be an ideal with J annihilated by the maximal ideal of A (so that J is a k-vector space). Set A_{0} := A/J. Pushing forward the exact sequence

(4.4.1)  0 → J ⊗ 𝒪_{ℳ_{1,1,k}} → 𝒪_{ℳ_{1,1,A}} → 𝒪_{ℳ_{1,1,A_{0}}} → 0

to 𝔸_{A}^{1} we obtain a commutative diagram

(4.4.2)  \begin{aligned}0 & ⟶ & (π_{∗}𝒪_{ℳ_{1,1,k}} ) ⊗ J & ⟶ & π_{∗}𝒪_{ℳ_{1,1,A}} & ⟶ & π_{∗}𝒪_{ℳ_{1,1,A_{0}}} \\ & & ^{a}↑ & & ^{b}↑ & & ↑^{c} \\ 0 & ⟶ & (𝒪_{𝔸_{k}^{1}}) ⊗ J & ⟶ & 𝒪_{𝔸_{A}^{1}} & ⟶ & 𝒪_{𝔸_{A_{0}}^{1}} & ⟶ & 0. \end{aligned}

By induction and the case when A is a field, we get that a and c are isomorphisms and therefore b is an isomorphism also. □

4.5. To complete the proof of 1.1 in the case when S is reduced, we make some general observations about the relationship between line bundles on a stack and line bundles on the coarse moduli space.

Let S be a noetherian scheme and 𝒳 → S a Deligne-Mumford stack over S. Let π : 𝒳 → X be the coarse moduli space, and assume that the formation of the coarse space X commutes with arbitrary base change on S and that X is reduced (we just saw that this holds for ℳ_{1,1} over a reduced scheme). For a field valued point x : Spec(k) → S let π_{x} : 𝒳_{x} → X_{x} denote the base change 𝒳 ×_{S} x → X ×_{S} x.

Proposition 4.6. Let L be a line bundle on 𝒳 such that for every field valued point x : Spec(k) → S the sheaf π_{x∗}(L|_{𝒳_{x}} ) is locally free of rank 1 and π_{x}^{∗}π_{x∗}(L|_{𝒳_{x}} ) → L|_{𝒳_{x}} is an isomorphism. If 𝒳 → X is flat, then the sheaf π_{∗}L is locally free of rank 1 on X and π^{∗}π_{∗}L → L is an isomorphism.

Proof. One immediately reduces to the case when X = Spec(R), Y = Spec(B) is a finite flat R scheme, Γ is a finite group acting on Y over X such that 𝒳 = [Y/Γ] (indeed étale locally on the coarse space every Deligne-Mumford stack can be presented in this way [8, 2.12]). Let M denote the B–module corresponding to L, so that M comes equipped with an action of Γ over the action on B. We can even assume that R is a local ring and that M is a free R–module (forgetting the B–module structure). We are then trying to compute the kernel of the map

M → ∏_{γ∈Γ} M, m ↦ (⋯, γ(m) − m, ⋯)_{γ∈Γ}.

We can also assume that S = Spec(Λ) is affine. □

Lemma 4.7. Let R be a reduced local Λ–algebra and let A ∈ M_{n×m}(R) be a matrix (which we view as a map R^{n} → R^{m}) with the property that for every x ∈ Spec(Λ) the matrix A(x) ∈ M_{n×m}(R ⊗_{Λ} k(x)) has kernel a free R ⊗_{Λ} k(x)–space of rank 1. Then Ker(A) is a free rank 1 module over R and for every x ∈ Spec(Λ) the natural map Ker(A)⊗_{Λ}k(x) → Ker(A(x)) is an isomorphism.

Proof. By induction on n. If n = 1, then the assertion is that A is a matrix with A(x) the zero matrix for all x ∈ Spec(Λ). Since R is reduced this implies that A is the zero matrix.

For the inductive step consider the system of m equations

∑_{i} a_{ij}X_{i} = 0

that we are trying to solve in R. If x ∈ Spec(Λ) is the image of the closed point of Spec(R), then A(x) is not zero since n ≥ 2. Since R is local some a_{ij} is invertible and so we can solve for the variable X_{i}. This gives a system of m − 1–equations in n − 1 variables which again has the property that for every point x ∈ Spec(Λ) the image in R ⊗ k(x) has a unique line of solutions. By induction we obtain the result. □

This completes the proof of (1.1 (i)).

5. Proof of (1.1 (ii))

Proposition 5.1. For any scheme S over ℤ[1/2] and any coherent 𝒪_{S}–module M, the sheaf R^{1}π_{∗}(𝒪_{ℳ_{1,1,S}} ⊗_{𝒪_{S}} M) is zero, where π : ℳ_{1,1,S} → 𝔸_{j,S}^{1} is the projection.

Proof. Using the theorem of formal functions one is reduced to the case when S is the spectrum of a field. Furthermore, if the characteristic is not 3 the result is immediate, so it suffices to consider S = Spec(k) with char(k) = 3, and M = k. We may further assume that k is algebraically closed.

The coherent sheaf R^{1}π_{∗}(𝒪_{ℳ_{1,1,k}} ) restrict to the zero sheaf on 𝔸_{k}^{1} −{0}, since over this open subset of 𝔸_{k}^{1} the stack ℳ_{1,1,k} is tame (the automorphism groups are {±1}). Let x̄ → ℳ_{1,1,k} be a geometric point mapping to 0 in 𝔸_{k}^{1}, and let A denote the completion of 𝒪_{ℳ_{1,1,k},x̄} along the maximal ideal. Let Γx̄ denote the stabilizer group scheme of x̄, so that Γx̄ acts on A. The ring of invariants B := A^{Γx̄} is equal to the completion of 𝔸_{k}^{1} at the origin. Let F denote the finite type B-module obtained by pulling back R^{1}π_{∗}(𝒪_{ℳ_{1,1,k}} ) to Spec(B). Then F is equal to the cohomology group H^{1}(Γx̄, A). We show that this group is zero. Since F is supported on the closed point of Spec(B), there exists an integer n such that j^{n}F = 0 (where j ∈ B is the uniformizer defined by the standard coordinate on 𝔸^{1}). To prove the proposition it therefore suffices to show that F is j-torsion free.

For this we use an explicit description of A and Γx̄ given by the Legendre family. Let

(5.1.1)  V = Spec(k[λ][1/λ(λ − 1)])

and let E_{V} → V be the elliptic curve

(5.1.2)  E_{V} : Y^{2}Z = X(X − Z)(X − λZ).

If μ denotes λ+1, then the j-invariant of E_{V} is equal to μ^{6}/(μ^{2}−1)^{2} (recall that char(k) = 3). The map V → ℳ_{1,1,k} defined by E_{V} is étale, so this defines an isomorphism A ≃ k[[μ]]. The group Γx̄ sits in an exact sequence

(5.1.3)  1 → {±1} → Γx̄ → S_{3} → 1,

and the action of Γx̄ on A ≃ k[[μ]] factors through the action of S_{3} on k[[μ]] given by the two automorphisms

(5.1.4)  α : μ ↦ −μ,

and

(5.1.5)  β : μ ↦ μ/(1 − μ) = μ(1 + μ + μ^{2} + ⋯).

Also note that the Leray spectral sequence

(5.1.6)  E_{2}^{pq} = H^{p}(S_{3}, H^{q}({±1}, A)) ⟹ H^{p+q}(Γx̄, A)

and the fact that H^{q}({±1}, A) = 0 for q > 0 (since 2 is invertible in k) implies that H^{1}(Γx̄, A) = H^{1}(S_{3}, A).

An element in H^{1}(S_{3}, A) can be represented by a set map ξ : S_{3} → k[[μ]] (written σ ↦ ξ_{σ}) such that for σ, τ ∈ S_{3} we have (recall the action is a right action)

(5.1.7)  ξ_{στ} = ξ_{σ}^{τ} + ξ_{τ}.

The class of ξ is trivial if there exists an element g ∈ k[[μ]] such that ξ_{σ} = g^{σ} − g for all σ ∈ S_{3}. Note that 5.1.7 implies that it suffices to check the equalities ξ_{σ} = g^{σ} − g for a set of generators σ ∈ S_{3}.

If ξ represents a class in H^{1}(S_{3}, A) annihilated by j, there exists an element g ∈ k[[μ]] such that

(5.1.8)  \frac{μ^{6}}{(μ^{2} − 1)^{2}} ξ_{σ} = g^{σ} − g

for all σ ∈ S_{3}. To prove that H^{1}(S_{3}, A) is j-torsion free, it therefore suffices to show that for such a ξ we can choose g to have μ-adic valuation ≥ 6 (since A is j-torsion free).

For this note that we can without loss of generality assume that g has no constant term, and then write

(5.1.9)  g = a_{1}μ + a_{2}μ^{2} + a_{3}μ^{3} + a_{4}μ^{4} + a_{5}μ^{5} + g_{≥6},

where g_{≥6} has μ-adic valuation ≥ 6. We have

(5.1.10)  \frac{μ^{6}}{(μ^{2} − 1)^{2}} ξ_{α} = 2a_{1}μ + 2a_{3}μ^{3} + 2a_{5}μ^{5} + (g_{≥6}^{α} − g_{≥6})

which implies that a_{1} = a_{3} = a_{5} = 0. Then

(5.1.11)  \frac{μ^{6}}{(μ^{2} − 1)^{2}} ξ_{β} = 2a_{2}μ^{3} + (higher order terms)

which gives a_{2} = 0. Finally using this we see that

(5.1.12)  \frac{μ^{6}}{(μ^{2} − 1)^{2}} ξ_{β} = a_{4}μ^{5} + (higher order terms)

which implies that a_{4} = 0 as desired. This completes the proof of 5.1. □

5.2. Now let us prove 1.1 for a connected ℤ[1/2]-scheme S. We need to show that if L is a line bundle on ℳ_{1,1,S} such that for any field valued point s ∈ S the fiber L_{s} on ℳ_{1,1,s} descends to 𝔸_{j,s}^{1} then L descends to 𝔸_{j,S}^{1}. By a standard limit argument it suffices to consider the case when S is noetherian and even affine, say S = Spec(Λ). Let J ⊂ Λ denote the nilradical. We would like to inductively show that if the result holds for over Λ/J^{r} then it also holds for Λ/J^{r+1}. In other words, let L_{0} denote a line bundle on 𝔸_{j,Λ/J^{r}}^{1} and L̃ a lifting of π^{∗}L_{0} to ℳ_{1,1,Λ/J^{r+1}}. Then we want to show that L̃ is pulled back from a lifting of L_{0} to 𝔸_{j,Λ/J^{r+1}}^{1}. By standard deformation theory this is equivalent to showing that the map

0 = H^{1}(𝔸_{Λ}^{1}, J^{r}/J^{r+1}) → H^{1}(ℳ_{1,1,Λ}, J^{r}/J^{r+1})

is an isomorphism. Equivalently that H^{1}(ℳ_{1,1,Λ}, J^{r}/J^{r+1}) is zero. Since 𝔸_{j,Λ}^{1} is affine, the group H^{1}(ℳ_{1,1,Λ}, J^{r}/J^{r+1}) is zero if and only if the sheaf R^{1}π_{∗}(J^{r}/J^{r+1} ⊗ 𝒪_{ℳ_{1,1,Λ}} ) is zero on 𝔸_{j,Λ}^{1} which follows from 5.1. This completes the proof of 1.1. □

6. Computations in characteristic 2

Proposition 6.1. Let k be a field of characteristic 2, and let π : \overline{ℳ} _{1,1,k} → ℙ_{k}^{1} be the morphism defined by the j-invariant. Then R^{1}π_{∗}𝒪_{\overline{ℳ} _{1,1,k}} is a line bundle on ℙ_{k}^{1} of negative degree.

Proof. We may without loss of generality assume that k is algebraically closed.

Let 𝒰_{∞} ⊂ \overline{ℳ} _{1,1,k} denote the open substack \overline{ℳ} _{1,1,k} ×_{ℙ_{j}^{1}} 𝔸_{1/j}^{1} (the complement of j = 0), and let 𝒰_{0} = ℳ_{1,1,k} ⊂ \overline{ℳ} _{1,1,k} denote the complement of j = ∞. Let U_{∞}, U_{0} ⊂ ℙ_{j}^{1} be the coarse moduli spaces (the standard open cover of ℙ_{j}^{1}).

The stack 𝒰_{∞} is a ℤ/(2)–gerbe over U_{∞}. Now in general, if f : 𝒢 → X is a ℤ/(2)–gerbe in characteristic 2, the sheaf R^{1}f_{∗}𝒪_{𝒢} is locally free of rank 1 and in fact canonically trivialized. This can be seen as follows. Etale locally on X, we have 𝒢 = X × B(ℤ/(2)). Computing in this local situation, one sees that R^{1}f_{∗}(ℤ/(2)) is a locally constant sheaf of groups étale locally isomorphic to ℤ/(2), and the natural map R^{1}f_{∗}(ℤ/(2)) ⊗_{ℤ/(2)} 𝒪_{X} → R^{1}f_{∗}𝒪_{𝒢} (which exists since we are in characteristic 2) is an isomorphism. Since a group of order 2 admits no nontrivial automorphisms there is a canonical isomorphism ℤ/(2) ≃ R^{1}f_{∗}(ℤ/(2)) which induces a canonical trivialization of R^{1}f_{∗}𝒪_{𝒢}. In the case of 𝒢 = X × B(ℤ/(2)) and X = Spec(A) we have

H^{1}(𝒢 , 𝒪_{𝒢}) ≃ Hom_{Gp}(ℤ/(2), A)

and the trivialization is given by the homomorphism sending 1 ∈ ℤ/(2) to 1 ∈ A.

Lemma 6.2. The sheaf R^{1}π_{∗}𝒪_{\overline{ℳ} _{1,1,k}} is locally free of rank 1 on ℙ_{j}^{1}.

Proof. By finiteness of coherent cohomology for stacks the sheaf is in any case coherent. Since ℙ_{j}^{1} is a smooth curve it therefore suffices to show that it is torsion free. Furthermore, the only issue is at the point j = 0. Since the formation of cohomology commutes with flat base change, it suffices to show that

(6.2.1)  H^{1}(\overline{ℳ} _{1,1,k} ×_{ℙ_{j}^{1}} Spec(k[[j]]), 𝒪_{\overline{ℳ} _{1,1,k}×_{ℙ^{1}_{j}}Spec(k[[j]])})

is j–torsion free.

For this we use the so-called Hesse presentation of ℳ_{1,1,k}. Let

(6.2.2)  V = Spec(k[μ, ω][1/(μ^{3} − 1)]/(ω^{2} + ω + 1)),

and let E_{V} → V be the elliptic curve given by the equation

(6.2.3)  X^{3} + Y^{3} + Z^{3} = μXY Z.

This is elliptic curve has a basis for its three-torsion group given by the points [1 : 0 : −1] and [−1 : ω : 0]. In fact, this is the universal elliptic curve with full level three structure. The j-invariant of E_{V} is μ^{12}/(μ^{3} − 1)^{3} (since we are in characteristic 2). In particular, the fiber over j = 0 is the curve X^{3} + Y^{3} + Z^{3} = 0.

Changing the choice of basis for the 3-torsion subgroup defined an action of GL_{2}(𝔽_{3}) on V such that ℳ_{1,1,k} ≃ [V/GL_{2}(𝔽_{3})]. A calculation shows that this action is described as follows:

(1) (μ, ω) ∗ (\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}) = (ωμ, ω).

(2) (μ, ω) ∗ (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) = (μ/(μ − 1), ω).

(3) (μ, ω) ∗ (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) = (μ, ω^{2}).

Putting this together one finds that

(6.2.4)  \overline{ℳ} _{1,1,k} ×_{ℙ_{j}^{1}} Spec(k[[j]]) ≃ [Spec(k[[μ]])/SL_{2}(𝔽_{3})],

where α = (\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}) acts by μ ↦ ζμ (for some fixed primitive cube root of unity ζ) and

β = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) acts by μ ↦ μ/(μ − 1).

As in the proof of 5.1, an element of 6.2.1 is given by a set map ξ : SL_{2}(𝔽_{3}) → k[[μ]] (written σ ↦ ξ_{σ}) such that for any two elements σ, τ ∈ SL_{2}(𝔽_{3}) we have

ξ_{στ} = ξ_{σ}^{τ} + ξ_{τ},

and the class of ξ is trivial if there exists an element g ∈ k[[μ]] such that for every σ we have ξ_{σ} = g^{σ} − g.

Now if 6.2.1 has j–torsion there exists a set map ξ as above and an element g ∈ k[[μ]] such that for all σ we have

\frac{μ^{12}}{μ^{3} − 1}ξ_{σ} = g^{σ} − g.

To prove that 6.2.1 is torsion free it suffices to show that we can choose g to be divisible by μ^{12}. For since k[[μ]] is an integral domain we then have

ξ_{σ} = (\frac{μ^{3} − 1}{μ^{12}}g)^{σ} − (\frac{μ^{3} − 1}{μ^{12}}g).

We can without loss of generality assume that g has no constant term. Write

g = a_{1}μ + a_{2}μ^{2} + ⋯ + a_{11}μ^{11} + g_{≥12}.

Then g^{α} −g has μ-adic valuation ≥ 12 (recall that α = (\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix})). Looking at the coefficients

a_{i} this implies that all but a_{3}, a_{6}, and a_{9} are zero, so

g = a_{3}μ^{3} + a_{6}μ^{6} + a_{9}μ^{9} + g_{≥12}.

Similarly g^{β} − g has μ–adic valuation ≥ 12. Looking at the coefficient of μ^{4} in g^{β} − g one sees that a_{3} = 0. Then looking at the coefficent of μ^{8} one sees that a_{6} = 0, and finally looking at the coefficient of μ^{10} one sees that a_{9} = 0. □

Let M denote the cohomology group 6.2.1 (a k[[j]]–module) and let M_{η} denote M ⊗_{k[[j]]} k[[j]][1/j]. Let e_{∞} ∈ M_{η} denote the basis element defined by the canonical trivialization of R^{1}π_{∗}𝒪_{\overline{ℳ} _{1,1,k}} over 𝒰_{∞}. The lattice M ⊂ M_{η} defines a valuation ν on M_{η} and it suffices to show that ν(e_{∞}) < 0. Equivalently we have to show that for any element m ∈ M if we write m = he_{∞} in M_{η} then the j–adic valuation of h is positive.

For this we again use the presentation 6.2.4. An element m ∈ M is then represented by a map ξ : SL_{2}(𝔽_{3}) → k[[μ]]. The corresponding element in M_{η} can in terms of the basis e_{∞} be described as follows. First of all the element ξ_{β^{2}} ∈ k[[μ]] is then SL_{2}(𝔽_{3})–invariant since for any other element σ we have

ξ_{β^{2}}^{σ} + ξ_{σ} = ξ_{β^{2}σ} = ξ_{σβ^{2}} = ξ_{σ}^{β^{2}} + ξ_{β^{2}}

and β^{2} acts trivially on k[[μ]]. Therefore ξ_{β^{2}} is actually an element in k[[j]]. The image of ξ in M_{η} ≃ Hom(ℤ/(2), k[[j]][1/j]) is then equal to the homomorphism

ℤ/(2) → k[[j]][1/j], 1 ↦ ξ_{β^{2}}.

The class e_{∞} corresponds to the homomorphism sending 1 to 1 so we have to show that the j–adic valuation of ξ_{β^{2}} is positive. For this let f = ξ_{β}. Then

ξ_{β^{2}} = f^{β} + f = f(μ(1 + μ + μ^{2} + ⋯)) + f(μ).

Since we are in characteristic 2 it follows that the μ–adic valuation of ξ_{β^{2}} is at least 2, and therefore the j–adic valuation of ξ_{β^{2}} is also positive. □

Corollary 6.3. For any field k, we have H^{1}(\overline{ℳ} _{1,1,k}, 𝒪_{\overline{ℳ} _{1,1,k}}) = 0.

Proof. We have R^{1}π_{∗}𝒪_{\overline{ℳ} _{1,1,k}} = 0 when char(k) ≠ 2 (when char(k) = 3 this follows from section 5). It follows that

(6.3.1)  H^{0}(ℙ_{k}^{1}, R^{1}π_{∗}𝒪_{\overline{ℳ} _{1,1,k}}) = 0

in all characteristics. From the Leray spectral sequence we obtain

(6.3.2)  H^{1}(\overline{ℳ} _{1,1,k}, 𝒪_{\overline{ℳ} _{1,1,k}}) = H^{1}(ℙ_{k}^{1}, 𝒪_{ℙ_{k}^{1}}) = 0.
□

Remark 6.4. Note that if char(k) = 2, then the restriction of R^{1}π_{∗}𝒪_{ℳ_{1,1,k}} to 𝔸_{k}^{1} ⊂ ℙ_{k}^{1} is non-zero. From the Leray spectral sequence it follows that the map

(6.4.1)  0 = H^{1}(𝔸_{k}^{1}, 𝒪_{𝔸_{k}^{1}}) → H^{1}(ℳ_{1,1,k}, 𝒪_{ℳ_{1,1,k}} )

is not an isomorphism. Since the group H^{1}(ℳ_{1,1,k}, 𝒪_{ℳ_{1,1,k}} ) classifies deformations of the structure sheaf to ℳ_{1,1,k[ϵ]/(ϵ^{2})} this implies that there are line bundles on ℳ_{1,1,k[ϵ]/(ϵ^{2})} which are in the kernel of 4.3.1 but are nontrivial. This implies that 1.1 fails for S = Spec(k[ϵ]/(ϵ^{2})). More generally, 1.1 fails for any nonreduced affine scheme over 𝔽_{2}.

7. Proof of 1.3

7.1. In order to prove 1.3 it is easiest to prove a stronger statement that implies it. Let 𝒫ic(\overline{ℳ} _{1,1,S}) denote the Picard stack over S which to any S–scheme T associates the group if line bundles on \overline{ℳ} _{1,1,T}. By [1, 5.1], the stack 𝒫ic(\overline{ℳ} _{1,1,S}) is an algebraic stack (an Artin stack) over S. There is a morphism of stacks

(7.1.1)  ℤ × B𝔾_{m,S} → 𝒫ic(\overline{ℳ} _{1,1,S})

sending a pair (n, L) consisting of n ∈ ℤ and L a line bundle on S to λ^{n} ⊗_{𝒪_{S}} L on \overline{ℳ} _{1,1,S}. The following theorem implies 1.3 by passing evaluating both sides of 7.1.1 on S and passing to isomorphism classes.

Theorem 7.2. The morphism 7.1.1 is an isomorphism.

Proof. Note first that if n and n′ are integers and L and L′ are line bundles on S, then λ^{n} ⊗ L and λ^{n′} ⊗ L′ on \overline{ℳ} _{1,1,S} are isomorphic if and only if n = n′ and L ≃ L′. Indeed, if these two sheaves are isomorphic, then this implies that λ^{n−n′} descends to ℙ_{j}^{1}. By the case of a field this implies that n = n′. In this case we recover L and L′ from their pullbacks to \overline{ℳ} _{1,1,S} by pushing back down to S. Therefore, the functor 7.1.1 is fully faithful. It therefore suffices to show that for any cartesian diagram

(7.2.1)  \begin{matrix}P & ⟶ & S \\ ↓ & & ↓^{L} \\ ℤ × B𝔾_{m} & ⟶ & 𝒫ic_{\overline{ℳ} _{1,1,S}} \end{matrix}

the morphism of algebraic spaces P → S is an isomorphism. For this it suffices to consider the case when S is artinian local. Furthermore, we know the result in the case when S is the spectrum of a field by section 3. Since a line bundle on the spectrum of an artinian local ring is trivial, what we therefore need to show is that if S is an artinian local ring then any line bundle on \overline{ℳ} _{1,1,S} is isomorphic to λ^{n} for some n. Proceeding by induction on the length of S, it further suffices to consider the following. Let S = Spec(A), k the residue field of A, and let J ⊂ A be a square–zero ideal annihilated by the maximal ideal of A, and set A_{0} = A/J. Then any deformation of λ^{n} over \overline{ℳ} _{1,1,A_{0}} to \overline{ℳ} _{1,1,A} is isomorphic to λ^{n}. Using the exponential sequence

0 → J ⊗ 𝒪_{\overline{ℳ} _{1,1,k}} → 𝒪_{\overline{ℳ} _{1,1,A}}^{∗} → 𝒪_{\overline{ℳ} _{1,1,A_{0}}}^{∗} → 0

one sees that this amounts exactly to H^{1}(\overline{ℳ} _{1,1,k}, 𝒪_{\overline{ℳ} _{1,1,k}}) = 0. □

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