On Néron models, divisors and modular curves.

Bas Edixhoven^{∗}

March 26, 1999

1 Introduction.

For p a prime number, let X_{0}(p)_{ℚ} be the modular curve, over ℚ, parametrizing isogenies of degree p between elliptic curves, and let J_{0}(p)_{ℚ} be its jacobian variety. Let f: X_{0}(p)_{ℚ} → J_{0}(p)_{ℚ} be the morphism of varieties over ℚ that sends a point P to the class of the divisor P − ∞, where ∞ is the ℚ-valued point of X_{0}(p)_{ℚ} that corresponds to ℙ_{ℚ}^{1} with 0 and ∞ identified, equipped with the subgroup scheme μ_{p,ℚ}. We suppose that X_{0}(p)_{ℚ} has genus at least one, i.e., that p = 11 or p > 13. Then f is a closed immersion. Let 0 denote the other cusp of X_{0}(p)_{ℚ}. The ℚ-valued point f(0) of J_{0}(p)_{ℚ} is well known to be of order n, the numerator of (p − 1)/12, expressed in lowest terms. Let now X_{0}(p) denote the model over ℤ of X_{0}(p)_{ℚ} as described by [8] and [15]: it is the compactified coarse moduli space for generalized elliptic curves with finite locally free subgroup schemes of rank p that meet all irreducible components of all geometric fibres. Let X_{0}(p)^{∼} denote the minimal regular model of X_{0}(p), and let J_{0}(p) be the Néron model over ℤ of J_{0}(p)_{ℚ}. See [8] for a description of the semi-stable curve X_{0}(p)^{∼}. By the defining properties of J_{0}(p), the morphism f extends uniquely to a morphism f: X_{0}(p)^{∼,0} → J_{0}(p), where X_{0}(p)^{∼,0} is the open part of X_{0}(p)^{∼} where the morphism to Spec(ℤ) is smooth; X_{0}(p)^{∼,0} is the complement of the set of double points in the fibre X_{0}(p)_{p}^{∼} over 𝔽_{p}. Robert Coleman asked about how the image of X_{0}(p)_{p}^{∼,0} under f intersects C, the closure in J_{0}(p) of the group generated by f(0). We will prove, see Theorem 8.2, for all p for which X_{0}(p)_{ℚ} has genus at least two, that the intersection consists just of the two obvious elements f(0)_{p} and f(∞)_{p}. (Of course, when X_{0}(p)_{ℚ} has genus one, f is an isomorphism, hence the intersection is all of C_{p}.) This result is used by Coleman, Kaskel and Ribet in [4] to study the inverse image under f of the torsion subgroup of J_{0}(p)(ℂ).

To prove our result, we use Raynaud’s description (see [25] or [2, 9.5]) of the Néron model of a jacobian variety of a curve in terms of a regular model of the curve. (A good reference for jacobians of modular curves is [26].) It turns out that in order to describe the special fibre of such a Néron model, and not just the connected component of the identity element and the group of connected components, one needs to know a bit more than just the special fibre of the regular model of the curve. This extra information is related, in the semi-stable case, to the log structure induced on the special fibre of the curve (see Section 6), and also to the map from the space of global deformations to that of local deformations of a stable curve (see [7, p. 81]). For the curve X_{0}(p)^{∼} we describe this extra information in terms of a well known modular form of weight p + 1 on the supersingular elliptic curves. We do not use rigid uniformization, and our methods can be used as well for curves whose special fibre is not totally degenerate. A rigid analytic description of J_{0}(p)_{p} is given by de Shalit in [29].

∗partially supported by the Institut Universitaire de France, and by the European TMR Network Contract ERB FMRX 960006 “arithmetic algebraic geometry”.

In Sections 2 and 3 we consider a proper nodal curve X with smooth generic fibre X_{K} over a discrete valuation ring, and study the open part J of the Néron model of the jacobian variety given by the relative Picard functor of X. We give a description of the special fibre J_{k} of J in terms of divisors on X_{k} and some invariants c(x) at the double points of X_{k}. Some properties of these c(x) are discussed in Sections 4, 5 and 6. The c(x) for the modular curve X_{0}(p) are determined in Section 7. Coleman’s question mentioned above is answered in Section 8. Section 9 answers the question as to what extent the morphism from the smooth locus X^{0} of X to the Néron model of J_{K} is a closed immersion. For example, if X is regular, we show that this morphism is a closed immersion if and only if all double points of Xk̄ are non-disconnecting (see Corollary 9.8). Some of the results in this last section can also be found in Coleman’s preprint [3].

2 Raynaud’s description in the nodal case.

Let D be a complete discrete valuation ring, let π be a uniformizer, K the fraction field of D and k the residue field. We suppose that k is separably closed. Let X_{K} be a smooth, proper and geometrically irreducible curve over K, and let X be a proper flat model of it over D. We suppose that X is nodal, i.e., the only singularities of the geometric special fibre are ordinary double points. The irreducible components of X_{k} are absolutely irreducible. The singular points of X_{k} are k-rational points. At such a point x the complete local ring O_{X,x}^{∧} is isomorphic to one of the form D[[u, v]]/(uv − π^{e(x)}), for a unique integer e(x) ≥ 1. Let R: X^{∼} → X be the minimal resolution of singularities of X; for x in X_{k} a singular point, R^{−1}x is a chain of e(x) − 1 projective lines.

Let N: X_{k}^{nor} → X_{k} be the normalization morphism. We define S_{0} to be the set of irreducible components of X_{k}^{nor}. Note that N induces a bijection between S_{0} and the set of irreducible components of X_{k}. Let S_{1} be the set of singular points of X_{k}, and let S_{1}^{nor} be N^{−1}S_{1}. We will view S_{0} and S_{1} as the sets of vertices and edges of a graph; the end points of x in S_{1} are the elements of S_{0} containing at least one of the two elements of N^{−1}{x}.

According to [25, Thm. 8.2.1] or [2, Thm. 8.2/2], the relative Picard functor Pic_{X/D} is representable, and is smooth over D (but not of finite type, and not separated if X_{k} has more than one irreducible component). As usual, let Pic_{X/D}^{0} be the connected component of the identity element in Pic_{X/D}; its D-valued points correspond to line bundles on X whose restrictions to all irreducible components of X_{k} are of degree zero. As in [18], let Pic_{X/D}^{[0]} be the closure in Pic_{X/D} of Pic_{X_{K}/K}^{0}; Pic_{X/D}^{[0]}(D) is the set of isomorphism classes of line bundles on X of which the sum of the degrees of the restrictions to the irreducible components is zero. Let E be the closure in Pic_{X/D} of the identity element of Pic_{X_{K}/K}. Then E(D) is the set of isomorphism classes of line bundles on X whose restriction to X_{K} is trivial. Such line bundles are given by Cartier divisors with support in X_{k}. The group of Weil divisors with support in X_{k} is the group ℤ^{S_{0}}. Let F be the subgroup of ℤ^{S_{0}} consisting of those Weil divisors that define Cartier divisors. Of course, if X is regular, then F equals ℤ^{S_{0}}. Let us consider a Weil divisor a := ∑ a(C)C in ℤ^{S_{0}}. Then a defines a Cartier divisor if and only if a is principal at all x in S_{1}. Let x be in S_{1}, and let C and C′ in S_{0} be the endpoints of x. Then a is principal at x if and only if a(C) and a(C′) are equal modulo e(x) (see [26, p. 15]). It follows that we have an exact sequence:

(2.1)  0 → F → ℤ^{S_{0}} → ∏_{x∈S_{1}}(ℤ/e(x)ℤ)^{N^{−1}{x}}/(ℤ/e(x)ℤ),

where the ℤ/e(x)ℤ are diagonally embedded. By definition, we have a map from F to Pic_{X/D}^{[0]}(D), whose image is E(D). This map sends a in F to the invertible O_{X}-module O_{X}(a), whose sections on open U ⊂ X are rational functions f on U with div(f) + a ≥ 0.

By [25], or the proof of [2, Prop. 9.5/3], E is the espace étalé of the sheaf i_{∗}E(D), with i: Spec(k) → Spec(D) the closed immersion given by the canonical projection D → k. In particular, E(k) = E(D), and E(K) = 0. It follows that J := Pic_{X/D}^{[0]}/E is representable and separated, and that J represents the open part of the Néron model over D of Pic_{X_{K}/K}^{0} that corresponds to the line bundles on X_{K} that admit an extension to a line bundle on X. Note that if X is regular, then J is the Néron model itself.

Let Φ := J_{k}/J_{k}^{0} be the group of connected components of J_{k}. By [2, Thm. 9.7/1], J_{k}^{0} = Pic_{X_{k}/k}^{0}. We have the following complex of ℤ-modules:

(2.2)  0 → ℤ → F → ℤ^{S_{0}} → ℤ → 0.

The element 1 in ℤ is mapped to X_{k}, the principal divisor defined by π. The morphism from F to ℤ^{S_{0}} is the composite of F → Pic(X) and the morphism “multidegree” from Pic(X) to ℤ^{S_{0}} that sends a line bundle to its degrees on the irreducible components of X_{k} (note that this map is not the same as the one in (2.1)). Explicitly, for a in F and C in S_{0}, one has:

(2.3)  deg_{C}(O_{X}(a)) = ∑_{x∈C∩S_{1}^{nor}} \frac{a(C′_{x}) − a(C)}{e(x)},

where C′_{x} denotes the other branch passing through x (see (3.5)). The morphism ℤ^{S_{0}} → ℤ is simply the sum. The fact that the ℚ-valued intersection pairing on ℤ^{S_{0}} is negative definite on ℤ^{S_{0}}/ℤX_{k} shows that (2.2) is exact at F. The definition of J, and the fact that E(D) is the image of F, show that the homology of (2.2) at ℤ^{S_{0}} is Φ. See [2, Thm. 9.6/1] for this result in the case where X is regular.

3 A description of J_{k} in terms of divisors.

We keep the notation and hypotheses of the preceding section. The results in that section give us a description of J in terms of Pic_{X/D}. Further on, we will have to do computations in the special fibre J_{k} of J, without knowing too much about the D-scheme X, but in terms of the k-scheme X_{k} with some additional data.

In Section 2 we have seen that J_{k} has the following presentation:

(3.1)  F → Pic_{X_{k}/k}^{[0]} → J_{k} → 0,

with Pic_{X_{k}/k}^{[0]} the open part of Pic_{X_{k}/k} corresponding to line bundles on X_{k} with total degree zero, and with F the subgroup of ℤ^{S_{0}} consisting of the Weil divisors that are locally principal (we view F as a constant group scheme over k). This subgroup F is described in terms of S_{0} and the e(x) in (2.1). The only ingredient in (3.1) that does not only involve X_{k} and the e(x) is the map from F to Pic(X_{k}). The aim of this section is to describe Pic_{X_{k}/k} and the map from F to it, in terms of X_{k}, the e(x), and some extra data related to the reduction of X modulo some power of π (recall that π is a uniformizer of D).

Let us first describe Pic_{X_{k}/k}. The proof of Example 8 of Chapter 9 of [2] shows that we have an exact sequence:

(3.2) 0 → 𝔾_{m,k} → ⨁_{C∈S_{0}} 𝔾_{m,k} → ⨁_{x∈S_{1}} (𝔾_{m,k}^{N^{−1}{x}}/𝔾_{m,k}) → Pic_{X_{k}/k} → Pic_{X_{k}^{nor}/k} → 0.

Let us assume that S_{1} is not empty. Then (3.2) gives the following description of Pic_{X_{k}/k}(S) for all k-schemes S.

3.3 Proposition. Under the assumptions above, Pic_{X_{k}/k}(S) is the set of isomorphism classes of line bundles on X_{S} that are obtained as follows. Choose a line bundle L on X_{S}^{nor}, i.e., for each C in S_{0}, a line bundle L_{C} on C_{S}, with a given isomorphism ϕ_{x}: O_{S} → x^{∗}L for each x in S_{1}^{nor}. Then glue, at each x in S_{1}, the line bundles L_{C_{1}} and L_{C_{2}} via ϕ_{x_{1}} and ϕ_{x_{2}}, where {x_{1}, x_{2}} is N^{−1}{x} and where C_{1}, C_{2} are the elements of S_{0} with x_{1} ∈ C_{1} and x_{2} ∈ C_{2}. Let (L, ϕ) denote the line bundle on X_{S} obtained in this way. Two such line bundles (L, ϕ) and (L′, ϕ′) are isomorphic if and only if there exists an isomorphism α: L → L′ that is compatible with the glueing data in the sense that for all x in S_{1} we have

α_{x_{2}}^{∘}ϕ_{x_{2}}^{∘}ϕ_{x_{1}}^{−1} = ϕ′_{x_{2}}^{∘}ϕ′_{x_{1}}^{−1}^{∘}α_{x_{1}}.

3.4 Remark. Another way to state the previous proposition is to say that N: X_{k}^{nor} → X_{k} is a morphism of universal effective descent for line bundles (see [11, Déf. 1.7]). □

Our next aim is to describe the line bundles of the form O_{X}(a)|_{X_{k}}, with a in F, in the form of Proposition 3.3. Note that the divisor of π is X_{k} = ∑_{C} C, so that multiplication by π induces an isomorphism from O_{X} to O_{X}(−X_{k}). Let a be in F. For C in S_{0} we have:

(3.5) O_{X}(a)|_{C} ˜→ O_{X} ( ∑_{C′≠C}(a(C′) − a(C))C′)|_{C} = O_{C} ( ∑_{x∈C∩S_{1}^{nor}} \frac{a(C′_{x}) − a(C)}{e(x)}x) ,

where the first isomorphism is given by multiplication by π^{a(C)}, and where C′_{x} is the other branch passing through x. Now we have to see how these line bundles are glued at the singular points. So let x be in S_{1}, N^{−1}{x} = {x_{1}, x_{2}}, and let C_{1} and C_{2} be the elements of S_{0} containing x_{1} and x_{2}, respectively. Let a_{1} := a(C_{1}) and a_{2} := a(C_{2}). Then we have:

(3.6)  x_{1}^{∗}O_{C_{1}} (\frac{a_{2} − a_{1}}{e(x)} x_{1}) = Ω_{C_{1}}(x_{1})^{⊗ \frac{a_{1}−a_{2}}{e(x)}} ,

where Ω_{C_{1}} is the sheaf of relative 1-forms of C_{1} over k. Of course, we have a similar formula at x_{2}. It follows that multiplication by π^{a_{1}−a_{2}} defines an isomorphism:

(3.7)  Ω_{C_{2}}(x_{2})^{⊗ \frac{a_{2}−a_{1}}{e(x)}} → Ω_{C_{1}}(x_{1})^{⊗ \frac{a_{1}−a_{2}}{e(x)}} ,

or, equivalently, a non-zero element of (Ω_{C_{1}}(x_{1}) ⊗ Ω_{C_{2}}(x_{2}))^{⊗(a_{2}−a_{1})/e(x)}. A simple computation shows that if we choose an isomorphism D[[u, v]]/(uv − π^{e(x)}) → O_{X,x}^{∧}, such that u is zero on C_{1} and v is zero on C_{2}, then this non-zero element is simply the (a_{2} −a_{1})/e(x)th power of

(3.8)  c(x) := dv ⊗ du in Ω_{C_{1}}(x_{1}) ⊗ Ω_{C_{2}}(x_{2})

(we will see in the next section that c(x) does not depend on the choice of the isomorphism). This finishes our description of the map from F to Pic(X_{k}), and hence of the description of J_{k}. For the sake of notation, let us say that for a in F, we call L_{a} the line bundle on X_{k} that we just constructed (so L_{a} is isomorphic to O_{X}(a)|_{X_{k}}, but its description in terms of Proposition 3.3 depends on π). Some properties of the c(x) will be discussed in the next section. To finish this section, we describe J_{k} in terms of degree zero divisors on X_{k}.

Let Div(X_{k}) be the ℤ-module of Weil divisors on X_{k} with support outside the singular locus S_{1}. Since such divisors are locally principal, we have a morphism Div(X_{k}) → Pic(X_{k}), sending E in Div(X_{k}) to isomorphism class of the line bundle O_{X_{k}}(E). This morphism is easily seen to be surjective. Let Div^{[0]}(X_{k}) be the subgroup of Div(X_{k}) consisting of the divisors of degree zero. Recall that J_{k} is the quotient of Pic_{X_{k}/k}^{[0]} by the image of F. We get a morphism Div^{[0]}(X_{k}) → J_{k}(k), sending E to P_{E}, say. It remains to describe the kernel of E ↦ P_{E}.

So let E be in Div^{[0]}(X_{k}). Of course, P_{E} is zero if and only if there exists an a in F such that L_{a} is isomorphic to O_{X_{k}}(E). A necessary condition for P_{E} to be zero is that the image of P_{E} in Φ be zero. This last condition is equivalent to the multidegree of E in ℤ^{S_{0}} being in the image of F (see (2.2)); let us suppose now that that is the case.

Let a = ∑_{C} a(C)C be in F such that L_{a} and E have the same multidegree; a is unique up to adding multiples of X_{k}. Then P_{E} is zero if and only if L_{a} ⊗ O_{X_{k}}(−E) is trivial, or, equivalently, if L_{a} ⊗ O_{X_{k}}(−E) has a non-zero global section (note that it has degree zero on each C). Using the description in Proposition 3.3 of line bundles on X_{k}, the last condition amounts to saying that the L_{a,C} ⊗ O_{C}(−E|_{C}) have non-zero global sections f_{C} that are compatible at all x in S_{1}. By construction, see (3.5), each such f_{C} has to be a non-zero rational function on C, whose divisor is:

(3.9)  E|_{C} − ∑_{x∈C∩S_{1}^{nor}} \frac{a(C′_{x}) − a(C)}{e(x)}x.

Suppose now that there exist f_{C} with these divisors, or, equivalently, suppose that the L_{a,C} ⊗O_{C}(−E|_{C}) are trivial. Let f_{C} be such functions. In order to make the compatibility conditions for the f_{C} explicit, we introduce some terminology. For f a non-zero rational function on C in S_{0}, and x in C(k), we define the leading term f(x) of f at x to be the non-zero element of Ω_{C}(x)^{⊗n} = m_{x}^{n}/m_{x}^{n+1} given by f, where n is the order of f at x, and m_{x} the maximal ideal in O_{C,x}. For x in S_{1}, C_{1} and C_{2} the elements of S_{0} containing the preimages x_{1} and x_{2} of x under N: X_{k}^{nor} → X_{k}, the two rational functions f_{C_{1}} and f_{C_{2}} are compatible at x if and only if:

(3.10)  f_{C_{1}}(x_{1}) f_{C_{2}}(x_{2})^{−1}c(x)^{(a(C_{2})−a(C_{1}))/e(x)} = 1,

as can be seen from the definitions earlier in this section. The f_{C} are unique up to multiplication by elements from k^{∗}. The problem of whether or not one can scale the f_{C} such that they satisfy the compatibility conditions at all x in S_{1} is a problem concerning the structure of the graph (S_{0}, S_{1}) of X_{k}. The following proposition gives the final conclusion of the preceding discussion.

3.11 Proposition. Let E be in Div^{[0]}(X_{k}). Then P_{E} in J_{k}(k) is zero if and only if the following conditions are satisfied. First of all, the multidegree of E in ℤ^{S_{0}} has to be in the image of F; suppose this is so and let a be in F having the same image in ℤ^{S_{0}} as E. Secondly, for each C in S_{0}, the divisor in (3.9) has to be a principal divisor; suppose that this is so, and let, for each C, f_{C} be a rational function on C with that divisor. Thirdly, for every cycle (C_{0}, x_{0}, C_{1}, x_{1}, …, x_{m−1}, C_{m} = C_{0}) in the graph (S_{0}, S_{1}) we must have:

(3.12)  ∏_{i=0}^{m−1} f_{C_{i}}(x′_{i}) f_{C_{i+1}}(x′′_{i})^{−1}c(x_{i})^{(a(C_{i+1})−a(C_{i}))/e(x_{i})} = 1,

with x′_{i} and x′′_{i} in N^{−1}{x} defined by x′_{i} ∈ C_{i} and x′′_{i} ∈ C_{i+1}.

3.13 Remark. Of course, in the third condition in Proposition 3.11, it is sufficient to consider a set of cycles that generates the first homology group of the graph. □

4 Some properties of the c(x).

Let us briefly recall the definition, given in (3.8), of the c(x). In the situation of (3.8), D is a complete discrete valuation ring with separably closed residue field k and fraction field K, π in D is a uniformizer and X/D is a proper nodal curve over D with X_{K} smooth and geometrically irreducible. Suppose that x is a singular point of X_{k}. Then the Dalgebra O_{X,x}^{∧} is of the type D[[u, v]]/(uv − π^{e(x)}) for some unique integer e(x) > 0. Let x_{1} and x_{2} be the two points of X_{k}^{nor} lying over x, and let C_{1} and C_{2} be the irreducible components of X_{k}^{nor} that they lie on. Let u and v be elements in O_{X,x}^{∧} that vanish on C_{1} and C_{2}, respectively, that induce parameters, still denoted u and v, of O_{C_{2},x_{2}}^{∧} and O_{C_{1},x_{1}}^{∧}, respectively, and such that uv = π^{e(x)}. Then we claim that the non-zero element dv ⊗ du in Ω_{C_{1}}(x_{1}) ⊗ Ω_{C_{2}}(x_{2}) does not depend on the choice of u and v. To see this, suppose that u′, v′ also satisfy the conditions. Then u′ = λu and v′ = λ^{−1}v with λ in O_{X,x}^{∧} a unit. Let \overline{λ} be the image in k^{∗} of λ. Then du′ = \overline{λ}du, and dv′ = \overline{λ}^{−1}dv, proving our claim.

Hence the elements c(x) of (3.8) are independent of the choice of the isomorphism. Let us now see how they depend on the choice of π. A simple computation shows that if π′ = λπ, then c′(x) = \overline{λ}^{e(x)}c(x), where c′(x) is defined using π′. It follows that the construction of c(x) defines an isomorphism of k-vector spaces between Ω_{C_{1}}(x_{1})⊗Ω_{C_{2}}(x_{2}) and (m_{D}/m_{D}^{2})^{⊗e(x)}, where m_{D} is the maximal ideal of D.

The construction of c(x) makes it clear that c(x) depends only on the reduction of X mod π^{e(x)+1}. It follows that J_{k}, as defined in Section 2, depends only on X mod π^{e+1}, where e is the maximum of all the e(x). In particular, if X is regular, the c(x) and J_{k} depend only on X mod π^{2}.

The construction of the c(x) also makes sense in more general situations (which we won’t use in this article). For example, consider a semi-stable curve f: C → S in the sense of [6, 2.21], i.e., S is an arbitrary scheme and f is of finite presentation, flat, proper and all its geometric fibres are connected curves whose singularities are ordinary double points. Let Z := Sing(f) be the closed subscheme of C defined by the first Fitting ideal of Ω_{C/S}, and let g: Z → S be the inclusion, followed by f. As explained in [6, 2.21–2.23], g is of finite presentation, finite and unramified, and the kernel I of g^{−1}O_{S} → O_{Z} is locally principal. We assume that I/I^{2} is a faithful g^{−1}O_{S}/I-module (this is the case, for example, if S is integral and f generically smooth). Then O_{Z} ⊗_{g^{−1}O_{S}} I is an invertible O_{Z}-module. Let P be the tautological section of f_{Z}: C_{Z} → Z, and let b: C_{Z}^{∼} → C_{Z} be the blow up of C_{Z} in the ideal sheaf defined by P. Let Z′ be the fibered product of P and b, and denote P′ the morphism from Z′ to C_{Z}^{∼}. Then Z′ → Z is the etale ℤ/2ℤ-torsor given by the two branches of C_{Z} at P, and C_{Z}^{∼} → Z is smooth at P′. The global version of the construction of the c(x) is then an isomorphism of invertible O_{Z}-modules:

(4.1)  Norm_{Z′/Z} (P′^{∗}Ω_{C_{Z}^{∼}/Z}) ⟶ O_{Z} ⊗_{g^{−1}O_{S}} I.

To prove all this, one reduces to the noetherian case and uses the description of [6, 2.23]. We give a more geometric description of the global version of the c(x) for the universal stable curve of some genus g ≥ 2. So let \overline{M} be the algebraic stack over ℤ classifying such curves, and let f: \overline{C} → \overline{M} be the universal curve (see [7] and [22]). Put Z := Sing(f). Then Z is the disjoint union of irreducible smooth closed substacks Δ_{i}^{∗}, 0 ≤ i ≤ ⌊g/2⌋, of \overline{C}, of codimension two, and the morphism Z → \overline{M} is locally a closed immersion, defined by a locally principal ideal I in f^{−1}O\overline{M}. Then then conormal bundle I/I^{2} of the locally closed immersion f: Z → M is a line bundle on Z. There is another line bundle, say L, on Z, with the property that for x in Z(k) corresponding to a pair (C, P) with C a stable curve of genus g over the algebraically closed field k and P a singular point of C, L(x) = Ω_{C_{1}}(x) ⊗ Ω_{C_{2}}(x), with C_{1} and C_{2} the two branches at x. Then the global version of the c(x) gives an isomorphism between L and I/I^{2}. The existence of this isomorphism was already noted, over ℂ, in [17, p. 143] and in [24, p. 477].

5 The c(x) and certain morphisms.

Let X, D, etc., be as in the beginning of Section 2. In particular, R: X^{∼} → X is the minimal resolution of the singularities of X. We want to compare the c(x) on X and X^{∼}. So let x be singular in X(k), and suppose we have an isomorphism between O_{X,x}^{∧} and D[[u, v]]/(uv−π^{e}). Let C_{1} and C_{2} be the irreducible components of X^{nor} on which u and v vanish, respectively. Then R^{−1}{x} is the union of e − 1 smooth, proper and geometrically irreducible curves of genus zero, say L_{1}, …, L_{e−1}, arranged in a chain: if we let L_{0} := C_{1} and L_{e} := C_{2}, then L_{i} ∩ L_{j} is empty if |i − j| > 1, and L_{i−1} ∩ L_{i} is a unique point P_{i} for 1 ≤ i ≤ e. Let u_{i} := π^{i+1−e}u and v_{i} := π^{−i}v. Then O_{X^{∼},P_{i}}^{∧} is D[[u_{i}, v_{i}]]/(u_{i}v_{i} − π), v_{i} is a parameter of L_{i−1} at P_{i} and u_{i} is a parameter of L_{i} at P_{i}. So we have c(P_{i}) = dv_{i} ⊗du_{i}. In order to relate the c(P_{i}) to c(x) we use the following lemma, the proof of which is obvious.

5.1 Lemma. On a two-pointed smooth projective curve of genus zero (L, P_{1}, P_{2}) over a field k, there is a unique non-zero element x in Ω_{L}(P_{1}) ⊗ Ω_{L}(P_{2}) with the following property: let f be a rational function on L with divisor P_{1}−P_{2}, then x = (df)P_{1}⊗(df^{−1})P_{2}, with (df)P_{1} and df^{−1}P_{2} the values of df and df^{−1} at P_{1} and P_{2}, respectively. This element does not depend on the numbering of the two points.

This lemma gives us an isomorphism

(5.2)  Ω_{C_{1}}(x) ⊗ Ω_{C_{2}}(x) → Ω_{L_{0}}(P_{1}) ⊗ Ω_{L_{1}}(P_{1}) ⊗ ⋯ ⊗ Ω_{L_{e−1}}(P_{e}) ⊗ Ω_{L_{e}}(P_{e}),

sending c(x) to c(P_{0}) ⋯ c(P_{e}). It follows that we have the following proposition.

5.3 Proposition. Let x_{0} and x_{e} be parameters of O_{C_{1},x}^{∧} and O_{C_{2},x}^{∧}, respectively. Then c(x) = dx_{0} ⊗ dx_{e} if and only if there exist coordinates x_{i} on the L_{i}, for 1 ≤ i < e such that c(P_{i}) = d(x_{i−1}^{−1}) ⊗ dx_{i} for 1 ≤ i ≤ e. If such coordinates exist, they are unique.

This proposition implies that knowing X_{k}^{∼} and the c(x) for the singular x on it, up to isomorphism, is equivalent to knowing X_{k} and the c(x) and e(x) in that case, up to isomorphism.

The last topic of this section is the behavior of the c(x) with respect to finite morphisms. Suppose that f: X → Y is a finite morphism of proper nodal curves over D. Let y be a singular point of Y_{k}, and let x be a point of X_{k} that lies over y. In [20, Prop. 5] it is explained that e(x) divides e(y), and that there exist isomorphisms from D[[u, v]]/(uv − π^{e(x)}) to O_{X,x}^{∧} and from D[[s, t]]/(st − π^{e(y)}) to O_{Y,y}^{∧}, such that, via f, s corresponds to u^{n} and t to v^{n}, with n := e(y)/e(x). This proves the following proposition.

5.4 Proposition. Let f: X → Y be a finite morphism of nodal curves over D. Let y in Y_{k} be a singular point, let x be in f^{−1}y, let C_{1} and C_{2} the two branches at x, and D_{1} and D_{2} the two branches at y. Let n be e(y)/e(x). Then f gives isomorphisms between Ω_{D_{i}}(y) and Ω_{C_{i}}(x)^{⊗n}. Under these isomorphisms, c(y) is mapped to c(x)^{n}.

6 The c(x) and log structures.

(This section will not be used in the rest of the article.) The elements c(x) as described in Sections 3–5 have a simple description in terms of log structures. The notations are as before. Recall from [14] that a log structure on a scheme Y consists of a morphism of sheaves of multiplicative monoids α: M_{Y} → O_{Y} on the small etale site of Y such that α induces an isomorphism from α^{−1}O_{Y}^{∗} to O_{Y}^{∗}. For our scheme X, let M_{X} be the log structure induced by its special fibre. This means that M_{X} is the subsheaf of O_{X} consisting of functions whose restriction to the generic fibre are invertible. Let Q_{X} be the sheaf of monoids M_{X}/O_{X}^{∗}. For x non-singular in X_{k}(k), the stalk Q_{X,x} is just ℕ, with generator the class of the uniformizer π of D. Let x in X_{k}(k) be singular. Then, locally at x in the etale topology, X is isomorphic to the spectrum of D[u, v]/(uv − π^{e(x)}). It follows that Q_{X,x} is the submonoid (e(x), 0)ℕ + (0, e(x))ℕ + (1, 1)ℕ of ℕ^{2}, generated by the classes of u, v and π. We give D the log structure induced by its closed point: M_{D} = D − {0}. Then we have a morphism of log schemes from (X, M_{X}) to (D, M_{D}). We will now base change to k, equipped with its log structure M_{k} induced from M_{D}: M_{k} is the fibred sum of M_{D} and k^{∗} over D^{∗}. Let (X_{k}, M_{X_{k}}) be the pull back of (X, M_{X}) to (k, M_{k}). Then M_{X_{k}} is the fibred sum of M_{X} and O_{X_{k}}^{∗} over O_{X}^{∗}|_{X_{k}}.

Let now x in X_{k}(k) be singular, and let M_{X_{k}}(x) be the fibred sum of M_{X_{k},x} and k^{∗} over O_{X_{k},x}^{∗}. Let C_{1} and C_{2} be the two branches of X_{k} at x, with u = 0 on C_{1} and v = 0 on C_{2}. We have an extension:

(6.1)  1 → k^{∗} → M_{X_{k}}(x) → Q_{x} → 0,

with Q_{x} as described above. That description shows that the k^{∗}-torsors induced on the inverse images of (e(x), 0) and (0, e(x)) correspond to the one dimensional k-vector spaces Ω_{C_{2}}(x) and Ω_{C_{1}}(x), respectively. Because of the monoid structure on M_{X_{k}}(x), the inverse image of (e(x), e(x)) corresponds to Ω_{C_{2}}(x)⊗_{k} Ω_{C_{1}}(x). But via the morphism from (X_{k}, M_{X_{k}}) to (k, M_{k}) it also corresponds to (m/m^{2})^{⊗e(x)}. The element c(x) is then just the image of π^{e(x)}. In the case that X is regular, the argument above shows that, conversely, the k-scheme X_{k} with the c(x) determine the log-scheme (X_{k}, M_{X_{k}}) over (k, M_{k}) up to unique isomorphism. Compare with [30].

7 The c(x) for X_{0}(p).

Let p be a prime number and let X_{0}(p) and X_{0}(p)^{∼} be the models over ℤ of modular curve X_{0}(p)_{ℚ} mentioned in the introduction. Let D be the ring of Witt vectors of an algebraic closure k of 𝔽_{p}, and let X := X_{0}(p)_{D} and X^{∼} := X_{0}(p)_{D}^{∼}. The aim of this section is to describe the c(x) for the singular x in X_{k}. We will first give such a description in terms of the Kodaira-Spencer map and a well known mod p modular form B on supersingular elliptic curves. After that we give an expression of some small power of c(x) in terms of the supersingular j-invariants mod p, using the fact that the point 0 − ∞ of J_{0}(p)_{ℚ} has order n, the numerator of (p − 1)/12. Note that knowing the c(x) for X is equivalent to knowing them for X^{∼}, by Section 5. The results of this section will not be used in the explicit computations of the next section; instead, when we really need the c(x), we will compute them using explicit formulas for the Hecke correspondence T_{2}.

Let n ≥ 3 be an integer prime to p, and let Y := \overline{M}(Γ(n), Γ_{0}(p))_{D}. Then Y is a regular projective nodal curve over D, and Y → X is the quotient for the faithful action of the group G := GL_{2}(ℤ/nℤ)/{1, −1}. The special fibre X_{k} of X consists of two irreducible components, X_{∞} and X_{0}, containing the cusps ∞ and 0, respectively. We define Y_{∞} to be the union of the irreducible components of Y_{k} that are mapped to X_{∞}, and Y_{0} the union of those that are mapped to X_{0}. Let Z denote the j-line over D, i.e., Z := X_{0}(1)_{D}. The constructions that associate, to an elliptic curve E/S/k, the isogenies F: E → E^{(p)} and V: E^{(p)} → E, induce isomorphisms Z_{k} → X_{∞} and Z_{k} → X_{0}, respectively, and isomorphisms from X(n)_{k} to Y_{∞} and Y_{0}. The singular points of X_{k} correspond to the F: E → E^{(p)} with E over k supersingular. Indeed, such an isogeny is isomorphic to V: E = E^{(p^{2})} → E^{(p)}. It follows that the singular point F: E → E^{(p)} on X_{k} corresponds to the pair (j(E), j(E)^{p}) of points on X_{∞} and X_{0}. A similar statement is of course true for Y_{k}.

Let P be a singular point of X_{k}, corresponding to a supersingular elliptic curve E over k, and let Q be in Y_{k}, lying over P. The Kodaira-Spencer map gives an isomorphism of line bundles on X(n)_{k} from Ω(cusps) to \underline{ω}^{⊗2}, where \underline{ω} is the pullback along the zero section of the bundle of relative differentials of the universal elliptic curve. This map, and the isomorphisms from Y_{∞} and Y_{0} to X(n)_{k}, induce isomorphisms of k-vector spaces:

(7.1)  Ω_{Y_{∞}}(Q) → \underline{ω}_{E}^{⊗2}, Ω_{Y_{0}}(Q) → \underline{ω}_{E^{(p)}}^{⊗2} = \underline{ω}_{E}^{⊗2p}.

Via these isomorphisms, c(Q) can be viewed as an element of \underline{ω}_{E}^{⊗2(p+1)}, and then c(P) = c(Q)^{e(P)} in \underline{ω}_{E}^{⊗2(p+1)e(P)}. Now there does exist, for each supersingular elliptic curve E over k, a non-zero element B(E) in \underline{ω}_{E}^{⊗(p+1)}, with the property that for ϕ: E → E′ any isogeny, one has deg(ϕ)B(E) = ϕ^{∗}B(E′). This property determines the B(E) up to a common scalar, since all supersingular elliptic curves over k are isogenous via isogenies of degree prime to p. Such B(E) were defined by Robert in [27, Thm. B] and used by Serre in [28]; see [9, Prop. 7.2] and [5, 5] for several constructions. We will prove below that, up to a common scalar, the c(Q) are the squares of the B(E). But before we do that, we give yet another construction of B.

So let E be a supersingular elliptic curve over k. Since V: E^{(p)} → E is the dual of F: E → E^{(p)}, we have a natural isomorphism between the kernel E^{(p)}[V] of V and the Cartier dual E[F]^{D} of the kernel of F ([15, 2.8]). But E^{(p)}[V] = E^{(p)}[F] = E[F]^{(p)}, hence we have a natural isomorphism λ_{E}: E[F]^{(p)} → E[F]^{D}. Now recall that the category of finite commutative group schemes over k that are annihilated by F and V is antiequivalent to the category of k-vector spaces by sending such a group scheme to its cotangent space at the origin, and that this converts Cartier duality into ordinary duality ([12, 3,Thm. 7.4]). Hence we get an isomorphism between \underline{ω}_{E}^{⊗p} and \underline{ω}_{E}^{⊗−1}, i.e., a non-zero element B(E) of \underline{ω}_{E}^{⊗(p+1)}. For ϕ: E → E′ an isogeny, one has λ_{E}^{∘}ϕ^{(p)t} = ϕ^{D}^{∘}λ_{E′}, hence ϕ^{D}^{∘}λ_{E′}^{∘}ϕ^{(p)} = deg(ϕ)λ_{E}, establishing ϕ^{∗}B(E′) = deg(ϕ)B(E).

7.2 Theorem. There exists an element μ in k^{∗} such that for all singular Q in Y_{k} we have c(Q) = μB(E)^{⊗2}.

Proof. Let us first note that, to define the c(Q), we have to choose a uniformizer of D, for which we take p, but that the choice does not matter for the existence of μ. To prove the proposition, it suffices to show that the c(Q) behave in the right way with respect to isogenies of degree prime to p.

So let Q be a singular point of Y_{k}, corresponding to a pair (E, α), with E a supersingular elliptic curve over k and α an isomorphism from (ℤ/nℤ)^{2} to E[n](k). Let ϕ: E → E′ be an isogeny of some degree d that is not divisible by p. Let Q′ be the singular point of Y_{k} corresponding to (E′, ϕ^{∘}α). Let C be the modular curve M(Γ_{0}(d), Γ(n), Γ_{0}(p))_{D}, and s and t the usual morphisms from C to Y (s just forgets the subgroup of rank d, t takes the quotient by it). Let R be the point of C_{k} corresponding to (E, ϕ, α). The morphisms s and t are etale, hence induce isomorphisms between O_{Y,Q}^{∧}, O_{C,R}^{∧} and O_{Y,Q′}^{∧}, preserving the labeling of the branches by ∞ and 0. It follows that c(Q) and c(Q′) correspond, via s and t repectively, to c(R).

For S → T be any morphism of schemes, and f: F → F′ any morphism of elliptic curves over S, the following diagram is commutative by [9, Lemma 7.4]:

(7.3)

\underline{ω}_{F/S}^{⊗2}  ⟵^{f^{∗}}  \underline{ω}_{F′/S}^{⊗2}
KS_{F/S} ↘  ↙ (deg f)KS_{F′/S}
Ω_{S/T}

Applying this to our isogeny ϕ at the point R of C_{∞} and of C_{0} shows that the following diagram, in which the vertical arrows are given by (7.1), is commutative:

(7.4)  \begin{matrix}Ω_{Y_{∞}}(Q′) ⊗ Ω_{Y_{0}}(Q′) & ⟶^{(t^{∘}s^{−1})^{∗}} & Ω_{Y_{∞}}(Q′) ⊗ Ω_{Y_{0}}(Q′) \\ ↓ & & ↓ \\ \underline{ω}_{E′}^{⊗2(p+1)} & ⟶^{ϕ^{∗}/d^{2}} & \underline{ω}_{E}^{⊗2(p+1)} \end{matrix}

This establishes that the c(Q) behave the same as the B(E) with respect to isogenies of degree prime to p. □

To finish this section, we will investigate the consequences of the fact that the point 0−∞ in J_{0}(p)(ℚ) has order n, the numerator of the fraction (p − 1)/12, expressed in lowest terms (see [18, II, Prop. 11.1]). These consequences will not be used in the following sections.

Let J denote Pic_{X/D}^{[0]}/E, as in the notation of Section 2. Then J is the image in the Néron model J_{0}(p)_{D} of Pic_{X}^{[0]}. Let P denote the element of J(D) given by 0 − ∞ (every line bundle of degree zero on X_{K} that admits an extension to a line bundle over X determines a unique element of J(D)). Let e be the least common multiple of the e(x), x ranging through the singular points of X_{k}. We recall from [8, VI, Thm. 6.9] that e(x) equals half the number of automorphisms of either of the two elliptic curves E and E^{(p)} over k corresponding to x. Hence all e(x) are equal to one, except when x has j-invariant 0 or 1728, in which case e(x) equals three or two, respectively, assuming p is at least five. We also recall that the elliptic curves over k with j-invariant equal to 0 or 1728 are supersingular if and only if p is congruent to −1 modulo three or four, respectively. It follows that n = (p − 1)e/12. The exact sequence (2.1) shows that E(D) is generated by the isomorphism class of the invertible O_{X}-module O_{X}(eX_{∞}). The “mass formula” of [15, Cor. 12.4.6] or [8, VI, 4.9.1] gives:

(7.5)  deg (O_{X}(eX_{∞})|_{X_{0}}) = ∑_{x∈S_{1}} e/e(x) = e ∑_{x∈S_{1}} 1/e(x) = e(p − 1)/12 = n.

We apply Proposition 3.11 to the divisor n(0 − ∞) on X_{k}. It follows that there exist rational functions f_{∞} and f_{0} on X_{∞} and X_{0} with divisors −n∞ + ∑_{x}(e/e(x))x and −n0 + ∑_{x}(e/e(x))x, resectively, such that at every x one has c(x)^{e/e(x)} = f_{∞}(x)f_{0}(x), where f_{∞}(x) and f_{0}(x) are to be interpreted as leading terms as in Section 3. This proves the following theorem.

7.6 Theorem. There exists a unique α in 𝔽_{p}^{∗} such that for every supersingular x in X_{k} one has:

c(x)^{e/e(x)} = α ∏_{y≠x}(j(y) − j(x))^{e/e(y)} ∏_{y≠x^{(p)}}(j(y) − j(x)^{p})^{e/e(y)} · (dj ⊗ dj)^{e/e(x)}
= α ∏_{y≠x}(j(y) − j(x))^{(p+1)e/e(y)} · (dj ⊗ dj)^{e/e(x)}.

7.7 Remark. Theorem 7.6 only gives us the c(x)^{e/e(x)}, but Proposition 3.11 says that these powers are sufficient to describe the special fibre J_{0}(p)_{p} of the Néron model of J_{0}(p)_{ℚ} completely. Hence it should be possible to improve the main theorem of [29] in the sense that one should be able to get rid of the factors ±1 in it, because of [29, 0.5]. □

8 The intersection mod p of X_{0}(p) and the cuspidal group.

Let p, k, X, X^{∼} be as in the previous section. As in the introduction, let 0 and ∞ denote the two cusps of X (we view them as D-valued points of X). Let J denote the Néron model over D of J_{0}(p)_{K}, and let f: X^{∼,0} → J be the morphism from the smooth locus of X^{∼} to J that extends the morphism from X_{K} to J_{K} sending a point P to the class of the divisor P −∞. Let c be f(0). Then c is of order n, the numerator of (p−1)/12, expressed in lowest terms. Let C be the closed subgroup scheme of J generated by c; it is called the cuspidal subgroup because its points correspond to classes of divisors supported on the cusps. In this section, we want to determine f^{−1}C_{k}, the inverse image under f of the fibre over k of C, in order to answer the question of Coleman mentioned in the introduction. There are two obvious elements in f^{−1}C_{k}: ∞_{k} and 0_{k}; the idea is that there should be nothing more. Let us note that Mazur’s theorem [19] on rational isogenies of prime degree implies that, for p bigger than 163, f^{−1}C_{K} consists just of ∞ and 0, because those are the only two ℚ-rational points of X_{0}(p). But a priori f^{−1}C_{k} could be bigger than the specialization of f^{−1}C_{K}.

We will now recall some facts about the geometry of X^{∼}, the group Φ := J_{k}/J_{k}^{0} and the map from C_{k} to Φ, that can be found in [18, III, Prop. 4.2]. As in the previous section, let X_{∞} and X_{0} denote the two irreducible components of X_{k}. The e(x) for X_{k} were described at the end of the previous section. It follows that, for p at least five, X_{k}^{∼} has at most five irreducible components: X_{∞} and X_{0}, one component F of genus zero (if p is −1 mod 4) originating from the singularity at x with j(x) = 1728, and two components G and H of genus zero (if p is −1 mod 3) originating from the singularity at x with j(x) = 0. (The cases p = 2 or 3 will be of no interest, because then X_{0}(p) has genus zero.) In order to fix the notation regarding G and H, we demand that G intersects X_{∞}. Figure 8.1 gives a picture in the case p = 23, where all five components are present. The intersection numbers of distinct components are as follows: X_{∞}·X_{0} equals the number of supersingular j-invariants in k − {0, 1728}; X_{∞}·F = X_{0}·F = 1; X_{∞}·G = 1; X_{0}·G = 0; X_{∞}·H = 0; X_{0}·H = 1; F ·G = F ·H = 0, G·H = 1. The self intersections can be computed from the fact that X_{k}^{∼} has trivial intersection with all components. From the resulting intersection matrices one can compute, as described in Section 2, that Φ is cyclic of order n, and generated by the image ϕ of X_{0} − X_{∞} (i.e., Φ is generated by the class of any invertible O_{X_{k}^{∼}}-module that has degree one on X_{0}, degree minus one on X_{∞}, and degree zero on the other components). It follows that the composition C_{k} → J_{k} → Φ is an isomorphism, hence every connected component of J_{k} contains exactly one element of C_{k}.

Of course, if X_{K} is of genus zero, J is just the trivial group scheme, and f^{−1}C_{k} is all Figure 8.1: A picture of X_{k}^{∼} when p = 23.

[figure: X_{∞}; X_{0}; G; H; F]

of X_{k}^{∼,0}. If X_{K} is of genus one, then f is an isomorphism (X^{∼,0} and J are both Néron models of X_{K}), hence f^{−1}C_{k} is isomorphic, via f, to C_{k}. So this solves the problem in these two cases, i.e., for p < 23. From now on we assume that X_{K} has genus at least two, i.e., that p ≥ 23.

8.2 Theorem. Let p ≥ 23 be prime. Then we have f^{−1}C_{k} = {∞_{k}, 0_{k}}.

The proof of this result will take some time. In the next section we will see that the morphism f: X_{k}^{∼,0} → J_{k} is a closed immersion, so that f^{−1}C_{k}, as a closed subscheme, is even reduced. We will first prove Theorem 8.2 for p > 71, the remaining cases will then be dealt with by hand or by simple computer computations.

8.3 First step of the proof of Theorem 8.2.

For the moment, let p ≥ 23 be prime. First we compute the map from the set of irreducible components of X_{k}^{∼} to Φ. This computation is done in [18, III, Prop. 4.2], but since we need the details of it later, we repeat it.

8.3.1 Lemma. Under f, X_{∞} is mapped to 0 in Φ, X_{0} is mapped to ϕ, F is mapped to (1/2)ϕ, G to (1/3)ϕ and H to (2/3)ϕ, (in each case the fraction makes sense mod n).

Proof. Of course, X_{∞} is mapped to 0 in Φ, because X_{∞} contains ∞_{k} and f(∞) = 0. Likewise, X_{0} is mapped to our generator ϕ of Φ, because it contains 0_{k}, and f(0) is the class of the divisor 0 − ∞, which has multidegree X_{0} − X_{∞}.

Suppose now that p is −1 mod 4. We want to compute the image of the component F. Note that n is then odd, so that we can invert 2 in order to compute in Φ using (2.2). Let P be in F (k). Then P − ∞ has multidegree F − X_{∞}. Note that O_{X^{∼}}(F) has multidegree X_{∞} + X_{0} − 2F. By the results of Section 2, the image of f(P) in Φ is the class of the multidegree F − X_{∞} + (1/2)(X_{∞} + X_{0} − 2F) = (1/2)(X_{0} − X_{∞}).

Finally, suppose that p is −1 mod 3. Then n is −1 mod 3. The intersection numbers above tell us that O_{X^{∼}}(G) and O_{X^{∼}}(H) have multidegrees X_{∞}−2G+H and X_{0}+G−2H, respectively. Let P be in G(k). It follows that the image of f(P) in Φ is the class of the multidegree

−X_{∞} + G + (1/3) (2(X_{∞} − 2G + H) + (X_{0} + G − 2H)) = (1/3)(X_{0} − X_{∞}).

A similar computation can be done for H. □

For Z an irreducible component of X_{k}^{∼}, let Z^{0} be Z minus the double points of X_{k}^{∼} lying on it.

8.3.2 Lemma. The only P in X_{∞}^{0}(k) with f(P) in C(k) is ∞_{k}.

Proof. Suppose P is in X_{∞}^{0}(k), with f(P) in C(k). Lemma 8.3.1 tells us that f(P) = 0 in J(k). Then Proposition 3.11 gives a rational function g on X_{∞} with divisor P − ∞, and with a constant value at all supersingular points. Since there are at least two distinct supersingular points, g is constant, and P = ∞. □

8.3.3 Lemma. The only P in X_{0}^{0}(k) with f(P) in C(k) is 0_{k}.

Proof. The proof is almost the same as the previous one. In this case one must have f(P) = c in C(k), and one applies Proposition 3.11 to the divisor P − 0 on X_{k}^{∼}. □

8.3.4 Lemma. Let p ≥ 23 be −1 mod 4 and suppose that there exists a supersingular j-invariant in 𝔽_{p} − {1728} and one in 𝔽_{p^{2}} − 𝔽_{p}. Then there is no point P in F^{0}(k) such that 2f(P) is in C(k).

Proof. Suppose that p is as in the Lemma, and that P in F^{0}(k) has 2f(P) in C(k). Then 2f(P) = c by Lemma 8.3.1. We apply Proposition 3.11 to the divisor E := 2P − 0 − ∞. It follows that there exist λ_{∞} and λ_{0} in k^{∗}, such that the functions f_{∞} = λ_{∞}(j − 1728) and f_{0} = λ_{0}(j − 1728) on X_{∞} and X_{0} satisfy f_{∞}(x) = f_{0}(x^{p}) for every supersingular j-invariant x in k other than 1728. The existence of such an x in 𝔽_{p} shows that λ_{∞} = λ_{0}. But the existence of such an x in 𝔽_{p^{2}} − 𝔽_{p} implies that λ_{∞} = −λ_{0}. □

8.3.5 Lemma. Let p ≥ 23 be −1 mod 3 and suppose that there exist two supersingular j-invariants x and y in k − {0} such that x^{p−2} ≠ y^{p−2}. Then there is no point P in G^{0}(k) or in H^{0}(k) such that 3f(P) is in C(k).

Proof. Suppose that p is as in the Lemma, and that P in G^{0}(k) has 3f(P) in C(k). Then 3f(P) = c by Lemma 8.3.1. We apply Proposition 3.11 to the divisor E := 3P − 2∞ − 0. It follows that O_{X^{∼}}(−2G − H)|_{X_{k}^{∼}} has a rational section with divisor E. The restrictions to X_{∞} and X_{0} of this section are of the form λ_{∞}j^{2} and λ_{0}j, respectively. It follows that for every supersingular x in k − {0} we have λ_{∞}x^{2} = λ_{0}x^{p}. This contradicts the hypotheses of the Lemma. The same result holds for H because of the w_{p}-operator. □

8.3.6 Lemma. For every prime p ≥ 23 with p = −1 mod 3 there do exist supersingular j-invariants x and y in k − {0} with x^{p−2} ≠ y^{p−2}.

Proof. Note that p − 2 and p^{2} − 1 have greatest common divisor 3, hence the required x and y exist if there are at least four distinct non-zero supersingular j-invariants in k. The mass formula of [15, Cor. 12.4.6] shows that the number of supersingular j-invariants is at least (p − 1)/12, which proves the Lemma for p > 49. The primes that are left are 23, 29, 41 and 47. Table 6 in [1] shows that in each of these cases there are at least two non-zero supersingular j-invariants x and y in 𝔽_{p}. For these we have x^{p−2} = x^{−1} ≠ y^{−1} = y^{p−2}. □

8.3.7 Lemma. For every prime p > 7 with p = −1 mod 4 there exists a supersingular j-invariant in 𝔽_{p} − {1728}.

Proof. There is a unique elliptic curve over ℂ (up to isomorphism) with endomorphism ring ℤ[2i]. Its j-invariant is 2^{3}3^{3}11^{3}, and we have 2^{3}3^{3}11^{3} − 1728 = 2^{3}3^{6}7^{2}. (An easy way to compute this is to use the explicit description of the Hecke correspondence T_{2} given below in Section 8.4.) □

8.3.8 Lemma. For every prime p > 71 there exists a supersingular j-invariant in k − 𝔽_{p}.

Proof. This is a result of Ogg, which can be found as a combination of the identities (21) and (24) in [23]. We recall the argument in a few lines. The 𝔽_{p}-rational supersingular points of X_{k} are precisely the fixed points of the Fricke involution w of X. It follows that all supersingular j-invariants in k are in 𝔽_{p} if and only if X_{k}^{+} := X_{k}/{1, w} has genus zero, hence if and only if X_{0}(p)/{1, w} is isomorphic to ℙ_{ℤ}^{1}. Suppose that this is the case. Then the inequalities:

10 = 2#ℙ^{1}(𝔽_{4}) ≥ #X_{0}(p)(𝔽_{4}) ≥ (p + 1)/12 + 2

show that p is at most 95. Table 6 of [1] then finishes the proof. □

Combining the previous lemmas in this section proves Theorem 8.2 for all primes p > 71: Lemmas 8.3.5 and 8.3.6 take care of points on G and H, and Lemmas 8.3.4, 8.3.7 and 8.3.8 take care of points on F. For the remaining primes p with 23 ≤ p ≤ 71 with p = −1 mod 4 we still have to show that no point P in F^{0}(k) has f(P) in C(k). Another inspection of Table 6 of [1] shows that the hypothesis of Lemma 8.3.4 is satisfied for 43 and 67. The only primes with which we still have to deal are 23, 31, 47, 59 and 71.

8.4 Second step in the proof of Theorem 8.2.

In this section we will deal with the remaining p for which we still have to prove Theorem 8.2: 23, 31, 47, 59 and 71. We recall that in these cases we only have to show that no P in F^{0}(k) has f(P) in C(k). The proof will use the full strength of Proposition 3.11 and the behavior of the c(x) under the Hecke correspondence T_{2}. Until now we only applied Proposition 3.11 in rather simple cases, not involving the c(x).

In order to apply Proposition 3.11, it suffices to know the c(x) up to a common scalar, for example, since the choice of a uniformizer of D is irrelevant. To compute the c(x) up to a common scalar, we just pick a supersingular point x_{0} with e(x_{0}) = 1 and choose c(x_{0}) = dj ⊗ dj. Then we use the Hecke correspondence T_{2} on X to find the c(x) at the other supersingular x. The following explicit description of the Hecke correspondence T_{2} on the affine j-line (over ℤ) is given in [21]: Y_{0}(2)_{ℤ} is the curve in 𝔸_{ℤ}^{2} given by the equation uv = 2^{12}, the morphisms s and t from Y_{0}(2)_{ℤ} to the j-line are given by s^{∗}j = (u + 16)^{3}/u and t^{∗}j = (v + 16)^{3}/v. Now consider the correspondence T_{2} on X; it is given by the morphisms s and t from X_{0}(2p)_{D} to X. In the beginning of Section 7 we gave isomorphisms between the j-line ℙ_{k}^{1} and the two irreducible components X_{∞} and X_{0} of X_{k}. Likewise, X_{0}(2p)_{k} is the union of X_{0}(2p)_{∞} and X_{0}(2p)_{0}, both isomorphic to X_{0}(2)_{k} via similar isomorphisms, and the induced correspondences on X_{∞} and X_{0} are given by the explicit formulas above.

8.4.1 Lemma. Let y in X_{0}(2p)(k) be supersingular, and put x := s(y), x′ := t(y). Then j −j(x) is a local coordinate on X_{∞} at x, j −j(x)^{p} is one on X_{0} at x, u−u(y) and v −v(y) are coordinates on X_{0}(2p)_{∞} at y, u − u(y)^{p} and v − v(y)^{p} are coordinates on X_{0}(2p)_{0} at y, and j − j(x′) and j − j(x′)^{p} are local coordinates at x′ on X_{∞} and X_{0}, respectively. Recall from Proposition 5.4 that e(y) divides e(x) and e(x′). Let α in k^{∗} be such that c(y) = α(du ⊗ du). Let β and γ in k^{∗} be given by:

s^{∗}(j − j(x)) = β(u − u(y))^{e(x)/e(y)} + higher order terms,
t^{∗}(j − j(x′)) = γ(v − v(y))^{e(x′)/e(y)} + higher order terms.

Then we have:

c(x) = α^{e(x)/e(y)}β^{−(p+1)}dj ⊗ dj,
c(x′) = (−αu(y)/v(y))^{e(x′)/e(y)}γ^{−(p+1)}dj ⊗ dj.

Proof. The local coordinates are obtained using the isomorphisms from the j-line to X_{∞} and X_{0} and from X_{0}(2)_{k} to X_{0}(2p)_{∞} and X_{0}(2p)_{0} that we discussed above. The rest is a direct application of Proposition 5.4. □

8.4.2 Remark. In practice, we will start with a supersingular x in X(k) with e(x) = 1. In order to find the various y’s in s^{−1}x, we factor the polynomial (T + 16)^{3} − j(x)T modulo p. (Actually, we will always have j(x) in 𝔽_{p}.) The factor β of Lemma 8.4.1 is then just the derivative (3(u(y)+16)^{2} −j(x))/u(y). Let us write c(x) = δ dj ⊗dj. Suppose that e(x) = 1 = e(x′). Then we simply have:

c(x′) = (\frac{j(x′) − 3(v(y) + 16)^{2}}{3(u(y) + 16)^{2} − j(x)} )^{p+1} δ dj ⊗ dj.

In all our computations, j(x), j(x′), u(y) and v(y) will be in 𝔽_{p}, so that we can replace the exponent p + 1 by 2. □

8.4.3 The case p = 31.

The supersingular j-values in k are 1728 = −8, 2 and 4 (see for example Table 6 in [1]). We have to check that no non-singular point on the component F is mapped to the cuspidal group. Suppose that P is in F^{0}(k) with f(P) in C(k). Lemma 8.3.1 shows that f(P) = 3c. Hence we apply Proposition 3.11 to the divisor −2∞ + 3·0 − P. Note that O_{X^{∼}}(X_{0}) has multidegree 2X_{∞} + F − 2X_{0}. Let x_{2} and x_{4} denote the supersingular points of X(k) with j-invariants 2 and 4, respectively. It follows that the functions f_{∞} = (j −2)(j −4) on X_{∞}, f_{0} = 1/(j +8)(j −2)(j −4) on X_{0} satisfy equation 3.12 for the cycle (X_{∞}, x_{2}, X_{0}, x_{4}, X_{∞}). As we remarked above, we only need to know c(x_{2}) and c(x_{4}) up to a common scalar, hence we simply put c(x_{2}) = dj ⊗ dj. We apply the method described in Remark 8.4.2. The y’s in s^{−1}x_{2} are given by u(y) ∈ {−15, 2, −4}. We take y with u(y) = −4; then j(t(y)) = 4. It follows that c(x_{4}) = −6dj ⊗ dj. The factor at x_{2} in equation 3.12 is:

((2 − 4)dj ⊗ ((2 + 8)(2 − 4))dj)c(x_{2})^{−1} = 9(dj ⊗ dj)c(x_{2})^{−1} = 9.

The analogous factor at x_{4} is:

((4 + 8)(4 − 2)dj ⊗ (4 − 2)dj)^{−1}c(x_{4}) = −6·17^{−1}(dj ⊗ dj)^{−1}(dj ⊗ dj) = −4.

Since −4·9 = −5 ≠ 1 in 𝔽_{31}, we have found a contradiction, so we have proved that no P in F^{0}(k) has image in C(k).

8.4.4 The case p = 47.

The supersingular j-invariants are 0, 1728 = −11, 9, 10 and −3. Suppose that P is in F^{0}(k) with f(P) in C(k). Then f(P) = 12c by Lemma 8.3.1. We apply Proposition 3.11 to the divisor E := −11∞+12·0−P. The divisor X_{∞}−2X_{0}−H has the same multidegree as E. Let x_{9} and x_{10} in X(k) be the points with j-invariants 9 and 10. The functions:

f_{∞} = j(j + 11)(j − 9)^{3}(j − 10)^{3}(j + 3)^{3} and f_{0} = (j(j + 11)^{2}(j − 9)^{3}(j − 10)^{3}(j + 3)^{3})^{−1}

then satisfy equation 3.12 for the cycle (X_{∞}, x_{9}, X_{0}, x_{10}, X_{∞}). We normalize the c(x) such that c(x_{9}) = dj ⊗ dj. In order to compute c(x_{10}), consider the point y on X_{0}(2)(k) with u(y) = −1. Then we have j(s(y)) = 9 and j(t(y)) = 10. One computes that c(x_{10}) = 3dj ⊗ dj, and that the factors at x_{9} and x_{10} in equation 3.12 equal 15 and −11, respectively. Since −11·15 = 23 ≠ 1, we have proved that no P in F^{0}(k) has f(P) in C(k).

8.4.5 The case p = 59.

The supersingular j-invariants are 0, 1728 = 17, 15, 28, −12 and −11. Suppose that P is in F^{0}(k) with f(P) in C(k). Then computations as in the previous two cases show that the functions:

f_{∞} = j(j − 17)(j − 15)^{3}(j − 28)^{3}(j + 12)^{3}(j + 11)^{3}
f_{0} = (j(j − 17)^{2}(j − 15)^{3}(j − 28)^{3}(j + 12)^{3}(j + 11)^{3})^{−1}

satisfy equation 3.12 for the cycle (X_{∞}, x_{15}, X_{0}, x_{28}, X_{∞}). We normalize the c(x) such that c(x_{15}) = dj ⊗ dj. In order to compute c(x_{28}), consider the point y on X_{0}(2)(k) with u(y) = −3. One finds that c(x_{28}) = 21dj ⊗ dj, and that the factors at x_{15} and x_{28} in equation 3.12 equal 35 and 10, respectively. Since 35·10 = −4 ≠ 1, we have proved that no P in F^{0}(k) has f(P) in C(k).

8.4.6 The case p = 71.

The supersingular j-invariants are 0, 1728 = 24, 17, −31, −30, −23 and −5. Suppose that P is in F^{0}(k) with f(P) in C(k). Computations as in the previous cases show that the functions:

f_{∞} = j(j − 24)(j − 17)^{3}(j + 31)^{3}(j + 30)^{3}(j + 23)^{3}(j + 5)^{3}
f_{0} = (j(j − 24)^{2}(j − 17)^{3}(j + 31)^{3}(j + 30)^{3}(j + 23)^{3}(j + 5)^{3})^{−1}

satisfy equation 3.12 for the cycle (X_{∞}, x_{−31}, X_{0}, x_{−23}, X_{∞}). We normalize the c(x) such that c(x_{−31}) = dj ⊗ dj, and find that c(x_{−23}) = −13dj ⊗ dj using the point y with u(y) = −14. The factors at x_{−31} and x_{−23} in equation 3.12 equal 49 and 38, respectively. Since 49·38 = 16 ≠ 1, we have proved that no P in F^{0}(k) has f(P) in C(k).

8.4.7 The case p = 23.

The supersingular j-invariants are 0, 1728 = 3 and −4. Suppose that P is in F^{0}(k) with f(P) in C(k). In this case we work with the model X′ of X obtained by blowing up X in its k-valued point with j-invariant 1728. We apply Proposition 3.11 to the divisor −5∞ + 6·0 − P. For a in the proposition one can take −3X_{0} − F. One finds that the functions:

f_{∞} = j(j − 3)(j + 4)^{3} and f_{0} = (j(j − 3)^{2}(j + 4)^{3})^{−1}

satisfy equation 3.12 for the cycle (X_{∞}, x_{0}, X_{0}, x_{−4}, X_{∞}). We normalize the c(x) such that c(x_{−4}) = dj ⊗ dj. Consider the point y with u(y) = −3. In the notation of Lemma 8.4.1 one finds: β = 6, α = β^{2} = 13 and γ = 1/7. It follows that c(x_{0}) = −3dj ⊗dj. The factors at x_{0} and x_{−4} in equation 3.12 equal 13 and −5, respectively. Since 13· − 5 = 4 ≠ 1, we have proved that no P in F^{0}(k) has f(P) in C(k).

9 Closed immersions.

We go back to the situation of Section 2. So let D be a complete discrete valuation ring with fraction field K, separably closed residue field k, and uniformizer π. Let X_{K} be a proper smooth geometrically irreducible curve, and X a proper flat nodal model of it. Let J := Pic_{X/D}^{[0]}/E be the open part of the Néron model J^{∼} of the jacobian of X_{K} over D corresponding to line bundles on X_{K} that admit an extension to a line bundle on X. Let X^{0} be the open part of X where the morphism to Spec(D) is smooth, i.e., X^{0} is the complement in X of the set of singular points of X_{k}. Let P be in X^{0}(D). Then we have a morphism f: X^{0} → J that sends, for S any D-scheme, Q in X^{0}(S) to the class of O_{X_{S}}(Q − P_{S}). Equivalently, f is the morphism one obtains by applying the Néron property to X^{0} and f_{K}: X_{K} → J_{K}, using that the image of X^{0} in J^{∼} is in J. In this section we study to what extent f is a closed immersion. The first observation in doing this is that the disconnecting double points behave differently from the non-disconnecting ones. Recall that a singular point x of X_{k} is called disconnecting if and only if X_{k} −{x} is not connected. We let X′ be the complement in X of the set of non-disconnecting double points of X_{k}.

9.1 Proposition. The morphism f: X^{0} → J extends to a morphism f: X′ → J.

Proof. Let i: Γ → X^{0} ×_{D} J be the graph of f, and let \overline{Γ} be its closure in X′ ×_{D} J. We have to show that the projection p from \overline{Γ} to X′ is an isomorphism. We will show, using the valuative criterion of properness ([13, II, 7.3.8 and 7.3.9]), that p is proper. Then we show that p is bijective, hence finite. The normality of X then implies that p is an isomorphism.

In order to prove that p is proper, it suffices to show that for D′ any complete discrete valuation ring with fraction field K′ and algebraically closed residue field k′, and any Q in X′(D′) with Q(Spec(K′)) in X_{K}, the element f(Q) in J(K′) extends uniquely to one in J(D′). (The fact that it is sufficient to consider Q in X′(D′) with Q(Spec(K′)) in the open dense subset X_{K} of X′ can be found for example, without proof, in [7, p. 103], and, with a proof, in the forthcoming book [16]. The proof of this fact starts with a reduction to the quasi-projective case, using Chow’s Lemma; then one considers a compactification.) Let D′ and Q in X′(D′) be as above. If Q is in X^{0}(D′), the condition is obviously satisfied (just compose Q with f). So we suppose that Q is not in X^{0}(D′). That means that Q(Spec(k′)) is a disconnecting double point, say x, of X_{k}. Base change from D to D′ reduces the whole problem to the case D′ = D. By definition, f_{K}(Q) extends in J^{∼}; we have to show that that extension is actually in J. Equivalently, we have to show that the connected component of J_{k}^{∼} to which f_{K}(Q) specializes is a connected component of J_{k}. Let F_{1}, …, F_{e(x)−1} be the chain of ℙ_{k}^{1}’s lying over x in the minimal resolution R: X^{∼} → X. Since x is disconnecting, X_{k}^{∼} can be written as Y ∪ Z ∪ (∪_{i}F_{i}), with Y and Z connected, disjoint, and Y ∪ Z the union of all irreducible components of X_{k}^{∼} other than the F_{i}. To fix who is who, we demand that P is in Y and that F_{1} meets Y. In X^{∼}, Q specializes to one of the F_{i}, say F_{i}. Let C be the irreducible component of X_{k} to which P specializes. Then the connected component of J_{k}^{∼} to which f(Q) specializes is, in the description of Section 2, the class of the multidegree F_{i} − C. Let a be the divisor on X^{∼} which is given as the sum of all irreducible components of Y, and the F_{j} with j < i. The multidegree of a equals F_{i} − F_{i−1}. It follows that the class of the multidegree F_{i} − C is the same as that of F_{0} − C, if F_{0} denotes the unique irreducible component of Y meeting F_{1}. Since F_{0} maps onto an irreducible component of X_{k}, we have shown that the class of F_{i} − C is represented by a multidegree with support in the set of irreducible components of X_{k}, hence that f(Q) extends to an element of J(D). This finishes the proof that p is proper.

Let us now show that p is bijective. So let x be a disconnecting double point of X_{k}. It suffices now to show that for every Q in X(K) specializing to x, f(Q) specializes to the same element in J(k) (the argument is similar to the proof of the generalization in [7] of the valuative criterion of properness mentioned above). Recall that J_{k} = Pic_{X_{k}/k}^{[0]}/E_{k}. Let Q and Q′ be two elements of X(K) specializing to x. The fact that the F_{i} are isomorphic to ℙ_{k}^{1} implies that f(Q) and f(Q′) specialize to the same element of J(k) if Q and Q′ specialize to the same F_{i}; one reduces to that situation using divisors of the type Y + ∑_{j<i} F_{j} as above. □

Now that we know that f extends to f: X′ → J, we want to know when this extension is a closed immersion. We begin by analyzing what happens in the special fibres. We denote by G the graph of X_{k} as in Section 2, with S_{0} as set of vertices and S_{1} as set of edges. We use the analogous notation G^{∼}, S_{0}^{∼} and S_{1}^{∼} for X_{k}^{∼}.

9.2 Proposition. Suppose that C_{1} and C_{2} are two distinct elements of S_{0} such that the multidegree C_{1} − C_{2} has image zero in Φ = J_{k}/J_{k}^{0}. Then there is a unique path in G from C_{1} to C_{2} such that every edge in that path is a disconnecting double point.

Proof. The graph G^{∼} is obtained from G by replacing each edge x by a chain of e(x) edges, hence it suffices to prove this proposition in the case X = X^{∼}. So we assume that X is regular. We choose an orientation on G, i.e., maps s and t from S_{1} to S_{0} such that, for every x in S_{1}, s(x) and t(x) are the vertices of x. Then we have maps s_{∗} and t_{∗} from ℤ^{S_{1}} to ℤ^{S_{0}} and s^{∗} and t^{∗} from ℤ^{S_{0}} to ℤ^{S_{1}} given by (s_{∗}f)(C) = ∑_{s(x)=C} f(x) and (s^{∗}f)x = f(s(x)), etc. We define the usual boundary and coboundary maps d_{∗} := t_{∗} − s_{∗} and d^{∗} := t^{∗} − s^{∗}. We recall from [10, 1] that we have the exact sequence:

(9.3)  0 ⟶ ℤ ⟶^{diag} ℤ^{S_{0}} ⟶^{d_{∗}d^{∗}} ℤ^{S_{0}}[+] ⟶ Φ ⟶ 0,

with ℤ^{S_{0}}[+] the kernel of ℤ^{S_{0}} → ℤ, f ↦ ∑_{C} f(C). In fact, this exact sequence is just a simple reformulation of Raynaud’s description of Φ from Section 2. (The hypothesis in [10] that the irreducible components of X_{k} are smooth is actually never used in that article, and the hypothesis that k be algebraically closed can be replaced with k separably closed.) All this is to indicate that Φ is the cokernel of a kind of Laplace operator. For g in ℤ^{S_{0}}[+], there exists h in ℚ^{S_{0}}, unique up to adding a constant function, such that g = d_{∗}d^{∗}h. Such functions g and h have the following electrodynamical interpretation. We view G as an electric circuit in which each edge has a resistance of one ohm, and in which at each C in S_{0} an electric current of g(C) ampère enters the circuit; then h is a function that gives at each C the potential in volts.

In our situation, we know that for g = C_{1} − C_{2} there exists such an h with integer values. Since the total current leaving the circuit is one, the current in each edge is then 0, 1 or −1. This implies that if we delete an edge through which there goes a non-zero current, the resulting graph is disconnected, which gives the conclusion we wanted.

For those readers (including the referee) who do not accept such physical arguments as mathematics, we give the following rigorous argument. Let V and E be the sets of vertices and edges. Say one ampère enters at x and leaves at y. Let V_{max} be the set of z in V such that h(z) = h(x) and z ≠ x. Let V_{min} be the set of z in V such that h(z) = h(y) and z ≠ y. Then V − {x, y} is the disjoint union of V_{max}, V_{min} and say V′. Since d_{∗}d^{∗}h is zero at all z in V − {x, y}, it follows that V_{max}, V_{min} and V′ are disconnected from each other in G − {x, y}. Since every edge connecting x to V′ carries a non-zero current, there is a unique such edge. Likewise for y. Induction on the number of elements of V (remove V_{max} and V_{min}, and x and y) finishes the proof. □

9.4 Proposition. The morphism f_{k}: X′_{k} → J_{k} is proper.

Proof. We apply the valuative criterion of properness. It suffices to consider the local rings of X_{k}^{nor} at closed points x′ whose image x in X_{k} is a non-disconnecting double point. Let x be such a point. We have to show that f_{k} does not extend over some neighborhood of x. Suppose first that x is an intersection point of two distinct irreducible components C and C′ of X_{k}. Proposition 9.2 implies that the images of C ∩ X′_{k} and C′ ∩ X′_{k} under f_{k} lie in two distinct connected components of J_{k}. It follows that f_{k} does not extend over x. To finish, suppose that x is a double point of an irreducible component C of X_{k}. Let C′ be the partial normalization of C, in which all singular points are normalized except x. As before, we may and do suppose that P specializes to C. Then f_{k}(C ∩ X′_{k}) lies in J_{k}^{0} = Pic_{X_{k}/k}^{0}. Let g: J_{k}^{0} → Pic_{C′/k}^{0} be the morphism obtained by restriction of line bundles. We claim that g^{∘}f_{k} does not extend over x. Suppose that it does. Then Pic_{C′/k}^{0} would be generated by the image of C′ in it, hence would be a complete variety, which it isn’t (it is an extension of Pic_{C^{nor}/k}^{0} by 𝔾_{m}). □

9.5 Proposition. The following two conditions are equivalent:

1. f_{k}: X′_{k} → J_{k} is injective;

2. every irreducible component of X_{k} isomorphic to ℙ_{k}^{1} contains a non-disconnecting double point.

If these conditions are satisfied, then f_{k} is a closed immersion.

Proof. Let us first show that the first condition implies the second. So assume that the second condition is not satisfied, and let C is an irreducible component of X_{k}, isomorphic to ℙ_{k}^{1} and not containing any non-disconnecting double point. Let C^{0} be denote the complement in C of the set of double points of X_{k} contained in C. Let Q and Q′ be two distinct elements of C^{0}(k). An easy application of Proposition 3.11 shows that f(Q) = f(Q′).

Assume now that the second condition is satisfied. We show that f_{k} is a closed immersion. Since we know that f_{k} is proper, it suffices to show that f_{k} is injective, and injective on tangent spaces. Let us first consider the restrictions of f_{k} to the irreducible components of X′_{k}, and prove that these are injective and injective on tangent spaces.

Let C be an irreducible component of X_{k}. Replacing P by an element of X^{0}(D) that specializes to C changes f by a translation, hence we may and do assume that P specializes to C. Then f_{k}(C ∩ X′_{k}) is contained in J_{k}^{0} = Pic_{X_{k}/k}^{0}. Since the double points of X_{k} contained in C ∩ X′_{k} are disconnecting they are smooth points of C, hence the morphism f_{k}: C ∩ X′_{k} → J_{k}^{0} is the unique extension of its restriction to C^{0}. If C is not isomorphic to ℙ_{k}^{1} then the composition of f_{k}: C∩X′_{k} → J_{k}^{0} with the morphism J_{k}^{0} → Pic_{C/k}^{0} induced by restriction of line bundles on X_{k} to C is injective and is injective on tangent spaces, hence the same is true for f_{k}: C ∩ X′_{k} → J_{k}^{0}. So suppose that C is isomorphic to ℙ_{k}^{1}. Then f_{k}(C ∩ X′_{k}) is contained in the maximal torus T of J_{k}^{0}. Since C does contain a non-disconnecting double point, there exists a connected component D of X_{k} −C whose closure in X_{k} meets C in at least two points, say x and y. Let g be the regular function on C − {x, y} with divisor x − y and g(P_{k}) = 1. The standard description of T says that the character group of T is the homology group H_{1}(G, ℤ) of the graph G. It follows that there is a character χ: T → 𝔾_{m} such that f_{k}: C ∩ X′_{k} → T composed with χ is the function g. Hence also in this case the restriction of f is injective and injective on tangent spaces.

To finish the proof, it suffices to prove that f_{k} is injective, and injective on the tangent spaces at the double points of X′_{k}. Suppose that Q and Q′ in X′(k) have f_{k}(Q) = f_{k}(Q′). Proposition 9.2 then implies that Q and Q′ lie on irreducible components C and C′ of X_{k} that are connected in G by a path consisting of disconnecting double points. Let C_{1}, …, C_{n} be the irreducible components in this path, and let Y denote their union, as a reduced closed subscheme of X_{k}. After replacing P by a point that specializes to Y ∩ X^{0}, f_{k} sends Y ∩ X′ into J_{k}^{0}. Let h: J_{k}^{0} → Pic_{Y/k}^{0} be the morphism obtained by restriction of line bundles. Since the double points in the path above are disconnecting, the morphism from Pic_{Y/k}^{0} to the product of the Pic_{C_{i}/k}^{0} is an isomorphism. For each i, let p_{i} be the projection from Pic_{Y/k}^{0} to Pic_{C_{i}/k}^{0}. Proposition 3.11 implies that the restriction of p_{i}hf_{k} to C_{j} ∩ X′ is constant if i ≠ j, and an immersion if i = j. This proves that Q = Q′, and also that f_{k} is injective on the tangent spaces at the double points. □

9.6 Proposition. The following three conditions are equivalent:

1. the morphism f: X′ → J is proper;

2. for every prime number p, every non-disconnecting double point x of X_{k} is contained in a loop in the graph G such that v_{p}(e(x)) ≤ v_{p}(e(y)) for every y in that loop, where v_{p} denotes the p-adic valuation;

3. for every finite extension K′ of K and every Q of X(K′) specializing to a nondisconnecting double point, f(Q) specializes to a connected component of J_{D′}^{∼} that is not in J_{D′}, with D′ denoting the integral closure of D in K′, J_{D′} the pullback of J to D′, and J_{D′}^{∼} the Néron model over D′ of J_{D′}.

In particular, the morphism f: X′ → J is proper if X is regular at all non-disconnecting double points of X_{k}.

Proof. We apply the valuative criterion of properness. Arguments similar to those in the proof of Proposition 9.1 show that conditions 1 and 3 are equivalent. So it remains to show that conditions 2 and 3 are equivalent.

We begin by giving a useful description of Φ^{∼}/Φ. Let M denote the first homology group H_{1}(G, ℤ) of the graph G; note that it is canonically isomorphic to H_{1}(G^{∼}, ℤ). We have:

(9.6.1) M = ker(d_{∗}: ℤ^{S_{1}^{∼}} → ℤ^{S_{0}^{∼}}), and M^{∨} = Hom_{ℤ}(M, ℤ) = coker(d^{∗}: ℤ^{S_{0}^{∼}} → ℤ^{S_{1}^{∼}}).

Section 1 of [10] shows that:

(9.6.2)  Φ^{∼} = ℤ^{S_{1}^{∼}}/(ker(d_{∗}) + im(d^{∗})) = ℤ^{S_{0}^{∼}}[+]/im(d_{∗}d^{∗}).

Raynaud’s description of Φ in Section 2 shows that Φ is the image in Φ^{∼} of the subgroup ℤ^{S_{0}}[+] of ℤ^{S_{0}^{∼}}[+]. The inverse image under d_{∗} of this subgroup is the relative homology group H_{1}(G^{∼}, S_{0}, ℤ). Diagram 1.12 of [10] says that the map from ℤ^{S_{1}^{∼}} to Φ^{∼} factors through the map ℤ^{S_{1}^{∼}} → ℤ^{S_{1}} that sends x in S_{1}^{∼} to R(x) (recall that R: X^{∼} → X is the resolution morphism). The image of H_{1}(G^{∼}, S_{0}, ℤ) in ℤ^{S_{1}} is the set of elements a with a(x) a multiple of e(x) for all x in S_{1}. Let q denote the quotient map from ℤ^{S_{1}} to ⊕_{x}ℤ/e(x)ℤ. Then we have:

(9.6.3)  Φ^{∼}/Φ = (⊕_{x}ℤ/e(x)ℤ)/qd^{∗}ℤ^{S_{0}}.

9.6.4 Lemma. Let Q be in X(K), specializing to a non-disconnecting double point y of X_{k}. Let i be the integer such that Q specializes to the ith irreducible component in the chain of projective lines replacing y in X^{∼}, using the orientation of G as in [10, 2.2]. Then the image of f(Q) in Φ^{∼}/Φ is represented by the element ī in the factor at y of ⊕_{x}ℤ/e(x)ℤ.

Proof. This follows directly from [10, 2.3]. □

Lemma 9.6.4 implies that condition 3 above is equivalent to the condition that, for every non-disconnecting double point y of X_{k}, the map from ℤ/e(y)ℤ to Φ^{∼}/Φ is injective. This injectivity is equivalent to the injectivity after tensoring with ℤ_{p}, for all p, hence also to surjectivity in the other direction after applying Hom(·, ℚ_{p}/ℤ_{p}), for all p. It follows that the map from ℤ/e(y)ℤ to Φ^{∼}/Φ is injective if and only if for every p there exists an element g in H_{1}(G, ℚ_{p}/ℤ_{p}) such that g(y) is the class of e(y)^{−1} and has g(x) in e(x)^{−1}ℤ_{p}/ℤ_{p} for all x in S_{1}. Using that for any abelian group A one has H_{1}(G, A) = A⊗H_{1}(G, ℤ), the existence of such a g is seen to be equivalent to the existence of a loop in G, containing y, such that v_{p}(e(y)) ≤ v_{p}(e(x)) for all x in that loop. This finishes the proof of Proposition 9.6. □

9.7 Proposition. Suppose that X_{K} has genus at least one. The morphism f: X′ → J is a closed immersion if and only if the following two conditions are satisfied:

1. every irreducible component of X_{k} that is isomorphic to ℙ_{k}^{1} contains a non-disconnec-ting double point;

2. for every prime number p, every non-disconnecting double point x of X_{k} is contained in a loop in the graph G such that v_{p}(e(x)) ≤ v_{p}(e(y)) for every y in that loop.

Proof. Apply Propositions 9.5 and 9.6. □

9.8 Corollary. Let X be a regular proper nodal curve over a discrete valuation ring D, with smooth geometrically irreducible generic fibre X_{K} of non-zero genus, with a given P in X(K). Let f_{K}: X_{K} → J_{K} be the usual closed immersion sending P to zero. Let f: X^{0} → J be the induced morphism from the complement in X of the set of singular points of the special fibre X_{k} of X to the Néron model over D of J_{K}. Then f is a closed immersion if and only if all double points of Xk̄ are non-disconnecting, with k̄ some algebraic closure of k.

Acknowledgements. I would like to thank Torsten Ekedahl for pointing out to me the relation with log structures, as given in Section 6.

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