Appendix.

Bas Edixhoven

May 23, 2000

1 Various descriptions of groups of connected components.

Let R be a complete discrete valuation ring, with fraction field K and algebraically closed residue field k. Let XK be a smooth, proper, geometrically connected curve over K. Suppose that X is a nodal model of XK over R, i.e., X is a proper flat R-scheme, its generic fibre is XK and the only singularities of the special fibre Xk are ordinary double points. We will suppose, for simplicity, that the irreducible components of Xk are smooth. Let JK be the jacobian of XK, J the Néron model over R of JK and Φ := Jk/J{o \atop k} the group of connected components of Jk. Let _{X} → X be the minimal resolution of X. We associate a graph _{G} to _{X} as follows. The set of vertices is the set _{C}of irreducible components of _{Xk}, the set of edges is the set _{S} of singular points of _{Xk}. The vertices meeting an edge x are the two irreducible components containing x. The elements of _{C} are Cartier divisors on _{X}, hence for any two of them, say C and C′, we have an intersection number (C · C′). This intersection pairing defines a morphism of free Z-modules:

(1.1)  α: Z_{C} ⟶ Z_{C},  f ↦ (C ↦ ∑_{C′}(C · C′) f(C′)) .

A theorem of Raynaud (see [1, Thm. 9.6/1]) says that Φ is canonically isomorphic to the homology of the complex:

(1.2)  Z_{C} \underrightarrow{α} Z_{C} \underrightarrow{+} Z,

where the map “+” sends f to ∑C f(C). The isomorphism works as follows. Let P be in J(K), let D be a divisor on _{X} such that P is the class of the restriction of D to XK. Then P specializes to the element of Φ given by C ↦ (C · D).

Grothendieck gave another description of Φ in terms of the monodromy pairing in [6, Thm. 11.5]. See also [7]. Our aim is to compare these two descriptions. We choose an orientation on the graph _{G}, i.e, we choose two maps s and t from _{S} to _{C} (s for source and t for target) such that for all edges x, s(x) and

t(x) are the vertices meeting x. We get induced maps:

(1.3)  s∗, t∗: Z_{S} ⟶ Z_{C},  s∗, t∗: Z_{C} ⟶ Z_{S},

where (s∗f)C = ∑s(x)=C f(x), (s∗f)x = f(s(x)), etc. Note that s∗ is the adjoint, with respect to the standard inner products on Z_{S} and Z_{C}, of s∗. With this notation, we can define the usual boundary and coboundary maps:

(1.4)  d∗ := t∗ − s∗: Z_{S} ⟶ Z_{C},  d∗ := t∗ − s∗: Z_{C} ⟶ Z_{S}.

By definition, ker(d∗) is the homology group H1(_{G}, Z) and coker(d∗) = H1(_{G}, Z) (the isomorphisms depend on the orientation chosen). Since _{G} is connected, ker(d∗) = H0(_{G}, Z) is the diagonal in Z_{C}, and coker(d∗) = H0(_{G}, Z) is Z, via the map +: Z_{C} → Z. A simple computation shows:

(1.5)  α = t∗s∗ + s∗t∗ − s∗s∗ − t∗t∗ = −(t∗ − s∗)(t∗ − s∗) = −d∗d∗.

Let T be the maximal torus of J{o \atop k}, and let M := Hom(T, Gm,k) be its character group. We denote the Z-linear dual of M by M∨. It is well known, e.g. [6, 12.3.7] or [1, §9.2, Example 8], that M and H1(_{G}, Z) are canonically isomorphic. Let us give an isomorphism. The torus T is the kernel of the pullback map Pic_{Xk}/k → PicX{nor \atop k}/k, where nor: X{nor \atop k} → _{Xk} is the normalization map. For each x in _{S} we can distinguish the two elements of nor−1\{x\} because one of

them lies on s(x) and one on t(x); we denote these two points by sx and tx. Let f be a k∗-valued function on _{S}. Then we can construct a line bundle on _{Xk} whose class is in T (k) as follows: take the trivial line bundle on X{nor \atop k}, and, for each x in _{S}, identify the fiber at sx with the fibre at tx via multiplication by f(x). This construction induces an isomorphism between M∨ and H1(_{G}, Z) which does not depend on the orientation chosen. Passing to duals gives the desired isomorphism. From (1.5) we get the following commutative diagram:

(1.6)

0  0
↘  ↓
Z  \underrightarrow{0}  M
id ↗  ↘ diag  ↓  ↘i
0  ⟶  Z  \underrightarrow{diag}  Z_{C}  \underrightarrow{d∗}  Z_{S}  ⟶  M∨  →  0
↘−α  ↓d∗
Z_{C}[+]  ↓
↘
↓  Φ
↙  ↘
0  0

As a consequence, we have the short exact sequence:

(1.7)  0 ⟶ M \underrightarrow{i} M∨ ⟶ Φ ⟶ 0.

A computation shows that for all m1 and m2 in M:

(1.8)  (i(m1)) (m2) = ∑_{x∈_{S}}m1(x) m2(x),

which is exactly the value at (m1, m2) of the monodromy pairing, see [6, Thm. 12.5] and [7]. Equations (1.7) and (1.8) together are Grothendieck’s description of Φ. It follows that the diagram (1.6) gives a translation between Raynaud’s and Grothendieck’s descriptions of Φ.

Our next objective is to give the relation between the descriptions of M in terms of _{X} and X. Let G be the graph associated to X: the set of vertices is the set C of irreducible components of Xk, the set of edges is the set S of singular points of Xk, the vertices meeting an edge x are the two components containing x. Let r: _{X} → X denote the resolution morphism; it is the composition of blowing ups in the singular points. For x in S let w(x) be the “width” of the singularity at x: if π is a uniformizer of R then, locally at x for the etale topology, X is given by the equation uv = πw(x). Let x be in S. It is well known that r−1{x} is a chain of w(x) − 1 projective lines. In terms of graphs, this means that _{G} is obtained from G by replacing each edge x of G by a path of w(x) edges:

(1.9)  [figure: becomes]

It is clear that an orientation on G induces an orientation on _{G}. We will assume that the orientation that we already have on _{G} comes from an orientation on G; the corresponding two maps S → C will also be denoted by s and t. It is also clear that H1(G, Z) and H1(_{G}, Z) are canonically isomorphic, and that this isomorphism is compatible with the identifications of both of them with M (there is an isomorphism between M and H1(G, Z) defined just as for _{G}).

The picture (1.9) suggests to consider the following maps:

(1.10)  hC: ZC ⟶ Z_{C}, C ↦ _{C},  hS: ZS ⟶ Z_{S}, x ↦ ∑_{y∈r−1x} y,

with _{C} the strict transform of C. By construction, we get a commutative diagram:

(1.11)  \begin{aligned}0 & ⟶ & M & ⟶ & ZS & \underrightarrow{d∗} & ZC & \underrightarrow{+} & Z & ⟶ & 0 \\ & & ↓id & & ↓hS & & ↓hC & & ↓id \\ 0 & ⟶ & M & ⟶ & Z_{S} & \underrightarrow{d∗} & Z_{C} & \underrightarrow{+} & Z & ⟶ & 0 \end{aligned}

Dualizing gives another such diagram:

(1.12)

0  ⟶  Z  \underrightarrow{diag}  Z_{C}  \underrightarrow{d∗}  Z_{S}  ⟶  M∨  ⟶  0
↓id  ↓h{∨ \atop C}  ↓h{∨ \atop S}  ↓id
0  ⟶  Z  \underrightarrow{diag}  ZC  \underrightarrow{d∗}  ZS  ⟶  M∨  ⟶  0

In this diagram h{∨ \atop S} and h{∨ \atop C} are given by:

(1.13)  h{∨ \atop S}: Z_{S} ⟶ ZS , y ↦ r(y),  h{∨ \atop C}: Z_{C} ⟶ ZC , C ↦ r∗(C),

where r∗(C) is zero if r(C) is a point, and r(C) otherwise.

The map hS from (1.10) is injective; the standard inner product on Z_{S} induces the following inner product on ZS:

(1.14)  ⟨·, ·⟩: ZS × ZS ⟶ Z, (f, g) ↦ ∑_{x∈S} w(x) f(x) g(x).

It follows that, when we view M as a submodule of ZS and M∨ as a quotient of it, the map (1.7) i: M → M∨ is given by:

(1.15)  (i(m1))(m2) = ∑_{x∈S} w(x) m1(x) m2(x).

The discussion up to here allows us to translate between Raynaud’s description of Φ in terms of _{X} and Grothendieck’s description in terms of the Picard–Lefschetz formula for X.

2 Specialization of divisors of degree zero.

We keep the notation of the previous section. Since X and _{X} are proper over R, we can identify X(K), X(R) and _{X}(R). The closure in _{X} of an effective divisor on XK is a Cartier divisor. Extending this by linearity we associate to each divisor D on XK a Cartier divisor _{D} on _{X}.

Suppose now that D is a divisor of degree zero on XK. Then the class of D under linear equivalence corresponds to an element, say [D], of J(K) = J(R). Let ϕ(D) denote the image of [D] in Φ. By Raynaud’s description of Φ, see (1.2), ϕ(D) is given by the element (C ↦ (C · _{D})) of Z_{C}. Our aim is now to get an expression for ϕ(D) in terms of ZC and ZS. For simplicity, we suppose that D has support in the set of K-rational points of X (note that since X is semi-stable, one can always reduce to this case by extending K). Let us write:

(2.1)  D = ∑_{P} nP P,

where the sum ranges through the set X(K) (of course, almost all nP are zero, and their sum is zero). Each P in X(K) specializes to a unique element c(P) of _{C}, since _{X} is regular. With this notation, ϕ(D) is given by the element ∑P nP c(P) of Z_{C}.

Suppose now that P is such that c(P) is not in C. Let x in S be the image of c(P) under the morphism r: _{X} → X. Then we have the following situation:

(2.2)  [figure: x_{1}; x_{2}; x_{m(P)}; x_{w(x)}; c(P); in _{G}.; x; s(x); t(x); in G,]

Note that the integer m(P) is defined by the condition that c(P) is the m(P)th projective line in the chain of projective lines from s(x) to t(x) in _{X}. Using (1.6) it follows that ϕ(D) is represented by the sum of the following two contributions:

(2.3)  ∑_{c(P)∈C} nP c(P) + ∑_{c(P)∉C} nP s(r(c(P))) and ∑_{c(P)∉C} nP m(P) r(c(P)),

where the first contribution comes from the subgroup ZC [+] of Z_{C}[+] and the second from ZS.

The last thing we want to do in this section is to give a useful interpretation of the integers m(P). Suppose that P in X(K) is such that c(P) is not in C. Let x denote the element r(c(P)) of S. Let π be a uniformizer of R. Let u and v be elements of the complete local ring _{OX,x} such that _{OX,x} = R[[u, v]]/(uv−πw(x)). By interchanging u and v, if necessary, we may and do suppose that (u, π) is the ideal of the branch s(x). Let c0, …, cw(x) be the irreducible components of r−1x, numbered such that s(xi) = ci−1 and t(xi) = ci. For the divisor div(v) of v on the completion of _{X} along r−1x one has:

(2.4)  div(v) = ∑_{i=0}^{w(x)} i ci.

To see this, note that blowing up R[u, v]/(uv − πn) in the ideal (u, v, π) gives the singularity R[π−1u, π−1v]/(π−1uπ−1v − πn−2). Equation (2.4) gives the following interpretation of m(P):

(2.5)  m(P) = P · (m(P)c(P)) = P · ∑_{i=0}^{w(x)} i ci = P · div(v) = valR(P ∗(v)),

where valR denotes the valuation on R such that valR(π) = 1, and where P ∗ is the morphism of rings _{OX,x} → R induced by the morphism of schemes P: Spec(R) → X.

3 Modular curves.

The aim of this section is to give a moduli theoretic interpretation of the integer m(P) in (2.5), in the case where XK is a certain modular curve.

From now on, let p be a prime number. We suppose that K has characteristic zero and that k has characteristic p. Let X be the compactified coarse moduli scheme for the algebraic stack with objects ((α: E → E′)/S/R, β), with S an R-scheme, E and E′ elliptic curves over S, α an isogeny of degree p, and β some prime-to-p level structure (of a fixed type) on E. By [5, V.1] or [9, 13.4], the R-scheme X is proper and flat, XK is smooth and the only singularities of Xk are ordinary double points. We suppose that the level structure β is such that XK is geometrically irreducible. Then X satisfies the hypotheses of the beginning of Section 1. The special fibre Xk has two irreducible components; we call them C∞ and C0, with C∞ containing the cusp ∞ and C0 the cusp 0. Over C∞, the isogeny α is isomorphic to the relative Frobenius F: E → E(p). Over C0, α is isomorphic to the Verschiebung V : (E′)(p) → E′. We choose the following orientation on the graph G of X: for every double point x of Xk, we have s(x) = C0 and t(x) = C∞.

Let ((α: E → E′)/R, β) be an R-valued point of the stack mentioned above, and let P in X(R) denote the induced R-valued point of X. We suppose that P specializes to a singular point x of Xk; this just means that Ek is supersingular. Let A be the deformation ring of the object corresponding to x (we consider deformations over complete local R- algebras with residue field k). By [5, V, 1.19] or [9, Thms. 5.1.1 and 13.4.7], we can take y and z in A such that A = R[[y, z]]/(yz − p). Let G be the group of k- automorphisms of the object corresponding to x. Then G acts on A, and by [5, I, 8.2.1] we have _{OX,x} = AG. The two branches at x of Xk are fixed by G, hence for all g in G, g(y)/y and g(x)/x are units in A. Let \overline{G} be the image of G in Aut(A), and |\overline{G}| its order. We get a subring R[[u, v]]/(uv − p|\overline{G}|) of AG by sending u to ∏g∈\overline{G} g(y) and v to ∏g∈\overline{G} g(z). This subring is normal, and the field extension given by the fraction fields of this subring and of A has degree |\overline{G}|. It follows that:

(3.1)  _{OX,x} = R[[u, v]]/(uv − p|\overline{G}|).

As in Section 2, we assume that u vanishes on the branch C0, and v on C∞. Let ((αuniv: Euniv → E{′ \atop univ})/A, βuniv) denote the universal deformation. Then z is an equation for the maximal closed subscheme of Spec(A/pA) over which E{′ \atop univ} is isomorphic to E{(p) \atop univ} (use that the analogous statement is true for the deformation ring over the Witt vectors of k; see [9, Thm. 13.4.7]). Since P in X(R) arises from the object ((α: E → E′)/R, β), the morphism P ∗: _{OX,x} → R is induced from a morphism P ∗: A → R. The element P ∗(z) is an equation for the maximal closed subscheme of Spec(R/pR) over which E′ and E(p) are isomorphic. In other words, valR(P ∗(z)) is the maximal integer n such that E′ and E(p) are isomorphic over R/πnR (as before, π is a uniformizer of R). We have:

(3.2)  valR(P ∗(v)) = valR(P ∗(v)) = ∑_{g∈\overline{G}} valR(P ∗(g(z))) = |\overline{G}| · valR(P ∗(z)).

Equation 3.1 implies that w(x) = |\overline{G}| · valR(p). Putting everything together gives:

(3.3)  m(P) = \frac{w(x)}{valR(p)} · max\{n ≤ valR(p) | E′ ≅ E(p) over R/πnR\}.

4 Shimura curves.

We will now adapt the arguments of Section 3 to the case of certain Shimura curves.

Let p and K be as in Section 3. Let BQ be an indefinite quaternion algebra over Q, of discriminant prime to p. Let B be a maximal order in BQ. First we consider the algebraic stack M with objects (E/S/R, i: B → EndS (E), β), where S is an R-scheme, E/S an abelian scheme of relative dimension two, i a morphism of rings, and β some prime-to-p level structure. See [2] for the proof that this category is indeed an algebraic stack, and that it is proper and smooth of relative dimension one over R. We suppose that the level structure β is such that the fibres of M/R are geometrically irreducible. Combining [2, p. 54] with [3, III, 1.5] gives that each such E/S with B-action has a canonical principal polarization (the unique principal ∗- polarization).

In [5, Introduction, §7] it is explained how the results of [5] and [9] for modular curves imply results for M. Let e be a non-trivial idempotent in Zp ⊗ B (recall that B is split at p). Let (E/S, i, β) be an object of M. Since Zp ⊗ B acts on the p-divisible group E[p∞] of E, we get a direct sum decomposition

(4.1)  E[p∞] = eE[p∞] ⊕ (1 − e)E[p∞].

The two terms in this decomposition are isomorphic, since e and 1 − e are conjugates. The construction G ↦ eG defines an equivalence between the category of p-divisible S-groups with an action by B and the category of p-divisible S-groups. Via this equivalence, we can apply the Drinfeldian level structures of [9, §1.9] to M. We will only consider the level structure corresponding to the Γ0(p) level structure in the modular curve case: it associates to (E/S/R, i) the set of finite locally free subgroup schemes of rank p of eE[p∞], or, equivalently, the set of isogenies α: (E, i) → (E′, i′) (up to isomorphism) of degree p2, with i′: B → End(E′) and α compatible with i and i′.

Let X be the coarse moduli space for the algebraic stack with objects ((α: E → E′)/S/R, β), with S an R- scheme, E and E′ abelian S-schemes of relative dimension two, equipped with an action by B, α as isogeny of degree p2 and β a prime-to-p level structure as above. Let x be a closed point of Xk. Let _{x} := ((α: E → E′)/k, β) be the object corresponding to x. Then eE[p∞] is a p-divisible group over k, of height two and with associated formal group of dimension one. The classification of p-divisible groups over algebraically closed fields implies that eE[p∞] is isomorphic to the p- divisible group of some elliptic curve F over k (in fact, if eE[p∞] is purely local, then any supersingular elliptic curve will do, otherwise any ordinary elliptic curve does the job). By the theorem of Serre and Tate [8] and the equivalence above, the deformation theory of _{x} is equivalent to that of (F/k, e ker(α)) (again, we consider deformations to complete local R-algebras). It follows that the deformation ring of _{x} is isomorphic to that of (F/k, e ker(α)). It follows that the only singularities of Xk are ordinary double points. One now easily shows, by constructing an ample divisor, that the algebraic space X is in fact a projective curve over R, satisfying the hypotheses of the beginning of Section 1. (An other way to see that X is a quasi-projective scheme is to prove that the morphism from M to the stack of two-dimensional principally polarized abelian varieties is representable and projective (use Hilbert schemes); see [4, §5.3].) We orient the graph G of X by demanding that for each singular point x of Xk, s(x) is the branch on which α is generically etale. The following proposition is now a direct consequence of the results of the preceding sections. 4.2 Proposition. Let P in X(R) be induced by an object ((α: E → E′)/R, β). Suppose that P specializes to a singular point x of Xk. Then the integer m(P) of (2.5) is given by:

m(P) = \frac{w(x)}{valR(p)} · max\{n ≤ valR(p) | E′ ≅ E(p) over R/πnR\}.

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