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ANNALES DE L’INSTITUT FOURIER

BAS EDIXHOVENMinimal resolution and stable reduction of X0(N)

Annales de l’institut Fourier, tome 40, no 1 (1990), p. 31-67<http://www.numdam.org/item?id=AIF_1990__40_1_31_0>

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Ann. Inst. Fourier, Grenoble40, 1 (1990), 31-67

MINIMAL RESOLUTIONAND STABLE REDUCTION OF X()(N)

by Bas EDIXHOVEN

Contents

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321. THE MINIMAL RESOLUTION OF XQ^N) AT p > 5. . . . . . . . . . . . . . . . 33

1.1. Outline of the results and computations in this chapter . . . . . 331.2. Xo(pnN) is regular at i n f in i ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3. Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4. Graph and local equations of XQ^N) 0z ~Fp . . . . . . . . . . . . . . 471.5. Xo(p2), for example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.6. Global structure of XQ(pnN) 0z Fp . . . . . . . . . . . . . . . . . . . . . . . . 50

2. THE STABLE REDUCTION OF Xo(p2 N) AT p > 5. . . . . . . . . . . . . . . . . . . 55

2.1. The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2. Formal computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3. Determination of the vertical components . . . . . . . . . . . . . . . . . . . 602.4. The coarse case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Key-words : Modular curves - Minimal models - Stable models.A.M.S. Classification : 11G18.

32 BAS EDIXHOVEN

INTRODUCTION

Let XQ^N) be the compactified coarse moduli scheme over Z of ellipticcurves together with a cyclic subgroup of order N. In chapter 1 the minimalresolution of Xo{N) over Z[j] is determined. In chapter 2 we describe thestable reduction of Xo(p2N) at p for p > 5, where p is a prime that doesnot divide N. It would be very interesting to known the stable reductionof XQ(pnN) at p, but wild ramification keeps me from finding it.

In a forthcoming chapter the actions of the Hecke algebra and theinertia group on the stable reduction will be studied. I hope that this(faithful) representation of the Hecke algebra will lead to an efficientalgorithm for computing all Weil curves with a given conductor. Such analgorithm has already been found in the case of a prime (and maybe alsosquare free) conductor by J.F. Mestre [12]. Until I heard from his resultsI was developing the same theory. Also the relation to A. Pizers Brandtmatrix representation of the Hecke algebra [13] [14] should become clear.Finally I want to thank dr. B. van Geemen for the idea of reducing Heckeoperators mod p, and for his proofreading.

After this text was written in November 1986, the actions of theHecke algebra and the inertia group have been studied in the authors thesis(Stable models of modular curves and applications, Utrecht, June 1989, tobe published). This resulted in an analog of the so-called "graph method" ofMestre and Oesterle [12], and also gave new information concerning strongmodular parametrizations of elliptic curves. Another application was givenin an article (to appear in Asterisque) to the computation of the actionof the Hecke algebra on the groups of connected components of reductionsmod p of Neron models over Z of jacobians of modular curves. Again,I would like to thank my thesis adviser F. Oort and especially B. vanGeemen for their advice and support.

NOTATIONS

In this paper, we will freely use notations introduced in the book ofKatz and Mazur [9]. For the convenience of the reader, we will list theplaces where a definition of some of these notations can be found. A fewother remarks should be made. Often, "irreducible component" will mean

MINIMAL RESOLUTION AND STABLE REDUCTION 33

the reduced subscheme corresponding to it. We will also use the expression"local equation" for what should really be called a formal local equation.

(Ell) modular stack of elliptic curves [9] 4.1.

(Ell/.R) modular stack of elliptic curves over base schemes overSpec(fi) [9] 4.13.

P moduli problem : a contravariant functor from (Ell) or(Ell/.R) to (Sets), see [9] 4.2.

M(P) (fine) moduli scheme associated to a representable moduliproblem, [9] 4.3.

M(P) coarse moduli scheme associated to a moduli problem Pwhich is relatively representable and affine, [9] 8.1 If P isrepresentable, then it is just M(P).

M(P) compactified coarse moduli scheme, [9] 8.6.

M(P) compactified moduli scheme, [9] 8.6.

[ro(AQ] the moduli problem that assigns to E / S the set of To(N)-structures on E / S , see [9] 3.4 and 5.1.

[(a, 6)-cyclic] the moduli problem on (Ell/Fp), where p is a prime,assigning to E / S the set of (a, -cyclic subgroups of E / S ,see [9] 13.4.3 and 13.4.5.

[Ig(j/1)] the moduli problem on (Ell/Fp) that assigns to E / S / F pthe set of Igusa-structures of level pn on E / S , see [9] 12.3.

[Exig^,^)] the moduli problem on (Ell/Fp) that assigns to E / S / F pthe set of i-exotic Igusa-structures of level p71 on E / S , see[9] 12.10.5.1.

1. THE MINIMAL RESOLUTION OF Xo(pnN) AT p > 5

1.1. Outline of the results and computations in this chapter.

I.I.I. Let p > 3 be a prime, N > 0 an integer that is not divisibleby p, and n > 1. Let [^o(pnN)] be the category fibered in groupoidsover the category (Sch) of schemes which classifies cyclic pnN-lsogemesbetween elliptic curves (n.b. not between generalized elliptic curves). Then[^Q(pnN)} is an algebraic stack because of [3] Ch. Ill, Thm. 2.5, and [9]Thm. 6.6.1.

34 BAS EDIXHOVEN

Remark 1.1.1.1. — The Mro(N) which is described in [4] Ch. Ill, §1,is not an algebraic stack. For a p-gon with its p-torsion subgroup is anobject of A1ro(p2)' This object has infinitesimal automorphisms. Note thatin the M^ of [3] Ch. Ill, Thm. 2.5, n-gons may only occur at points wherethe characteristic does not divide 71.

Let Xo(pnN) be the compactified coarse moduli scheme M^r^p71^)as constructed in [9] Ch. 8, and let XQ^N) be the minimal resolution ofXQ^N). We will give a complete description of XQ(pnN) 0z Fp in termsofXo(A00zFp.

Description 1.1.1.2. — Tie graph and local equations ofXo(pnN) (g)zFp can be found in (1.4). The non-reduced irreducible components ofXo^N^z Fp are isomorphic to the corresponding infinitesimal neighbo-rhoods of the zero sections of their normal bundles. The conormal bundleof the reduced (a, b)-component (with a, b > 0) is :

("^(^^zF/CUSps))^10-^^),

where D can be found in Table 1.6.3.2.

1.1.2. The computation runs as follows. First we show that XQ^N) isregular at the cusps, so there is no problem at infinity. Then we introducean auxiliary level structure P to view M([ro(j?71^)]) as a quotient of aregular scheme M(P, [ro^lV)]) by the action of a finite group.

The only non-trivial stabilizer groups we encounter are cyclic of order2 or 3. Since p > 5 the quotients we consider commute with all base changeswe will perform, and the quotient of a closed subscheme will be its imagein the quotient.

The scheme M{P, [ro^AOD^z Fp is completely described in termsof.M(P, [ro(AQ])0z Fp in [9] Thm. 13.4.7. This description yields enoughinformation to get the result.

1.1.3. We use the notation of [9] Thm. 13.4.7 : M^P^TQ^N)}) 0zFp is the disjoint union, with crossings at the supersingular points, ofthe n + 1 schemes M(P, [^o(N)}, [(a,6)-cyclic]) for a + b = n. Themultiplicity of the (a, -component is ^(p"1111 ' ), so only the (n,0) andthe (0, n)-components are reduced. By taking the quotient we see thatXQ^N) (g)z Fp consists of n + 1 irreducible components all isomorphicto XQ^N) 0z Fp which intersect at every supersingular point and nowhereelse. The multiplicity of the (a, -component is ^(p"11"^'^). It follows that


Xo(pnN) -+ Spec(Z) is smooth on the ordinary part of the (n,0) and(0, -components; since Spec(Z) is regular, XQ^N) is regular at thosepoints.

Consider the points in the finite part of Xo(pnN) 0z Fp which :

1. have extra automorphisms, and

2. are supersingular or have a -^ 0 9^ b.

It is easily checked that these points are isolated in XQ(pnN) for theproperty of having extra automorphisms. By purity of branch locus, theymust be singular in the surface XQ^N). We will not use this, because itfollows directly from the calculations we will do.

1.1.4. The first observation we make is that the points on M(P, [^o(pTW)])which are candidates to become singular in the quotient, are all singular inM(P, [TO^N)]) 0z Fp. Hence at such a point, the cotangent space of thesurface equals the cotangent space of the p-fibre. Since we know all aboutthis fibre, we know the actions of the stabilizer groups on the cotangentspaces. We can then apply [15] Thm. I5 of Serre :

THEOREM 1.1.4.1. — Let A be a noetherian regular local ring withmaximal ideal m and residue Geld k. Let G be a finite subgroup ofAui(A),and let A° denote the ring of G-invariants of A. Suppose that :

1. the characteristic ofk does not divide the order of G,

2. G acts trivially on k,

3. A is a finitely generated A°-module.

Then A° is regular if and only if the image ofG in Aut^; (m/m2) is generatedby pseudo-reflections.

Remark 1.1.4.2. — An element cr of Aut^Y), where V is a vectorspace over k, is called a pseudo-reflection if rank(l — cr) < 1.

1.1.5. We consider a point x in M(P^ [To^N)}) satisfying the conditions1 and 2 of 1.1.3. If we find that the image of x in the quotient X^^^N)is a singular point, then we replace M(P^o(pnN)}) by its blow upin x. The second observation we make is that in doing so we do notloose any essential information. The reason for this is the following. Thecompletion of the strict Henselization of M(V^ ^p^N)}) at x is of theform A := W[[x, y]]/(/), where x and y are local moduli of source and target

36 BAS EDIXHOVEN

(cf. [9] Thm. 13.4.7 with V := (P, [I^AQ])). We can write / = fo +p/iwith :

/Q = (x^ — y)(x — y ^ ) TT {x^ — yp )p-l, if x is supersingular,a+6=na,6>0

/o = (x^ — y? )p~l with a + b = n and a, b > 0, if x is ordinary.

Because A is regular, /i has to be a unit in M^[[^,2/]], and {x^y) is asystem of parameters at x. The blow up of A at x can be covered bytwo open affines. One of these is Spec(A), where A = W[[.r,^.r]][v]/(/)with f(x,v) = f(x,xv), and / = /o + pfi- On this part of the blowup, the exceptional divisor (this is the preimage of x) is described by theequation : x = 0. It follows that ^M[[.r]]/(/) is the completion of A alongthe exceptional divisor.

We see from this that after blowing up we still know the p-fibre +action, and we can repeat the procedure until the quotient will be regular.Then we know the p-fibre of the quotient, and after contracting —1-curveswe have the minimal resolution.

1.1.6. There is still more to know, namely the global structure of thenon-reduced irreducible components (the description in [9] Thm. 13.4.7only gives the fibre of the non-compactified moduli scheme). Following theconstruction of the minimal resolution we will determine the conormalbundles of the reduced (a, -components. We express these bundles interms of Xo(N) 0z Fp. The statements of description 1.1.1.2 then followfrom the fact that the degrees of these line bundles are sufficiently large.

1.2. Xo(pnN) is regular at infinity.

1.2.1. In this section we do not need the restrictions p > 3 and n > 1. Let Pbe the moduli problem ([r(12)], [ro(M)]) on (Ell), where M is an arbitrarynon-negative integer, and let G = 071/2 (Z/12Z). Then by [9] Proposition8.11.7 we have an isomorphism :

r^/

M^)z((g))———^Tate(<7)/Z((g))/{±l}.

Taking the quotient by G and applying [9] Proposition 7.3.1, Theorem 7.4.2and Corollary 8.11.9 gives :

M([ro(M)])z((,))——^[ro(M)]Tate(,)/Z((,)).


This implies that the finite Z [[^-scheme Cusps ([ro(M)]) is the normaliza-tion of Z[[q}} in the finite normal Z((g))-scheme [ro(M)]Tate(9)/z((g)).

Applying factorization into prime powers ([9] Lemma 3.5.1) we getan isomorphism :

[roÂQlTate^/Z^))———'[ô(^n)]Tate(g)/Z((g)) ><Z((9)) [ro(A^)]Tate(9)/Z((9)).

We will first compute the two factors of the right hand side.

1.2.2. By 1.2.1 and [9] Thm. 13.6.6, Cus^FoO^)]) is the normalizationofSpec(Z[[g]])in:

Spec(Z((g))) [J Spec(Z((g))[^]/(^n - q)) [J

]J Spec(Z((9))M/(^(^-l/â-l))).a+b=na,6>0

Of course the normalization of Z[[q}} in Z((q)) is Z[[q}] itself. The normali-zation of Z[[q]} in Z^q))^}/^ -q) is Z^n.rJ/^71 -q), because it is finiteover Z[[q}} and regular. lfa>b then -l/â~l = x^, with x = z / q P " ' ,and we must find the normalization of Z[[q}} in Spec(Z((q))[x]/{^ pb(x))),which is Spec(Z[[q]}[x}/(^pb(x))), because it is regular and finite over Z[[q}}.Now suppose b > a > 1, then ^b-l/â~l = .r^"1, with x = z ^ ' ' / q , i.e :zp == qx. So we must find the normalization of Z[[q}} in :

(Z((9))M/^(â-l))[^]/(^-o -qx),

which is :

(ZNIM/^^"1))^]/^"0-^),-because it is regular and finite over Z[[q}]. We have proved the followingproposition.

PROPOSITION 1.2.2.1. — The Z[[q\]-scheme Cusps ([FoO^)]) isj'somor-phic to :

Spec(Z[[g]])]JSpec(Z[[^-"]])[J [J Spec(Z[C^][[g]]) ]Ja-\-b=na>b>0

]J Spec(Z[Cp«][M]M/(^!'-a-Cp-g)),a+6=n6>a>0

and hence is regular.

38 BAS EDIXHOVEN

1.2.3. We see from the proposition above that locally etale on Spec(Z[l/p])the Z [[^-scheme Cusps([ro(p71)]) is isomorphic to a disjoint union ofschemes of the type Spec(Z[[gl/•f]]), with f^. This implies that locallyetale on Spec(Z[lyW]) the Z[[g]]-scheme Cusps([ro(AOD ls isomorphic toa disjoint union of schemes of the type Spec(Z[[gl//]]), with f\N. Now weform the fibre product :

X := CuspsOro^)]) xz[[,]] Cusps([ro(AO]) xz Z[l/^].

The normalization ofX is then CmipsQroQ^AQ]) Xz Z[1/7V].

We see that in the decomposition of X that comes from the de-composition of Cuspsdro^)]) (see Proposition 1.2.2.1), the (n,0), (0,n)and (a, (^-components with a > b are regular. For the (n,0) and (0,n)-components one uses the fact that Xo^) —^ Spec(Z) is smooth at thecusps of these two components. The (a, (^-components of X with b > a > 0are locally etale isomorphic to :

Spec(Z[Cpo][^]][^]/(^-a-C^^)),

so they cause a problem. One way to get out of this is to let the involutionWpn act. The action of Wpn on [^o(pn)} is defined by :

^ ' . E ^ E ^ ^ ^ ' . E ^ E ^ .

This action induces one on XQ(pnN), and it interchanges the (a,&) and(6, a)-components of the p-fibre, and hence interchanges the (a, 6) and (6, a)-components of the scheme of cusps. Then it follows that the componentswith b > a > 0 are regular too. Of course, we could also have directly com-puted the normalization. Anyhow, we have proved the following theorem.

THEOREM 1.2.3.1. - The scheme Xo(N) is regular at infinity forallN.

1.3. Computations.

1.3.1. In this section we will do the computations that are outlined in 1.1.5.By the blow up X of a scheme X in a sheaf of ideals I we will mean theX-scheme Proj(©^>or1), as in [5] II 8.1.3. It is clear from this definitionthat blowing up commutes with flat base change : if Y —^ X is flat, thenthe fibered product Y x^ X is the blow up of Y in the inverse image sheaf


of ideals of I . We will use this in the case where X is the spectrum of anoetherian ring A, and Y is the spectrum of some completion A of A.

In our application, X = Spec(A), where A is a regular local ring ofdimension 2, and X is the blow up of X in the sheaf of ideals of the closedpoint x. Let rr, y be a set of parameters of A. According to [5] IV Proposition19.4.11, X is the closed subscheme of P^ = Proj(A[(7,y]) defined by theequation xV — yU = 0, and X is regular. The scheme X is covered by thetwo open affines -D+(i7) and D+{V), where

D^(V) = Spec(A[u}/(x - yu)\ D^U) = Spec(AH/(z/ - xv)).

On D^-{V) and D+((7), the inverse image ofx, which we call the exceptionaldivisor, is described by the equations : y = 0 and x = 0, respectively.

To be even more specific, we will have A = W[[x,!/]]/(/), wheref = /Q +j?/i, with /i a unit in IV[[.K,I/]] and /o as in 1.1.5. In that case,

A[v}/(y - xv) = W[[x,y]}[v}/(y - xvj) ^W[[x,xv\}[v\l{f\

where / = f(x,xv). Note that the completion of this ring along theexceptional divisor is given by TyMIMlA/). For the other open affine,we have an analogous formula. It follows that we can repeat this blowingup procedure at an arbitrary point of the exceptional divisor.

1.3.2. The case j = 1728, supersingular (hence p '= —1(4:)).

Let k := Fp. The point x of the moduli stack [Fo^)] we considercorresponds to : F71 : E —>- Ey where E is the elliptic curve over k givenby the Weierstrass equation : Y2 = X3 - X, which has Autjk(E) ^ Z/(4),with generator [i}^ : X —> —X,Y —> iY, with i € fc* of order 4. The groupAut^(x) is then cyclic of order 4 with generator :

prz

E ———— E

(-IFMpn

E ———> E

The elliptic curve over k[[t}} given by Y2 = X3 -X-K is a universal formaldeformation of E, and [i] acts on it by : X i- —X, Y i-> zV, t \-> —t

40 BAS EDIXHOVEN

(note that the equation is preserved). It follows that t is a coordinate ofthe universal formal deformation space of E, such that [i] act on t by :t^ -t.

Let x and y be the local moduli of source and target at x, bothcorresponding to t. Then [i] acts on x and y by : x \-> —re, y i- —y. Theuniversal formal deformation space of x is isomorphic to the completion ofthe strict Henselization of M(P,[^Q(pnN)}) at some point over x, whereP is some moduli problem that is etale over (Ell). As in 1.1.5, we haveA = W[[x,y\}l(f), with / = fo+pfi, fi a unit in W[[x,y}}, and

/o=(^1 - y ) ( x - y ^ 1 ) n (^-^-y-1.a+b==na,6>0

Figure 1.3.2.1 gives a picture of the special fibre.

The action of [z] on the cotangent space at x is not a pseudo-reflection,hence we blow up A in its maximal ideal. As explained above, the resultis covered by the two open affines D^(V) and D^.(U). We visualize thesituation in figure 1.3.2.2 :

v = 1, coordinates y, u = x / y

a>b n-ha=b

pi

•

(

\a =

^= b . a <b

u = 1, x, v = y / xFigure 1.3.2.2.

The action of [z] on u and v is given by : u ^ u, v ^ v. The locus offixed points is now exactly the exceptional divisor. It follows from these


formulas that, at every point of the exceptional divisor, the action of [i] onthe cotangent space is given by a pseudo-reflection. By Theorem 1.1.4.1,the quotient by [i] is regular. Since D^(V) and D^(U) are stable under theaction of [z], this quotient is covered by the quotients of Dj^(V) and D^.(U)by the action of [i]. Coordinates are given by y2, u and rr2, v, respectively.It is a matter of administration to express /o in these coordinates. Wevisualize the quotient in figure 1.3.2.3.

coordinates < = y2, u = —

u-t

-i+-2 if n S 0(2)

a=ba>b ( i fn=0(2)) a<b

/5 = 0

V ..-

.„ ^t=0

1

P1, selfinter-/ section —2,

/ multiplicity:/f p+1 -1+-

2 ' p

n-l

p 2 i f n s

coordinates 5 = a;2, v = v-

Figure 1.3.2.3.

The multiplicities of the (a, (^-components are unchanged. The selfinter-section of the quotient of the exceptional divisor can be obtained from theintersection theory in [2] exp. X.

1.3.3. The case j = 1728, ordinary (hence p = 1(4),), (a, b)-component.

We suppose that a ^ 0, b 0. This point x of the moduli stack [FoO^)],together with a generator [i] of Autfc(x) corresponds to :

ybpa771 ^ T?Hi ———> n/

ybpa

E ————> E

42 BAS EDIXHOVEN

The computation in this case is the same as the one in 1.3.2, the onlydifference is that the special fibre has only one component at x. There arenow three cases corresponding to a > ft, a = b and a < b. We draw threemore pictures.

p-'-i-i/=0

s = 0 P1, selfintersection : —2

multiplicity : •—— •p^"1

coordinates s = a;2, v = '

Figure 1.3.3.1. Case a > b > 0.

coordinates t = y2, u =

u=l

s = 0 t = Q P1, selfintersection : -2— i

multiplicity : p—1 .p°-1 = p""1 .^ ^ - - T - - ^ 1

v=l

coordinates s = x2^ v = y-x

Figure 1.3.3.2. Case a = b (if n = 0(2)).

MINIMAL RESOLUTION AND STABLE REDUCTION

coordinates i = y2, u = —y

43

t=Q P1, selfintersection : -2

multiplicity : ^—— •p0-1

^--i

u-< =0

Figure 1.3.3.3. Case 0 < a < b.

1.3.4. The case j = 0, supersingular, n = 0(2) (hence p = -1(3)).

This point of the moduli stack [Fo^)] corresponds to : F71 : E —> E,where E is the elliptic curve over k given by the Weierstrass equation :y2 == X3 - 1, which has Auik(E) cyclic of order 6, with generator[p\: X -^ C~1^, Y -^ -V, with C e A;* of order 3. The group Autfc(x) iscyclic of order 6 with generator p :

E E

[P] [p}

E E

We can choose again a coordinate t of the universal formal deformationspace of E such that [p] : t ^-> (t. Namely, in this case a universal formaldeformation is given by : V2 = X3 + tX - 1, and [p\ acts on it by :X t-^ C"1^? Y —Y' i t C^- As in 1.3.2, we take the local moduli x andy of source and target to be equal to t. Then the deformation space of x is^[M/(/o+^/i),with:

/o = (x^ - y)(x - y ^ ) IJ (^a-l - "y-1,a+&=0a,6>0

44 BAS EDIXHOVEN

and [p] acts on it by : x \-^ C^x^y \-> C,y. Since [p\ does not act on thecotangent space of x by a pseudo-reflection, we blow up in x. The situation isnow the same as in figure 1.3.2.2 : the locus of fixed points is the exceptionaldivisor, and at every fixed point, [p] acts on the cotangent space by apseudo-reflection. Figure 1.3.4.1 gives a picture of the quotient.

coordinates t = y3, u = —

= 0

a > b a =

V5 = 0

< -

= 6 a <b.., .< = 0

)

' p1--!

b u - < 3 =0

P1, selfinter. : -3

+1 -l+n-multiplicity : *—— • p 2<u

coordinates s = a;3, v = - •x

Figure 1.3.4.1.

1.3.5. The case j = 0, supersingular, n '= 1(2) (still p = —1(3)).

This point x of [Fo^)], together with a generator [p\ of Aut^(x) corres-ponds to :

EF"

E

W [P~

E E

From this we get the action [p\ : x ^-> C,x^y ^—> C,y on the local moduli ofsource and target. We blow up because the action on the cotangent spaceof the deformation space of x is not by a pseudo-reflection. The resultingsituation is pictured in figure 1.3.5.1.

We are left with two fixed points. There the action is not given by apseudo-reflection. We blow up in these points. Figure 1.3.5.2 gives the newsituation.

MINIMAL RESOLUTION AND STABLE REDUCTION

coordinates y, u = —V

45

a

fixed point ^^( x ^ ( x ^^action : < „ —^ v i—^ (v

> b a <

f \y = 0a:=0

k , A

b

n-l

multiplicity : 2 • p 2

PI, selfintersection : —1

Y^s fixed point

action J^f^1 ti 1—* L ~ U

coordinates .r, v = —xFigure 1.3.5.1.

n~l

P1, selfintersection : -1, multiplicity : 3p 2

/__________________\ P1, selfintersection : -3

multiplicity : 2p

Figure 1.3.5.2.

Now the action at the fixed points is given by pseudo-reflections, so wetake the quotient. The situation then looks like the one in figure 1.3.5.2,but the selfintersections -1 have turned into -3, and the -3 has turnedinto -1 (because the quotient map has degree 3 on this P^). We contractthis last one, see for example [10] Thm. 27.1. The selfintersections of theremaining two Pj^'s are -2. The final state is pictured in figure 1.3.5.3.

P1,selfintersection : — 2

n-i

Figure 1.3.5.3.

46 BAS EDIXHOVEN

1.3.6. The case j = 0, ordinary (hence p = 1(3)^, on the (a, b)-component.

We suppose that a / 0, & 7^ O. The point x together with the automorphism[p] corresponds to :

Eybpa

E

[p] [p]

Eybpa

E

Hence the action to consider is [p}^ '. x \-^ C,x,y \-^ (y. The computation isthe same as in 1.3.4, but now the divisor has only one component, and wehave the three cases a > &, a = b and a < b. We draw three pictures.

P--I

^ j - v=0

5 = 0 P1, selfinter. : -3

mult.:^!.^-1

coord. s = a;3, v = y

Figure 1.3.6.1. Case a > b > 0.

coord. t = y3, u = -

^ ^0 P1, selfinter. : -3^lt.:E^l.p-i= l.^-i

v = l

coord. s = .c3, t; = w-xFigure 1.3.6.2. Case a = b ( i fn == 0(2)).


coord. t = y3, u = —i—————————————

^ = 0 p1' selfinter. : -3——————————————— ,mult.'.^.p0-1

Figure 1.3.6.3. Case 0 < a < b.

1.4. Graph and local equations of Xo^N)^ Fy

1.4.1. By "graph of Xo(pnN) 0z Fp" we mean the data :

1. the irreducible components,

2. their multiplicities,

3. where they intersect.

The local equations then give the intersection numbers. We describe stepby step how to get all this. Let k = Fp.

1.4.2. Step 1. Take the disjoint union of n + 1 copies of Xo(N) 0z k, andlet $ be the morphism :

^: ]J Xo(AQ0z k-^Xo(N)^ka+6=n

given by :

the identity morphism if a > &,

(absolute Frobenius)6"" 0 id^ if a < b.

Give the (a, -component multiplicity ^(p1"111^6)).

1.4.3. Step 2. At every supersingular point x of XQ^N) 0z k contract<&~1 x to one point. The local equation at such a point is :

( " - y)^ - yP") ] (^°-1 - -Y-1 = 0.

a+6=na,6>0

48 BAS EDIXHOVEN

1.4.4. Step 3. Let x e Xo{N)(k) be a point with AuU(x) ^ Z/4Z (these arethe points in Xo(7V)(fc) corresponding to an elliptic curve with j-invariant123, together with a cyclic subgroup of order N which is invariant underthe automorphism of order 4). If x is supersingular (this corresponds top = —1(4)) then replace the unique point lying over x by figure 1.3.2.3. Ifx is ordinary (p = 1(4)) then let {xa,b\a + b = n} be the set of points lyingx, and replace Xa,b by figure

1.3.3.1. if a > & > ( ) ,

1.3.3.2. if a = ft,

1.3.3.3. i f 0 < a < 6 .

The multiplicities are left unchanged.

1.4.5. Step 4. Let x 6 Xo(AO(&) be a point with Aut^) ^ Z/6Z (inthis case j(E) = 0, and the cyclic TV-group has to be invariant under theautomorphism of order 6). If x is supersingular (p =. —1(3)) then replacethe unique point lying over x by figure

1.3.4.1, if nE=0(2 ) ,

1.3.5.3, if n = 1(2).

If x is ordinary (p =. 1(3)) then let {xo^|a + b = n} be the set of pointslying over x, and replace Xa,& by figure

1.3.6.1. i f a > & > 0 ,

1.3.6.2. if a =b ,

1.3.6.3. i f 0 < a < b .

1.5. Xo(p2), for example.

1.5.1. The case Xo(pN) is done in [3] VI §6, see also the appendix of [11].We will apply section 1.4 in the case Xo(p2), that is, we give the results.


1.5.2. The casep = 1(12). Writer = 12A:+1. The number of supersingular^'-invariants is A;, and j = 0,123 are both ordinary. Figure 1.5.2.1 gives apicture of Xo(p2) 0z Fp-

p-1j = 1728 2 @ genus = -S^-l

there are^ of these.

j =0

p-13 ^

Figure 1.5.2.1.

In this picture the numbers in circles denote selfintersections, the othernumbers denote multiplicities. The irreducible components are just pro-jective lines over Fp. From this picture one can compute the genus of^o(p2) z Fp5 and hence the genus ofXo(p2) 0z C (cf. [2] X, compare [8]V, exc. 1.3). It is a nice verification to show that it is indeed 12A;2 — 3k — 1.

1.5.3. The case p = 5(12). Write p = 12k + 5. See figure 1.5.3.1.

1 p-1 1

there arek of these.

j == 1728

J = 0

genus = 12k2 + 5Jb.

Figure 1.5.3.1.

50 BAS EDIXHOVEN

1.5.4. The case p = 7(12). Write p = 12k + 7. See figure 1.5.4.1.1 p-1 1

^=1728 p-^-

j = 0


genus = 12JP+9Jb+l.

Figure 1.5.4.1.

1.5.5. The case p = 11(12). Write p = 12Jfc + 11. See figure 1.5.5.1.

\ p? 1 }

j = 1728

3 = 0


genus = 12Jk2 + 17k + 6.

Figure 1.5.5.1.

1.6. Global structure of Xo^TV) <8>z Fp.

1.6.1. In this section we determine the global structure of the non-reducedirreducible components in XQ^N) 0z Tp and ~M(P, [Fo^)]) 0z Fp interms of Xo(N) <S>z Fp and A^CP) 0z Fp. We will apply the followingtheorem.

THEOREM 1.6.1.1. Let X —> S be an integral regular two dimensionalscheme flat and proper over the spectrum of a discrete valuation ring. Lets be the closed point of S, and let Xs be the special fibre. Let Y be an


irreducible component ofXg and suppose that Y is smooth over k := k(s),let N be the multiplicity ofY in Xg, and let 1 be the sheaf of ideals ofYin X. Let Yn be the n-th infinitesimal neighborhood o fy=Yo m X. Finallysuppose that there is given a projection :

YN-I^YQ idyo,

and that -YY > 2.genus(Y) — 2. Then there exists an isomorphism :

YI —^ SpecyJ0y,©Z/Z2©...©ZN-l/^)\ /

Yo

(this means that YN-I -/- projection is isomorphic to the N — 1-th infinite-simal neighborhood of the zero section in the normal bundle on Yo).

Proof. — Let Z —> YQ be the normal bundle of YQ in X :

Z = Specy,(Symm(J/T2)) = SpecyjOy, @I/I2 @I2/!3 ©...).

For N = 1 the proof is trivial, hence we assume that N > 1. The projectiony^v-i —> Yo gives us a projection Vi —> Yo. This results in an isomorphismof sheaves of rings Oy^ ^ Oyo l/l2. This means that we have a closedimmersion :

9i '.Yi -^ Z\ /

Yo

We have to show that this closed immersion can be lifted to Y^-i- In orderto do this, we use obstruction theory as in [7] §4, i.e. we apply [6] exp.Ill Cor. 5.2 + the first alinea following this. Suppose that we have a Yo-morphism g^ '. Yn —> Z lifting ^i, with 1 < n <N — 1. Note that such aOn is automatically a closed immersion. We will now show that gn can belifted to a Vo-morphism fi^+i ; Yn-^-i —^ Z. Let P(gn) be the sheaf of setson Yo with

P(9n)(U) = {liftings g^\ of gn over U}

for every open subset U of Yo. According to [6] exp. Ill Cor. 5.2, P(gn) isa G-torsor, where

G^g^Y^^o^^/I^2.

We have now to show that P(gn) has a global section, i.e., that P(gn)is trivial as a G-torsor. The totality of G-torsors (up to isomorphism) is

52 BAS EDIXHOVEN

parametrized by ^(YQ^G), hence a sufficient condition for the existenceof a lifting gî is that H^Vo, G) = 0. Therefore we compute G. Note tnatsince jî/p^ ig a Oy-module, 6' is a Oy -module too. Let go :Yo —^ Zbe the zero section. Then we have :

G = g^Z/Y^Y Ôy I^/I^2 = (I/I2)" ^Qy (I/I2)^1 = (I/I2)^.

Note that G is an invertible Oy-module. The calculation of G gives :deg(G) = n deg(Z/Z2) = n(-YY). We can write X, = NY + R asdivisors on X, with R effective and not containing Y. Then it follows that0 = Y.Xs = N(Y.Y)+Y.R. Since Y.R is non-negative, -Y.Y is non-negativetoo. We find that deg(C?) > -Y.Y > 2.genus(V) - 2. By Serre duality onY this implies that ^(YQ.G) = 0.

1.6.2. It remains to determine the conormal bundles of the irreduciblecomponents of the curves Xo(pnN) z Fp and M(P, [FoO^)]) 0z Fp,and to show that the conditions ofThm. 1.6.1.1 are satisfied. Let C denotethe conormal bundle of a component Y of type (a, b) with a, b > 0. Theintersection of Y with the other components gives us ^(^"""^'^deg^),and hence the degree of C itself.

In the rigidified case [9] Thm. 13.4.7 tells us that M{P, (a, 6)-cyclic)is isomorphic to the ^(j/11"1^^))-!!! infinitesimal neighborhood of the graphof F^"6! (Frobenius iterated |a - b\ times) in M(P) Xz M(P) Xz Fp.This gives us an isomorphism :

^\M{P)Wp————^MiP^Fp)î-b\

and hence :r——ô1 \®P ( D }L / ^ ^ M W W p ) ^^

with D a cuspidal divisor. Since we know the degree of £, we know thedegree of D. A modular interpretation of the cusps would be helpful at thispoint, but since we do not have one, we proceed in a not so elegant way.

For V representable finite etale Galois over (Ell/Zp) the divisor D hasto be equally distributed over the cusps. Let us compute the degree of C onthe (a, (^-component Y in this case. Since this degree for (a, 6) is the sameas for (&,a), we may suppose that b > a > 0. Let m be the multiplicity ofV, then we have :

deg(/:) = -YY = m-\Y-mY) = m-^Y.Xs-mY) = m-\Y. mcC\C^Y


where the sum is over the components C of Xs other than V, and wherethe me denote their multiplicities. Hence :

mdeg(£)= mc(Y.C) = ^ m^slâ'.C^Y 0<a'<n

a'â

where rria' denotes the multiplicity of the (a',n - a'^component, wheres is the number of super-singular points on V, and where la,a' is thelocal intersection number of the (a, b) and (a',n - a'^components at asupersingular point. According to [9] Thm. 13.4.7, ma' is the minimum of^(p^) and ^(p^^). From the local equation for XQ^N) 0z Fp at sucha point (see [9] Thm. 13.4.7) it follows that

( pn-2a ^ o < a' < a < jlaâ' = p71-20' if a < a' < j

1 if t<a ' .

Evaluating the sum gives :

deg(£) = 2m-15p7^-a-l = 2g p6-0.p- 1

We can compute s using the Kodaira-Spencer isomorphism of [9] Thm.10.13.11 and the Hasse invariant [9] Thm. 12.4.3 :

^w, (cusps)——^, O^^^(,.,.)__^(P-I).

From these two formulas we see that :

25

= ("^(p)^ (^P8))-p - 1

It follows that for P representable finite etale Galois over (Ell/Zp) we havethat :

1.6.2.1 C——-(n^^ (cusps))^-

Let us now consider the moduli problem (P,[ro(N)]). The morphismfrom this moduli problem to P is finite and etale. It follows that themorphism from M(P,[To(N)}) to M(P) can only be ramified along thecusps. From [9] Thm. 8.6.8 if follows that locally etale the morphism fromCusps^, [ro(AQ]) to Cusps(P) is of the form :

Spec(Zp[[(^]]) -. Spec(Zp[[g1^]]),

|a-b|

54 BAS EDIXHOVEN

with d and e both prime to p. This implies that the pullback of theconormal bundle of the (a, (^-component of.M(P, [Fo^)]) is the conormalbundle of the (a, -component of M(P, [To^N)]), and that the pullbackof "^(P)0F, ^P8) is "^(P,[roW])^ ^P^ It follows that we have

the isomorphism 1.6.2.1 for P replaced by {P, [ro(AQ].

It is now obvious that the only non-obvious condition of Thm. 1.6.1.1(-Y.Y > 2g(Y) — 2) is satisfied in the rigidified case.

1.6.3 Following the construction of Xo(pnN) as the quotient of a blow upofA7(P, [To^N)]) we get on the (a, -component of "KQ^N) 0z Fp :

1.6.3.1 ^——^("XoW^ (cusps))^10"^),

with D a divisor supported on the points in the finite part which have extraautomorphisms. In order to compute D, we have to see what happens to Cand Q1 when we blow up or take a quotient. Let P be a point on the (a, 6)-component such that its image in XQ^N) has extra automorphisms. Afterd blow ups in P, the conormal bundle of the (a, (^-component is isomorphic(Zariski locally at P) to (0— p ^'"""(dP), and the stabilizerGp of P acts on the cotangent space at P by pseudo-reflections. Let z be alocal coordinate on the (a, -component at P, and let e be the order of Gp.Then u := z6 is a local coordinate on the quotient of the (a, (^-component,and (cb)^0"61, as a section of (^—p [ro(^)])®F )<g>plo~61 (dp)^ has a zero at

p of order d -h (e — I)?!""6!. It follows that on the quotient it has a zero of

order ——v———-———. The results in the various cases are summarized inethe following table, which gives the multiplicity of P in D.

Table 1.6.3.2

j = 1728J = 0 ordinary (p = 1(3))

supersingular (p = —1(3)) n = 0(2)n =E 1(2)

(pl»-b| + 1)/2

(2p|a-i>l + 1)/3

(2pl»-<>l + i)/32(pl'»-"l +1)/3

If is now obvious from the isomorphism 1.6.3.1, plus the fact that D in Table1.6.3.2 is an effective divisor, that the condition "-Y.Y > 2g(Y) - V ofThm. 1.6.1.1 is satisfied.


2. THE STABLE REDUCTION OF X^N) AT p > 5

2.1. The result.

2.1.1. THEOREM. — Let P be a representable finite etale moduliproblem over (EU/W), where W is the ring of Witt vectors over k := Fp.We suppose that p > 5. Then the curve M(P, [^(p2)}) over W acquiresstable reduction over W[z]/(z(<p2~1^2 - p). The special fibre of the stablemodel is as follows :

k supersingularpoints

(M(P)^wk)^~^ \ 4 'M(P)^>wk j4(P)(^wk

A7(P,[Wp)]/±l)

The horizontal curves are all isomorphic to the smooth model C ofthe singular curve given by the equation : yP^~1 = x(x - 1)P~1. The pointson C over x = 0, x = oo are glued to the outer components. The two pointsover x =1 are glued to the two middle components.

2.1.2. THEOREM. — Let k := Fp with p > 5, W := W(k). Let?be relatively representable finite etale over (EllyW), e.g. P = [ro(N)} with(p, N) = 1. Then the curve M(P, [To(p2)}) over W acquires stable reductionover Ty[^]/(^^2-1)/2 -p). The special fibre of the stable model is as follows :

56 BAS EDIXHOVEN

)®w t)C")

M(P,\ /[wi/

CT(P)

^±1)

011

< supersingularpoints

M(P) ®iv Jb

Tie horizontal component over a supersingular point x on M(P) 0 kis isomorphic to the smooth model of the singular curve given by :

y^1 = x(x - \Y~1 if Autjfc(x) ^ Z/(2)^(p+i)/2 ^ _ ^p-i ^ Autfc(x) ^ Z/(4)y(p+i)/3 ^ ^ _ ^p-i ^ Autfc(x) ^ Z/(6).

TAe points over x = 0, rr = oo are g'2ued ^o the ou^er components. The twopoints over x = 1 are giued to the two middle components.

2.1.3. The proof of these theorems is given in sections 2.2, 2.3 and 2.4.

2.2. Formal computations.

2.2.1. For simplicity we work over W := W(k), where k := Fp. Let Pdenote a representable moduli problem on (Ell/lV), which is finite etaleover (Ell/WQ. Let XQ denote ~M{P, [ro(jp2)]). This is a regular IV-scheme.The finite part of its special fibre is described in [9] Thm. 13.4.7. This fibreis a Cartier divisor on XQ. To get normal crossings it suffices to blow uponce in each supersingular point. Let Xi denote this blow up. The special


fibre of X\ is a Cartier divisor with normal crossings. Its pictures is infigure 2.2.1.1.

1 p - 1 1

exceptional curves .

=p+i

components : (^ Q\ (1,1) (0,2)

Figure 2.2.1.1. Picture of X\ 0^ fc. The numbers denote the multiplicities.

2.2.2. The least common multiple of the multiplicities of the componentsof Xi 0M. k is (p2 - 1)/2. Let W^ := W[t]/(t^2-1^2 - p),ând let t € W^denote the image of t. Define X^ := X-i <S>w W\, and let X^ —»• X^ be itsnormalization. This X^ will be our stable model, and therefore we wantto compute its special fibre. First we do this formally, i.e. we compute thecomplete local rings of X^.

Let x be a closed point of Xi ®w k = X^ <S)Wi k. Then Ox^ is ofthe form W[[x,y\}|(xayb - p), with a,b (E {0, l,p - l,p -h 1}, a ^ 0 and{0,6} ^ {l ,p- 1}. This gives 6x^ ^ W^x^]}/^ - 2-1)/2). Wemust find its normalization, since (by [5] IV, 7.8.2 and 7.8.3 (vii))

X2 Xx2 Spec(0x2,x) = Spec (this normalization).

We treat the different cases separately.

2.2.2.1 The case b = 0, a = 1.

The ring in question is regular, hence normal.

2.2.2.2 The case b = 0, a = p - 1.

We can write :2-i.^-D/2^ Jj (^ - C^4-1)/2).

Ce^-i(î)X1

From this it follows :

X; xx2 Spec(Bx2,J = U Spec(î[[^,y]]/(rc - (p+l)/2)).Ce/^-i(î)

58 BAS EDIXHOVEN

In this case the normalization is regular too.

2.2.2.3 The case b = 0, a = p -hi.

Just as in (2.2.2.2) we get :

X2 xx, Spec(Ox.,J = ]J Spec(TVi[[rr,i/]]/Or - C^-^2)).Ce/^+iW)

2.2.2.4 TAe case b = 1, a = p + 1.

We can split the base change Spec(Wi) —> Spec(iy) up in two. LetW'2 := W^]/^^1 — p), and let z denote the image of z in W^. Firstwe compute the normalization of W^x^y^/^x^y ~ ^p+l). To do this,we blow it up along the ideal (rr,^), i.e. we set z := ux. This gives usW^x^yû^/^z—ux^y—u^1), or, equivalently : W^[[x,y]}/(z—ux). Afterthe base change Spec(lVi) -^ Spec^) we get Wi[[x,u]}/(ux - t^~1^2).Its singular locus is given by u = x = t = 0, so it is the closed point. Theexceptional divisor of the minimal resolution is a chain of projective lines.This implies that TVi[[a:,^]]/(ita; — ^P~1)/2) is normal. We have :

X, xx, Spec(Ox.,J = Spec(lViM]/(^ - -1)/2)),

with y = p+l. In this case the normalization is not regular.

2.2.2.5 The casea=p- 1, b=p^-1.

We split the base change Spec(Wrl) —> Spec(Ty) up in three. Let W^ :=W[z]/(z2 —p), and let z denote the image of z in W^. We write xp~lyp~^l —z2 = (^(p-i)/2y(p+i)/2 ^ ^(p-i)/2^(p+i)/2 _^ ^ T^ normalization ofSpec(W^[[x,y}}/(xp~lyp^l — z2)) is isomorphic to the disjoint union of twocopies of Spec(W^[[x,y}}/(x(<p~1^2y(-p+1^2 — z)) (here one uses that eitherp—}- or p+]- is odd). Let W^ := W^[s\l{s^1^2 - z), and let s denote

the image of s in Ws. We blow up W3[[x,y]}/(x^p-l)/2y(p+1^2 - 5^+1)/2))along (y,5), i.e. we set s = uy. This gives lV3[[.r,y,n]]/(5 — uy,x^F~1^2 —y(p+i)/2^ g blow it up along (re, it), i.e. we set x = vu. This givesW3[[y,u,v]}/(s-uy,v(p-1^2-u) = tM^]]/^-^"1^2!/). We blow up^[b/^]]/^"1^2 - 'y^"1^2^) along (t,v), i.e. we set ^ = uv. This givesWi[[2/,'y,iA]]/(< — m;,'^"1^2 — y) = Wi[[u^v}]/(t — uv), which is regular.Hence we have :

^Xx.Spec(Bx.,J = Spec(W,[[û}}/(tûv))\^Spec(W,[[v,u]]/(t-uv)^


2.2.3. Now look at figure 2.2.1.1, and consider X^ (S)\Vi k —^ X\ ®w k. Fromthe computations in (2.2.2) it follows that :

1. over each of the outer two vertical components lies one component,and the map is an isomorphism,

2. over each of the horizontal components lies one component, the maphas degree p + 1 and ramifies in the following way :

X)£±J-

, X)^1 2! X)^1r^NEzLip+ll ^) 2

——•——————•——————•——3. over the (l,l)-component there are one or two components, and the

ramification is as follows :

X)^2 X)^p - 1 ! x)^ x)^

It follows that there are two possible pictures

Our task it :

• to determine the horizontal components,

• to decide between the two pictures, and to determine the verticalcomponent (s).

2.2.4. We will now compute the horizontal components. Let x be asupersingular point of XQ ^w k. We write 0Xo,x = ^[[^^^/(/o -t-P/i)»with /o = (^p2 - y)(x ~ y)p~l(x - yP2), and /i a unit of W[[x, y}}. We blowup 0Xo,x ln ^s maximal ideal. As explained in 1.3.1, an affine open partof the result is W[[v,xv]][v]/(f), with f(x,v) = f(x,vx) and /= fo +p/i.Taking the completion of this ring along the exceptional divisors gives

60 BAS EDIXHOVEN

^MN]/(7). Let TVi := W[z}/(z^1 -p), and let z denote the imageof z in Wi. After extension of scalars to W\ we have : WiH^HA./o +pfi)'We blow this up along the ideal (x,z), i.e. we write x := uz. We get :

w^^/^ûzr^-vî-vy-î-vpûzr2-1)^/^where the completion is with respect to the principle ideal (uz), andfi = /lO^^) = fi(uz,vuz). The curve we are looking for is describedby the equation z = 0. Substituting z = 0 in the equation above yields^+i(--y)(l _ y)P-i + /i (0,0) = 0. It follows that (in new coordinates) a(singular) model of our smooth curve is given by the equation :

y^1 =x(x-l)p-l.

2.3. Determination of the vertical components.

2.3.1. In this section we determine the "vertical components(s)" that arisein the computations of 2.2.2. These vertical components are evolutionproducts of the (1,1 component of ~M(P,[To(p2)}) <^w k. They alreadylive in the normalization of ~M(P, [Fo(p2)]) ^>w Wi, where W\ is ramifiedover W of degree p - 1. Since they are stable, they will not evolve anyfurther, whatever base changes and normalizations we let them undergo.We will determine them in two different ways.

2.3.2. The first method. Let W^ := W[^}, W^ is ramified over W of degreep(p-l).. A remarked in 2.3.1, we must determine the irreducible componentsof the special fibre of the normalization of ~M(P, [ro(p2)]) ^w Wi. Thisnormalization is :

^(PWr^/G, with € ? = / ( * ^ I cSW/p^).

We see this as follows : the normalization of ^\4(P,[r(p2)]) (g>^ W^ isUce/^(Wi) (^ PW^"]^ now take quotients.

Note that the order of G is divisible by p. Therefore the special fibreof the quotient may not be the same as the quotient of the special fibre.However the morphism :

2.3.2.1. (A7(P, W)^}) ®w, k)/G -^ (M(P, Wr^/G) 0^ k


is_ surjective and radicial ([9] A7.2.1). The Ad(P) ®w ^-schemeM^P^n?2)^}) 0^ k is described in [9] Thm. 13.7.6. It is the disjointunion, with crossings at the supersingular points, of the ~M(P) <^w k-schemes M(P, [Exig^2^)]), indexed by P^Z/Q^2)). The group G actsonPl(Z/(p2))by:

ft s \( x : y ) I p ^ I ^(tx'.sx+t^y).

This action has 4 orbits. They are generated by (0 : 1), (1 : 0), (p : 1) and(dp : 1), with d a non-square in Fp. The first two orbits correspond to the(2,0) and (0,2)-components of [ro(p2)] 0fc. The other two give the curve(s)we are looking for. At this moment we see that the (1,1 component getsreplaced by two components. Hence the first of the two pictures in 2.2.3 isthe correct one.

The stabilizer group of (p : 1) and (dp : 1) is { ( t s ) e G; t2 = ll.I V 0 " / J

This group acts on M(P, [Exlg(j?2,2)]). The subgroup /[1 *")l acts

trivially. We are left with the group (±1 + pZ)/(l + j^Z) C (Z/j^Z)*.There is an exact sequence :

{1} -. (1 +pZ)/(l +^Z) (±1 +pZ)/(l +p2Z) -^ {±1} -^ {1}.

_We want to determine the quotient of the ~M{P) 0^ Jfc-scheme.^(^[Exig^2^)]) by the kernel. Consider the following diagram from[9] Thm. 12.10.6. :

M^P^wk^^W)}) ——. M(P,[Exig(p^2)})

^ Pr \ pr<2 pr

M((P (S)w A;)^"2)) -F^ M(P^wk)

62 BAS EDIXHOVEN

By [9] Thm. 12.6.1(3) we get an isomorphism :

M((P ^w k^^Mp)}) ——^ ^((^[ExIg(p2,2)])/((l+^Z)/(l+p2Z))

\ pr2 i/ pr

~M{V ®W k)

Taking the quotient by {±1} gives :

M((P^w k^^Mp)])^!}) —— A^MExIg^)])/^! 4-pZ)/(l +p2Z))

\ pr2 y pr

~A4(P <S>w k)

This is a component of the source of the morphism 2.3.2.1. Bothsource and target of this M(P) ^w A;-morphism are reduced. A degreeconsideration over .M(P) <S)w k then implies that the component in thetarget is M(P, [Ig(p)]/{d=l}). We have proved :

THEOREM 2.3.2.2. — In the stable reduction ofM(P,[To(p2)}) the(1, l)-component gives two components, each M.{V)^w k-isomorphic toM(P^g(p)}/{±!}).

2.3.3. The second method. This method is a global version of the formalcomputations of (2.2). Consider M(P,[To(p2)]). We blow it up in thesupersingular points. Let X be the formal completion of this blow up alongthe (1,1 component. We claim the existence of a global function / on Xsuch that f2 = p- unit. The divisor of such a / must be (cf. 2.2.1.1) :

(P-l)/2±

(P+1)/2

(P+l)/2


This can be checked in the special fibre itself. Let Z be the ideal of thereduced (l,l)-component of X, and let Xn be the subscheme of X definedby I^. Then Theorem 1.6.1.1 gives :

Xp_2——^Spec^(Oxo © r © ... © /7-2),

with C := I / I 2 the conormal bundle of XQ in X. The isomorphism 1.6.2.1plus the Kodaira-Spencer isomorphism and the Hasse invariant give :.C——^18' 1. Let A € ^^(Xo) be the Hasse invariant. Its divisor isexactly the supersingular locus. Let

A(p+i)/2 := A^1)/2 e a^1^-1)/2^) -^(Xo).

Then A(p+i)/2 is a global function on Xp_2 which has the right divisor.We will now try to lift this function to one on X which still has the rightdivisor.

Let J be the ideal of the reduced horizontal components. Let n >p— 1. We suppose that we have a lifting fn-i to Xn-i. We want to lift it toa fn on Xn, with fn a global section of (Z7^1 + j(p+i)/2)/jn+i. Considerthe diagram :

p+i p+i(Jn+l+J- 2 )^n+l _, (Jn+J- 2 ^n

II IIP+l p+1 p+1 p4-l p+1 p+1

^ 2 •Z71/^ 2 •Z^1 -. J 2 /^ 2 •Z^1 -^ J 2 /J 2 .J^

P+1

/ / r 2 .7-n\|v^ -L ^|Xo

Since H^Xo, J^+^^lxo) = {0} for n > (p-1)/2, the required fnexists. This gives us a / on X with f2 = p- unit, and /|xp-2 = A(p+i)/2- Thetwo vertical components of the stable reduction are obtained by extractingthe {(p - l)/2)-th root of /. This amounts to extracting the ((p - 1)/2)-root of the Hasse invariant. By [9] Thm. 12.8.2 we see that the verticalcomponents are M(P, [Ig(p)]/{=bl}).

2.4. The coarse case.

Let P be relatively representable finite etale over (EllyW). Let V berepresentable finite etale Galois (with group G) over (Ell/TV). Then by

64

construction :

BAS EDIXHOVEN

M(P, [ro(p2)]) = M^, [ro(^)])/G.

Let X^/W^ be the stable model of ~M(p,V, [ro(p2)]) as constructed in2.2.2. From the construction it follows that G operates on X^fW\. There-fore X^IG is a model ofM(P, [ro^2)])^^^!. The only singularities of itsspecial fibre are ordinary double points. We will call this model the stablemodel of M(77, [ro(j?2)]) over TYi. Strictly spoken this is not always true,for the genus can be less than two, and also the condition on the rationalcomponents need not be satisfied. Following the computations of 2.2.4 and2.3, the statements in Thm. 2.1.2 about the components of {X^/G) 0^ kare easily checked.

2.5. Examples.

2.5.1. We apply Thm. 2.1.2 to Xo(p2). We draw pictures in the variouscases for p mod 12. The numbers denote the genera of the components.

#=k

total genus = 12k2 - 3k - 1

03Jk2

)

2 -3k^^

+1

0)

6Jk

ci,

6k

Figure 2.5.1.1. p= 12k+1.3k2-k

0 ^^

#=Jb

.7=0

6k +2

6k+2

2k

total genus = 12k2 + 5JkFigure 2.5.1.2. p = 12k + 5.


#=^

j = 1728

total genus = 12Jb2 + 9Jk + 1

C33fc2

^"^ C3

C 1 i 0

/? I i Q

<•) I. i I

Figure 2.5.1.3. p = 12fc + 7.

0

^ - 1728 ———

j - 0 ———

3Al;2+2i^^ C)

^* L i K.

0 L i 0

total genus = 12JP + 17Jk + 6

Figure 2.5.1.4. p = 12k + 11.

2.5.2. The case Xo(72). We have the picture :

0 0 0 0

The elliptic curve over k is the smooth model of : y2 = x{x — I)2. Theautomorphism of order 4 : {x,y) ^ {x,iy) fixes the regular point (0,0),hence the elliptic curve has j = 2633 = -1(7). From Table 1 of [1]we extract : Xo(72) over Q is the elliptic curve given by the equation :y2 + xy = x3 — x2 — 2x — 1. Its ^'-invariants is —S^3, and indeed-S^3 = -1(7).

66 BAS EDIXHOVEN

2.5.3. The case Xo(352). Let p = 5. We have the picture :0 0 0 0

The genus 2 curve is the smooth model C of: y6 = x(x-l)^. Another modelof C is : y2 = x6 - 1. In these coordinates a basis for the global differentialforms is : 0:1 := ( d x ) / y , := x(dx)/y. The involution W : (x,y) (-x,y)acts by : lV*(ji = -0:1, W*^ = ^2. The quotient C/W is the ellipticcurve : v2 = u3 - 1, it has j = 0. It follows that up to isogeny the jacobianof C is the product of two elliptic curves, one of which is isogenous to thecurve with j = 0. Since j = 0 is the only supersingular j-invariant mod 5,it has j = 0. The automorphism (x,y) (x^^iyx^3) of C interchangesthe two eigenspaces of W acting on the differentials. It follows that bothelliptic curves have j = 0. Table 1 of [1] gives us two strong Weil curveswith conductor 3.52 having potentially good reduction at 5 :

75A: y 2 + y = x 3 - x 2 - 8 x - 7 j = -212.3-1.52 = 0(5)75B : y2 -h y = x3 + x2 + 2x + 4 j = 212.3-5.5 EE 0(5)

BIBLIOGRAPHY

[1] B.J. BIRCH and W. KUYK, Modular functions of one variable IV, SpringerLecture Notes in Mathematics, 476 (1975).

[2] P. DELIGNE and N. KATZ, Seminaire de geometric algebrique 7 II, SpringerLecture Notes in Mathematics, 340 (1973).

[3] P. DELIGNE and M. RAPOPORT, Les sch^mas de modules des courbes elliptiques.In Modular Functions of One Variable II, Springer Lecture Notes in Mathematics349 (1973).

[4] B.H. GROSS and D.B. ZAGIER, Heegner points and derivatives of L-series InventMath., 84 (1986), 225-320

[5] A. GROTHENDIECK, Elements de geometric algebrique, Ch. I, II, III, IV,Publications Mathematiques de 1'I.H.E.S, 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967).

[6] A. GROTHENDIECK, Seminaire de geometric algebrique I : Revetements Staleset groupe fondamental, Springer Lecture Notes in Mathematics, 224 (1971).

[7] R. HARTSHORNE, Curves with high selfintersection on algebraic surfaces, Publi-cations Mathematiques de 1'I.H.E.S, 36 (1969).

[8] R. HARTSHORNE, Algebraic geometry, Springer Graduate Texts in Mathematics,52 (1977).

[9] N.M. KATZ and B. MAZUR, Arithmetic moduli of elliptic curves, Annals ofMathematics Studies, Princeton University Press, 108 (1985).


[10] J. LIPMAN, Rational singularities, with applications to algebraic surfaces andunique factorization, Publications Mathematiques de 1'I.H.E.S, 36 (1969).

[11] B. MAZUR, Modular curves and the Eisenstein ideal, Publications Mathematiquesde 1'I.H.E.S, 47 (1977), 33-186.

[12] J.F. MESTRE, Courbes de Weil et courbes supersingulieres, Seminaire de theoriedes nombres 1984-1985, Universite de Bordeaux 1.

[13] A. PIZER, An algorithm for computing modular forms on FQ(N), Journal ofAlgebra, 64 (1980), 340-390.

[14] A. PIZER, Theta series and modular forms of level ^M, Compositio Math., 40(1980), 177-241.

[15] J.-P. SERRE, Colloque d'algebre, 6-7 mai 1967, ENSJF.

Manuscrit recu Ie 4 mai 1988,revise Ie 5 janvier 1990.

Bas EDIXHOVEN,University of California at BerkeleyDept. of MathematicsBerkeley, CA 94720 (U.S.A.).

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