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Proc. London Math. Soc. (3) 95 (2007) 545–566 C

e2007 London Mathematical Society

doi:10.1112/plms/pdm013

DRINFELD MODULES AND TORSION IN THE CHOW GROUPS

OF CERTAIN THREEFOLDS

CHAD SCHOEN and JAAP TOP

Abstract

Let E→Bbe an elliptic surface deﬁned over the algebraic closure of a ﬁnite ﬁeld of characteristic greater

than 5. Let Wbe a resolution of singularities of E×BE. We show that the l-adic Abel–Jacobi map from

the l-power-torsion in the second Chow group of Wto H3(W, Zl(2)) ⊗Ql/Zlis an isomorphism for almost all

primes l. A main tool in the proof is the assertion that certain CM-cycles in ﬁbres of W→Bare torsion, which

is proven using results from the theory of Drinfeld modular curves.

Introduction

Let ¯

Fbe an algebraic closure of a ﬁnite ﬁeld Fof characteristic p. For a smooth projective

variety V/F, let Zr

rat(V¯

F)⊂Zr

alg(V¯

F)⊂Zr

hom(V¯

F) denote the groups of codimension ralgebraic

cycles on V¯

Fwhich are rationally (respectively algebraically, respectively homologically)

equivalent to zero. The group CHr

hom(V¯

F):=Zr

hom(V¯

F)/Z r

rat(V¯

F) is a fundamental invariant

of V¯

Fabout which very little is currently known. For each prime number ldistinct from pthere

is an l-adic Abel–Jacobi map

ar

V,l :CHr

hom(V¯

F)⊗Zl−→ H2r−1(V¯

F,Zl(r)) ⊗Ql/Zl.(0.1)

In the cases r= 1 and r= dim(V), one knows that ar

V,l is an isomorphism. It is interesting to

ask the following.

Question 0.1. Is ar

V,l an isomorphism for all rand all l=p?

In this paper we consider the map αr

V,l obtained by restricting ar

V,l to the torsion subgroup

of CHr

hom(V¯

F)⊗Zl. Note that CHr(V¯

F)tors ⊗Zlis in fact a subgroup of CHr

hom(V¯

F)⊗Zl

whenever H2r(V¯

F,Zl(r)) is torsion free which occurs for almost all l[13]. Quite generally,

α2

V,l is injective (cf. §1). The main purpose of this paper is to show that α2

V,l is surjective for

almost all lfor a particular class of threefolds which we now describe.

Let Bbe a smooth, geometrically irreducible, projective curve over F. Let :E→Bbe a

non-isotrivial elliptic surface. Suppose that Vis a smooth projective variety which is birational

to E×BE.Ifis semi-stable, we may construct such a Vby blowing up the reduced singular

locus of E×BE. In order to be able to deal with the non-semi-stable case as well, we assume

that char(F)>5. Now Vexists by a general theorem on resolution of singularities [1].

Theorem 0.2. For all except possibly ﬁnitely many primes l, the l-adic Abel–Jacobi map

gives rise to an isomorphism

α2

V,l :CH2(V¯

F)tors ⊗Zl−→ H3(V¯

F,Zl(2)) ⊗Ql/Zl.

Received 13 September 2005; revised 19 December 2006; published online 21 June 2007.

2000 Mathematics Subject Classiﬁcation 14C25 (primary), 11G09, 11G16 (secondary).

The ﬁrst author gratefully acknowledges support from NSA (MDA904-97-1-0041), NSF (DMS-9306733, DMS-

9970500, DMS-0200012), and a Duke University Planning Grant for International Research.

546 CHAD SCHOEN AND JAAP TOP

We now give a brief outline of the proof of Theorem 0.2 together with an overview of the

contents of each section of the paper.

The ﬁrst section is devoted to the injectivity of α2

V,l. This is established by relating α2

V,l

to Bloch’s cycle class map on torsion cycles, which is known to be injective for cycles of

codimension 2.

To prove the surjectivity of α2

V,l, it is shown in Section 2 that it suﬃces to treat the case in

which the elliptic surface is semi-stable. In this case there is an explicit subgroup

CH2

CM(V¯

F)⊂CH2

hom(V¯

F),

generated by so-called complex multiplication cycles, with the property that the Abel–Jacobi

map restricted to CH2

CM(V¯

F)+CH2

alg(V¯

F) is surjective for almost all primes l[26,§11]. Since

CH2

alg(V¯

F) is a torsion group, the theorem will be proved if one can show that complex

multiplication (CM) cycles lie in CH2(V¯

F)tors. Complex multiplication cycles live in the open

subvariety ˙

V¯

F⊂V¯

F, obtained by removing the singular ﬁbers of the map V¯

F→B¯

F.

In Section 3 it is shown that it suﬃces to prove that CM cycles live in CH2(˙

V¯

F)tors. The

problem of verifying this is a function-ﬁeld analog of a problem studied by Flach [11] and

Mildenhall [23] in the context of elliptic curves over Q. In order to adapt these techniques to

our situation we must replace moduli schemes for elliptic curves with Drinfeld modular curves.

Thus Section 4 recalls brieﬂy the notions of Drinfeld module, full level structure, and

Γ0-level structure. Essential facts about the modular varieties for full level structure and their

compactiﬁcations are recalled in Section 5. Compactiﬁed coarse moduli spaces for Γ0-level

structure appear as quotients of the compactiﬁed full level structure varieties in Section 6.

Key components of the method of Flach and Mildenhall are the Manin–Drinfeld Theorem,

Atkin–Lehner automorphisms and a detailed understanding of the singular ﬁbers in the moduli

schemes for a Γ0-level structure. These topics are treated brieﬂy in Sections 7, 8 and 9. The fact

that an elliptic surface of conductor x∞cwith split multiplicative reduction at the point x∞is

dominated by a Drinfeld modular variety for a Γ0(c)-level structure is recalled in Section 10.

With these tools in hand one can now proceed, in the ﬁnal section, to prove the theorem:

roughly speaking, representatives z∈Z2(˙

V¯

F) of multiples of generators of CH2

CM(V¯

F) are shown

to be supported on the images of certain Drinfeld modular varieties. With the help of the

Manin–Drinfeld theorem, rational functions on these modular varieties are constructed whose

divisors give rise to multiples of the cycles z. This shows that CM cycles give torsion elements

in CH2(˙

V¯

F) and completes the proof.

We now describe brieﬂy how our result relates to Beilinson’s conjecture [2, 1.0], which states

that

CHr

hom(V¯

F)⊂CHr(V¯

F)tors

for any smooth projective variety over the algebraic closure of a ﬁnite ﬁeld. Recall that this

conjecture is known to hold for products of three curves over Fand closely related varieties

[31]. Here we are interested in the case that Vis birational to E×BEand the tautological

rational map f:V Bis a morphism. In this case there is an exact localization sequence,

b∈|B¯

F|0

CH1(f−1(b)) ib∗

−−−→CH2(V¯

F)u

−→ CH2(Eη×ηEη)−→ 0,

where η∈B¯

Fis the generic point. Our results imply that imib∗∩CH2

hom(V¯

F) is a torsion

group plus a ﬁnitely generated free abelian group. The free summand is generated by cycles

in the supersingular ﬁbers and singular ﬁbers where the reduction is not semi-stable. We

do not know whether this summand is zero or not. The problem of showing that the group

u(CH2

hom(V¯

F)) is torsion appears to be closely related to a function ﬁeld analog of the following

well-known and diﬃcult conjecture.

TORSION CM CYCLES AND DRINFELD MODULES 547

Conjecture 0.3 (Beilinson [2, 5.1], Bloch [4, Introduction]). Let Sbe a smooth,

irreducible, d-dimensional variety deﬁned over a number ﬁeld. Then the albanese map,

CHd

hom(S¯

Q)−→ AlbS(¯

Q),

is an isomorphism.

Acknowledgements. We wish to thank S. Bloch for asking whether complex multiplication

cycles are torsion when the base ﬁeld is ﬁnite. We are grateful to E. U. Gekeler for establishing

a form of the Manin–Drinfeld theorem suitable for the present context [15]. It is furthermore a

pleasure to thank Marius van der Put, Gert-Jan van der Heiden and Lenny Taelman for helpful

conversations concerning Drinfeld modular schemes.

Notational conventions. Throughout:

∗tors is the torsion subgroup of an abelian group ∗;

Zr(V) is the free abelian group on irreducible codimension rsubvarieties of a variety V;

CHr(V)=Zr(V)/Z r

rat(V);

CHr

alg(V)=Zr

alg(V)/Z r

rat(V);

Zr

hom(V):=Ker

Zr(V)→l=char(F)H2r(V¯

F,Zl(r)),where Vis smooth over a ﬁeld F

with algebraic closure ¯

F;

Fis a ﬁnite ﬁeld of characteristic p.

In all sections except Section 1:

Xis a smooth, projective, geometrically connected curve;

π:Y→Xis a regular, relatively minimal, non-isotrivial elliptic surface with section;

Wis the blow-up of Y×XYalong the reduced singular locus.

Abuse of notation: if c:P→Qis a morphism of schemes over a ﬁeld k, and k⊂kis a

ﬁeld extension, then the base changed morphism ck:Pk→Qkwill frequently be denoted c,

without the subscript k.

1. Injectivity of the l-adic Abel–Jacobi map

Let Ube a smooth, projective variety over a ﬁeld Fwhich is ﬁnitely generated over its prime

subﬁeld. Let ¯

Fbe a separable closure of F. The purpose of this section is to prove the following

result.

Proposition 1.1. For any prime l= char(F)the restriction of a2

U,l to CH2

hom(U¯

F)tors ⊗Zl

is injective.

Proof. The l-adic Abel–Jacobi map ar

U,l may be deﬁned in terms of extensions of Galois

modules or in terms of the ordinary cycle class map to H2r(UF,Zl(r)) for [F:F]<∞by

means of the Hochschild–Serre spectral sequence [19,§9]. To reconcile the form given below

with the l-adic Abel–Jacobi map mentioned in the introduction, see Remark 1.4. In general

the Abel–Jacobi map takes the form

ar

U,l :CHr

hom(U¯

F)lim

FCHr

hom(UF)−→ lim

FH1(GF,H),(1.1)

where the direct limit is taken over intermediate ﬁelds, F⊂F⊂¯

F, with [F:F]<∞,GF=

Gal( ¯

F/F), H=H2r−1(U¯

F,Zl(r))/tors, and group cohomology is taken with continuous

cocycles for the l-adic topology [34]. The torsion subgroup of the right-most term of (1.1)

is canonically identiﬁed with

lim

F(H2r−1(U¯

F,Zl(r)) ⊗Ql/Zl)GFH2r−1(U¯

F,Zl(r)) ⊗Ql/Zl.(1.2)

548 CHAD SCHOEN AND JAAP TOP

On the other hand, Bloch has deﬁned a cycle class map [3],

λr:CHr(U¯

F)tors ⊗Zl−→ H2r−1(U¯

F,Ql/Zl(r)),(1.3)

which is injective when r=2 [6, Corollaire 4]. As the right-hand side of (1.2) is naturally

identiﬁed with the kernel of H2r−1(U¯

F,Ql/Zl(r)) →H2r(U¯

F,Zl(r)), the proposition follows

from the following result.

Theorem 1.2. The restrictions of λrand ar

U,l to CHr

hom(U¯

F)tors ⊗Zlagree up to sign.

Proof. Let Hr(Ql/Zl(r)) denote the Zariski sheaf associated to the presheaf whose value

on an open V⊂UFis the group Hr

et(V, Ql/Zl(r)). Consider the diagram

CHr(UF)tors ⊗Zl

ρF

//H2r(UF,Zl(r))

Hr−1

Zar (UF,Hr(Ql/Zl(r))) γF

//

αF

OO

H2r−1(UF,Ql/Zl(r))

βF

OO

(1.4)

which is commutative up to sign; see [6, Corollaire 1] or [5, 3.8]. We refer to [6]or[5] for the

deﬁnitions of the maps and the fact that αFis surjective. Bloch’s cycle class map is the unique

map,

λr:CHr(U¯

F)tors ⊗Zl−→ H2r−1(U¯

F,Ql/Zl(r)),

satisfying λr◦limFαF= limFγF; see [6, Corollaire 4] or [5, 4.3; 24, III.1.16]. Deﬁne

DF= KerH2r(UF,Zl(r)) fF

−→ H2r(U¯

F,Zl(r)),

EF= KerfF◦βF:H2r−1(UF,Ql/Zl(r)) →H2r(U¯

F,Zl(r)),

HF= Ker(fF◦βF◦γF).

Now (1.4) gives rise to the left-hand square in

CHr

hom(UF)tors ⊗Zl

ρF

//DF

eF

//H1(GF,H)

HF

γF

//

αF

OO

EF

cF

//

βF

OO

H2r−1(U¯

F,Ql(r))GF/HGF

dF

OO

(1.5)

The map eFarises from the Hochschild–Serre spectral sequence for cohomology with μ⊗r

lm

coeﬃcients by taking the projective limit with respect to m. The map limFeF◦ρFis the

restriction of the l-adic Abel–Jacobi map to CHr

hom(U¯

F)tors ⊗Zl. On the other hand, the

restriction of λrto CHr

hom(U¯

F)tors ⊗Zlfactors through limFcF. Thus the agreement up to

sign of the restrictions of a2

U,l and λrwill follow from the commutativity of the right-hand

square in (1.5).

To establish this commutativity we work with cohomology with ﬁnite coeﬃcients. Associated

to the short exact sequence

1−→ μ⊗r

lm−→ μ⊗r

ln+m−→ μ⊗r

ln−→ 1 (1.6)

are the Bockstein (coboundary) maps

βi

F(n, m):Hi(UF,μ

⊗r

ln)−→ Hi+1(UF,μ

⊗r

lm),(1.7)

analogous maps βi

¯

F(n, m) with ¯

Freplacing Fin (1.7), and the Galois coboundary map

δ(n, m) : Ker(β2r−1

¯

F(n, m))GF−→ H1(GF,Coker(β2r−2

¯

F(n, m))),

TORSION CM CYCLES AND DRINFELD MODULES 549

coming from the short exact sequence of GF-modules

0−→ Coker(β2r−2

¯

F(n, m)) −→ H2r−1(U¯

F,μ

⊗r

ln+m)−→ Ker(β2r−1

¯

F(n, m)) −→ 0.(1.8)

Deﬁne

L1H2r(UF,μ

⊗r

lm) = KerH2r(UF,μ

⊗r

lm)s

−→ H2r(U¯

F,μ

⊗r

lm),

H2r−1(UF,μ

⊗r

ln)0= Ker(s◦β2r−1

F(n, m)),

Hmn = Coker(β2r−2

¯

F(n, m)),

Kmn = Ker(β2r−1

¯

F(n, m)).

Now the commutativity of the right-hand square in (1.5) may be deduced from the following

lemma by applying an inverse limit in mand a direct limit in n.

Lemma 1.3. The following diagram commutes:

L1H2r(UF,μ

⊗r

lm)eF(n,m)

//H1(GF,H2r−1(U¯

F,μ

⊗r

lm)) //H1(GF,Hmn)

H2r−1(UF,μ

⊗r

ln)0

cF(n,m)

//

βF(n,m)

OO

KGF

mn

δ(n,m)

OO

where eF(n, m)comes from the Hochschild–Serre spectral sequence.

Proof. A similar commutativity result is proved in [19, 9.5]. We reduce the current problem

to the one treated there.

Take an injective resolution of (1.6) and apply the functor ∗, where :UF→Spec(F)

is the structure morphism. This gives an exact sequence of complexes of injective sheaves on

Spec(F)[24, III.1.2] which forms the ﬁrst row in the following commutative diagram with

exact rows [36, 1.5]:

0//A·u//B·//C·//0

0//B·v//cone(u)//

OO

A·[1] //0

0//cone(u)//cone(v)//

ξ

OO

B·[1] //0

(1.9)

Let f:SheavesonSpec(F)→Ab denote the global section functor. The Grothendieck

spectral sequence for the composite functor f◦∗may be identiﬁed with the Hochschild–Serre

spectral sequence: the category of etale sheaves of abelian groups on Spec(F) is equivalent to

the category of discrete GF-modules and fmay be identiﬁed with the functor which associates

to a GF-module its GF-invariant subgroup. In this context the long exact sequence of

cohomology groups H•(U¯

F,μ

⊗

l•) associated with (1.6) is identiﬁed with the long exact sequence

of etale sheaves on Spec(F),

···−→H

nA·−→ H nB·−→ H nC·−→ H n+1 A·−→··· .

By [36, 1.5] this is isomorphic to the cohomology sequence

−→ H n−1(cone(v)) ψn−1

−−−→H

n−1(B·[1]) ∂n−1

−−−→H

n(cone(u)) μn

−−−→ H n(cone(v)) −→ .

550 CHAD SCHOEN AND JAAP TOP

The map δ(n, m) of Lemma 1.3 is now identiﬁed with the coboundary map, δ:fY →R1fX,

associated to the short exact sequence with n=2r−1,

X:= im(ψn−1)→H

n−1(B·[1]) im(∂n−1) = Ker(μn)=:Y. (1.10)

The cohomology group H2r−1(UF,μ

⊗r

ln) is identiﬁed with the hypercohomology group

R2r−1fC·. The map βF(n, m) in Lemma 1.3 is induced by the map on the hypercohomology

R2r−1fC·→R2rfA·. Since the objects in (1.9) are f-acyclic, hypercohomology is just the

cohomology of fapplied to each complex. Since cone and fcommute, the long exact

hypercohomology sequence for the top row in (1.9) is isomorphic to that for the bottom row.

Thus βF(n, m) comes from the natural map ν:R2r−1f(cone(u)) →R2r−1f(cone(v)).

To go further we need the hypercohomology spectral sequence on RnfA·which may be

constructed as follows. Endow A·with the canonical ﬁltration [7, 1.4.6], and let A·→J·

be a ﬁltered quasi-isomorphism to a ﬁltered complex whose associated graded complexes are

f-acyclic. Then the spectral sequence of the ﬁltered complex fJ·is the hypercohomology

spectral sequence up to a shift of indices [7, 1.4]. Since ξis a ﬁltered quasi-isomorphism when

both complexes are given the canonical ﬁltration, the hypercohomology spectral sequences for

the functor fand the complexes A·[1] and cone(v) are isomorphic. Consider the commutative

diagram with n=2r−1,

Y//

κ0

Rnf(cone(u)) ν//

κu

Rnf(cone(v))

κv

fY //fHn(cone(u)) fμn

//fHn(cone(v))

(1.11)

where κuand κvare standard maps from the spectral sequences Y:= Ker(κv◦ν) and

fY Ker(fμn). Denote the ﬁltration on the hypercohomology Rnf(cone(v)) coming from

the hypercohomology spectral sequence by F•. Now the diagram in Lemma 1.3 is written in

the current notation as

F1Rnf(cone(v)) //R1fHn−1(cone(v)) R1fψ

//R1fX

Yκ0//

ν0

OO

f(Y)

δ

OO

(1.12)

where ν0is induced by ν. This diagram commutes by [19, 9.5] applied to the bottom row

in (1.9).

This completes the proof of Theorem 1.2 and of Proposition 1.1.

Remark 1.4. To relate the Abel–Jacobi maps (0.1) and (1.1) observe that when Fis a

ﬁnite ﬁeld,

H1(GF,H)⊗QlH⊗Ql/(1 −FrobF)=0,

since by the Weil conjectures (Deligne’s theorem) 1 is not an eigenvalue of the action of

Frobenius on H.ThusH1(GF,H) gets identiﬁed with its torsion subgroup which is naturally

identiﬁed with (H⊗Ql/Zl)GF.

Remark 1.5. The cycle class map

λr:CHr(U¯

F)tors ⊗Zl−→ H2r−1(U¯

F,Ql/Zl(r))

may fail to be injective for 2 <r<dim(U¯

F). So far this phenomenon has only been noted when

char(F) = 0. In light of Question 0.1 it would be interesting to understand the specialization

TORSION CM CYCLES AND DRINFELD MODULES 551

of Ker(λr) from Fto the algebraic closure of a ﬁnite ﬁeld. In this context the cycles in [35, 7.2]

seem especially interesting. (See also [27].)

2. Reduction to the semi-stable case

With the injectivity of the map α2

V,l established, Theorem 0.2 will follow once surjectivity for

almost all lis known. The purpose of this section is to show that surjectivity will hold for ﬁber

products of general non-isotrivial elliptic surfaces with section if it holds for ﬁber products of

semi-stable ones.

Let :E→Bbe a non-isotrivial elliptic surface over a ﬁnite ﬁeld, which we will assume to

have characteristic p>5. After replacing this ﬁeld with a ﬁnite extension denoted F,wesee

that the theory of semi-stable reduction [30, VII.5.4] gives a smooth, projective geometrically

irreducible curve X/F, and a regular, relatively minimal, non-isotrivial, semi-stable elliptic

surface with section π:Y→X, which ﬁts into a commutative diagram

Yh//___

π

EF

F

Xh//BF

(2.1)

where his a morphism and his a dominant rational map. Since Fis non-isotrivial, Ywill

have at least one singular ﬁber. The only singular points of Y×XYare ordinary double points

at points (y1,y

2), where each yiis a singular point of the ﬁber π−1(π(yi)). These singularities

are resolved by a single blow-up of the reduced singular locus. We denote this blow-up by

σ:W→Y×XY.ForVas in Theorem 0.2, the map hgives rise to a dominant rational map

ˆ

h:W VF. By Abhyankar’s theorem [1, p. 1, Dominance], which applies since char(F)>5,

ˆ

h¯

Ffactors as ˜

h◦κ−1,

W¯

F

κ

←− ˜

W¯

F

˜

h

−→ V¯

F,

where κis a sequence of blow-ups with smooth centers and ˜

his a surjective morphism.

Lemma 2.1. (i) The map α2

W,l is surjective if and only if α2

˜

W,l is.

(ii) If α2

˜

W,l is surjective, then so is α2

V,l.

Proof. (i) Blowing up a point changes neither the domain nor the target of α2

l. Blowing

up a smooth curve Cadds a direct summand CH1(C¯

F)tors ⊗Zlto the domain of α2

land a

summand H1(C¯

F,Zl(1)) ⊗Ql/Zlto the target. The cycle class map on the blow-up gives rise

to the Kummer theory isomorphism between these two groups [12, 6.7(d) and 3.3(b)].

(ii) By the pro jection formula the map

˜

h∗:H3(˜

W¯

F,Zl(2)) ⊗Ql/Zl−→ H3(V¯

F,Zl(2)) ⊗Ql/Zl

is surjective. The assertion follows from the functoriality of the Abel–Jacobi map with respect

to a proper direct image [3,§3; 25, 1.10].

3. Reduction to a theorem about complex multiplication cycles

In this section we recall the notion of complex multiplication (CM) cycle and reduce

Theorem 0.2 to an assertion about CM cycles.

Let π:Y→Xdenote a semi-stable, non-isotrivial elliptic surface with section over a ﬁnite

ﬁeld F. Recall that Wis the blow-up of the ﬁber product along the reduced singular locus.

The inclusion of the locus where π:Y→Xis smooth will be denoted j:˙

X→Xand base

552 CHAD SCHOEN AND JAAP TOP

change by jwill be indicated by adding a dot, ˙ , to the notation. Deﬁne Ξ := X−˙

X. This

set is non-empty since πis not isotrivial. Let f:W→Xbe the tautological map.

There are only ﬁnitely many points ¯x∈˙

X¯

Ffor which π−1(¯x) is a supersingular elliptic curve

[30, V.4]. The remaining closed points will be called complex multiplication (CM) points. Let

¯x∈˙

X¯

Fbe a CM point. Then End(π−1(¯x)) is an order in an imaginary quadratic number ﬁeld

and the N´eron–Severi group N1(f−1(¯x)) is a free Z-module of rank 4. Write Δ¯x⊂f−1(¯x)

π−1(¯x)×π−1(¯x) for the diagonal and N1

0(f−1(¯x)) ⊂N1(f−1(¯x)) for the rank 3 submodule

spanned by {π−1(¯x)×s(¯x),s(¯x)×π−1(¯x),Δ¯x}, where s:X→Yis a section of π.

Definition 3.1. Let ¯x∈˙

X¯

Fbe a CM point. A 1-dimensional cycle zsupported in the ﬁber

f−1(¯x) is called a complex multiplication (CM) cycle if the class of zin N1(f−1(¯x)) generates

the free Z-module N1

0(f−1(¯x))⊥of rank 1.

The subgroup of CH2(W¯

F) generated by the classes of all CM cycles in Z1(f−1(¯x)) as ¯x

ranges over all CM points will be denoted CH2

CM(W¯

F).

Lemma 3.2. CH2

CM(W¯

F)⊂CH2

hom(W¯

F).

Proof. From the non-isotriviality of πand the fact that complex multiplication cycles

annihilate N1

0(f−1(¯x)) one deduces that their cohomology classes annihilate H2(W¯

F,Ql(1))

under the cup product pairing [25, 5.4]. Since H4(W¯

F,Zl(2)) is torsion free for each l= char(F)

[28, 8.7(i)], complex multiplication cycles are homologous to zero.

Write mπfor the least common multiple of all nfor which πhas a singular ﬁber of Kodaira

type In. As mentioned in the introduction, the proof of Theorem 0.2 makes use of the following.

Proposition 3.3. If l2·5·p·mπ, then

a2

W,l (CH2

alg(W¯

F)+CH2

CM(W¯

F)) = H3(W¯

F,Zl(2)) ⊗Ql/Zl.

Proof. See [26, 11.3.2].

Let Ube a smooth, projective variety over a ﬁeld Fwith algebraic closure ¯

F. An additional

ingredient in the proof of Theorem 0.2 is the following well-known fact.

Proposition 3.4. If Fis a ﬁnite ﬁeld, then CHr

alg(U¯

F)is a torsion group.

Proof. It follows from the deﬁnition of algebraic equivalence, that CHr

alg(U¯

F) is isomorphic

to a quotient of CJac(C)( ¯

F), where the sum is over a (possibly inﬁnite) collection of curves.

When ¯

Fis the algebraic closure of a ﬁnite ﬁeld, the ¯

F-rational points in any abelian variety

form a torsion group.

Write ˙

W⊂Wfor the complement of the singular ﬁbers. Using techniques from the theory

of Drinfeld modules we will show in subsequent sections that complex multiplication cycles

give torsion elements in CH2(˙

W¯

F). In order to conclude that complex multiplication cycles are

torsion in CH2(W¯

F) we must study the exact localization sequence

CH1(f−1(Ξ¯

F)) −→ CH2(W¯

F)˜

j∗

−→ CH2(˙

W¯

F)−→ 0.

The result we need is the following.

Proposition 3.5. The kernel of the restriction of ˜

j∗to CH2

hom(W¯

F)is a torsion group.

TORSION CM CYCLES AND DRINFELD MODULES 553

Proof. By the localization sequence, one is reduced to proving that if z∈Z2

hom(W¯

F)is

supported on f−1(Ξ¯

F) then the class of zin CH2(W¯

F) is torsion.

Write z=ξ∈Ξ¯

Fzξwith zξ∈Z1(f−1(ξ)). Write s:X→Yfor the zero section. Deﬁne

divisors T1=Y×Xs(X) and T2=s(X)×XYon Y×XYminus the singular locus. We will

view these as divisors on W. Let T3⊂Wbe the strict transform of the diagonal in Y×XY.

Write ai,ξ for the intersection number Ti·zξ. Deﬁne

S1=−T1+T2+T3,

S2=T1−T2+T3,

S3=T1+T2−T3,

z

ξ=2zξ−

1i3

ai,ξSi·f−1(ξ).

Lemma 3.6. The cycle z

ξis contained in Z2

hom(W¯

F).

Proof. Since H4(W¯

F,Zl(2)) is torsion free for all l=p[28, 8.7(i)], it suﬃces to show that

z

ξ·H2(W¯

F,Ql(1)) = 0. Write qi:W→Yfor the composition of σwith the projection on the

ith factor in the ﬁber product Y×XY. It is not diﬃcult to show that q∗

i(H2(Y¯

F,Ql(1))), for

i∈{1,2}, together with the cohomology classes of the Tiand the components of the ﬁbers

generate H2(W¯

F,Ql(1)) [28, 7.1, 7.3(ii)–(iii), 7.9, 8.7(i)]. Deﬁne

LH2(Y¯

F,Ql(1)) := Ker[H2(Y¯

F,Ql(1)) →H2(π−1(ξ),Ql(1))].

Since the cohomology classes of the components of π−1(ξ) together with the cohomology

class of the section sgenerate a subgroup of H2(Y¯

F,Ql(1)) which maps surjectively to

H2(π−1(ξ),Ql(1)), one may in fact replace q∗

i(H2(Y¯

F,Ql(1))) with q∗

i(LH2(Y¯

F,Ql(1))) in the

description of a generating set for H2(W¯

F,Ql(1)). By the projection formula,

z

ξ·q∗

i(LH2(Y¯

F,Ql(1))) = 0.

Also z

ξis orthogonal to any ﬁber component. Finally, z

ξ·Tj= 0 for all jbecause

Si·Tj·f−1(ξ)=2δij .(3.1)

This proves Lemma 3.6.

Lemma 3.7. The class of z

ξin CH1(f−1(ξ)) is torsion.

Proof. Recall that π−1(ξ)⊂Yis a ﬁber of Kodaira type Imfor some m>0. The ﬁber

f−1(ξ) has 2m2irreducible components: m2of these correspond to irreducible components of

π−1(ξ)×π−1(ξ). The remaining ones arise as exceptional divisors in Wwhen the m2ordinary

double points of Y×XYwhich lie over ξare blown up.

Lemma 3.8. CH1(f−1(ξ)) Z2m2+2 ⊕Z/mZ.

Proof. See [29].

To prove Lemma 3.7, it suﬃces to show that intersection pairing gives a surjective map,

h: N.S.(W¯

F)⊗Q−→ Hom(CH1(f−1(ξ)),Q).

Let S⊂W¯

Fbe a very ample non-singular hypersurface whose intersection with each component

of f−1(ξ) is irreducible. The geometric generic ﬁber of g=f|Sis connected. We choose Sso

that it is smooth. Write Nξ(W¯

F)⊂N.S.(W¯

F)⊗Q(respectively Nξ(S)⊂N.S.(S)⊗Q) for the

554 CHAD SCHOEN AND JAAP TOP

subspace of the N´eron–Severi group generated by the components of ﬁbers over ξ. The left

kernel in the intersection pairing in the top row of

Nξ(S)⊗CH1(g−1(ξ)) //

Q

Nξ(W¯

F)

OO

⊗CH1(f−1(ξ)) //Q

(3.2)

has dimension 1 by the well-known theorem on intersections of components of singular ﬁbers

in a ﬁbred surface [9, Proposition 2.6]. Thus dim(h(Nξ(W¯

F))) dim(Nξ(W¯

F)) −1=2m2−1.

Observe that Ti·f−1(ξ)∈CH1(f−1(ξ)) lies in the right kernel of the bottom row of (3.2) since

on W¯

F,Ti·f−1(ξ) is numerically equivalent to Ti·f−1(x) for any x∈X¯

F. As the pairing

Span{Si}1i3⊗Span{Ti·f−1(ξ)}1i3−→ Q

is non-degenerate by (3.1), rank(h)2m2−1 + 3. By Lemma 3.8, his surjective.

To complete the proof of Proposition 3.5 we need only show that ξ∈Ξ¯

Fai,ξSi·f−1(ξ) gives

a torsion class in CH2(W¯

F). Since Ti·ξ∈Ξ¯

Fz

ξ= 0, it follows that ξ∈Ξ¯

Fai,ξ = 0. Since Fis

ﬁnite, ξ∈Ξ¯

Fai,ξξ∈CH1(X¯

F)tors. Applying f∗:CH1(X¯

F)→CH1(W¯

F) and intersecting with

Sigives the desired result.

4. Drinfeld modules and level structures

The proof of our main theorem requires a detailed understanding of the geometry of the

compactiﬁcation of moduli schemes for rank 2 Drinfeld modules with Γ0(c)-level structure. In

particular we need information about the singular ﬁbers (cf. Proposition 9.6), Atkin–Lehner

automorphisms (cf. Theorem 8.1), the Manin–Drinfeld theorem (cf. Theorem 7.1), and modular

parametrizations of elliptic curves (cf. Theorem 10.1).

In this section we recall the notion of Drinfeld module and various notions of level structure

on a Drinfeld module. The original source for the material in this section and the next several

sections is [10]. Additional sources include [8,16,21,32,18].

Let Xbe a smooth, projective, geometrically irreducible curve over a ﬁnite ﬁeld Fof

cardinality qand characteristic p. Let x∞∈Xbe a closed point. Set A=H0(X−{x∞},OX).

Fix an integer r>0. For a∈A\{0}, deﬁne da∈Zby the formula qda=#(A/aA)r. Then

da=−r·ordx∞(a)·[F(x∞):F].

Let Sbe an A-scheme.

Definition 4.1. A Drinfeld A-module over Sof rank rconsists of a group scheme G→S

and a ring homomorphism Φ : A→End(G), a→ Φa, satisfying the following conditions.

(a) There is a covering of Sby aﬃne open subsets such that for each U= Spec(B)inthe

cover, the restriction GUof Gis isomorphic to the additive group Ga,U = Spec(B[x]).

(b) For any a∈A\{0}, the resulting diagram

Ga,U //

Ga,U

GU

Φa,U

//GU

OO

yields an endomorphism of Ga,U = Spec(B[x]), with the property that it is given by an F-linear

polynomial n0bn(a)xqnsuch that b0:A→Bis the structure morphism to the A-algebra

B,bda(a)∈Bis a unit, and bn(a) is nilpotent for n>d

a.

TORSION CM CYCLES AND DRINFELD MODULES 555

A morphism, λ:Φ→Φ, of Drinfeld A-modules over Sis a homomorphism of S-group

schemes satisfying λ◦Φ(a)=Φ

(a)◦λfor all a∈A.

One thinks of a Drinfeld module as a way to describe an A-module structure on the group

scheme G. For a rank rmodule, this structure has the property that the a-torsion for a∈

A\{0}is a ﬁnite subgroup scheme of rank da=#(A/aA)r.

We now recall the notion of a Drinfeld level structure [10,20]. Let b⊂A, with b= (0), be

an ideal. Any element b∈bdeﬁnes an endomorphism multiplication by b(written Φb)ona

Drinfeld A-module (G,Φ). Its kernel is denoted G[b]; for b= 0 this is a subgroup scheme of

rank #(A/bA)r. The intersection ×GG[b] over all b∈bwill be written as G[b]. A full level

b-structure, also called a Γ(b)-structure, on a rank rDrinfeld A-module Gover Sis a group

homomorphism ϕ:(A/b)r→G(S) such that

G[b]=

x∈(A/b)r

[ϕ(x)].

This is interpreted as an equality of Cartier divisors.

Suppose now that r=2. A Γ

0(b)-structure is deﬁned as a ﬁnite ﬂat subgroup scheme

H⊂G[b] with an induced action of A/b. Moreover, His assumed to be cyclic and of constant

rank #(A/b). This means that a ﬁnite, faithfully ﬂat base change T→Sexists, and there is

a point P∈(G×ST)(T) such that

H×ST=

x∈A/b

[xP ],

interpreted as an equality of Cartier divisors.

5. Modular varieties for Γ(b)-level structure

In this section we gather together basic facts about Drinfeld modular varieties with full level

b-structure.

Suppose r>0 is an integer. Suppose b⊂Ais a non-zero ideal which is divisible by at least

two distinct primes. Consider the functor Mr(b) which assigns to an A-scheme Sthe set of

isomorphism classes of rank rDrinfeld A-modules over Sequipped with a full level b-structure.

Theorem 5.1. The functor Mr(b)is representable by a ﬁnite type aﬃne A-scheme Mr(b).

Proof. See [10, 5.3] or [21, 2.5] or Sa¨ıdi’s article [16, p. 24] or [32, 5.3.3].

For each maximal ideal m⊂Awrite ˆ

Amfor the m-adic completion of A. Let Kdenote

the quotient ﬁeld of A,K∞the completion at the place x∞,A∞⊂K∞the valuation ring,

π∞⊂A∞a uniformizer, and Cthe completion of the algebraic closure of K∞.

Theorem 5.2. (i) The scheme Mr(b)is regular of dimension r.

(ii) The structure map ςr(b):Mr(b)→Spec(A)is ﬂat and surjective. It is smooth over

Spec(A)\Spec(A/b).

(iii) The scheme Mr(b)is irreducible.

(iv) There is a natural left action of the ﬁnite group Pic(A)×GL(r, A/b)/F∗Id on Mr(b).

(v) The scheme M1(b)is isomorphic to the aﬃne scheme with coordinate ring the integral

closure of Ain the abelian extension K⊂K(b), which by class ﬁeld theory corresponds to the

following subgroup of the idele class group of K:

K∗⎛

⎝K∗

∞·

mb

ˆ

A∗

m·

m|b

(1 + bˆ

Am)⎞

⎠⊂I

K/K∗.

556 CHAD SCHOEN AND JAAP TOP

In particular, M1(b)is Galois over Spec(A)with Galois group Pic(A)×GL(1,A/b)/F∗Id, and

K(b)is the maximal abelian extension of Ktotally split at inﬁnity, with conductor b, in which

Fis algebraically closed.

(vi) The structure morphism factors:

ςr(b):Mr(b)ϑr(b)

−−−→M1(b)(b)

−−−→Spec(A).

Furthermore, ϑr(b)is compatible with group actions in (iv) and the homomorphism

(Id,det) : Pic(A)×GL(r, A/b)/F∗Id −→ Pic(A)×GL(1,A/b)/F∗Id.

Proof. (i) and (ii) See [10, 5C; 21, 3.4].

(iii) See [18, 5.10 and 5.3.1].

(iv) See [10, 5D; 21, 3.5]. For deﬁniteness we note that a matrix g∈GL(r, A/b) acts on

column vectors in (A/b)rfrom the left. In the notation of §4 the action on level structures is

by g∗ϕ=ϕ◦g−1.

(v) See [10,§8] and the article by Rust and Scheja [16, Lecture 4, 6.4–6.6]; compare [16,

Lecture 9, 2.1] where the special case A=Fq[T] is considered.

(vi) In [18, 5.4] van der Heiden gives the argument in the case that bis a principal ideal.

To treat the general case take the quotient by the group Gin [18, 5.3.1].

As we are primarily concerned with moduli of Drinfeld modules of rank 2, we will generally

drop the superscript rfrom the notation when r= 2 (for example, M(b)=M2(b)). We now

review the (relative) ‘compactiﬁcation’ of M(b).

Theorem 5.3. (i) There exists an irreducible, regular scheme M(b), proper and ﬂat over

Spec(A), containing M(b)as an open dense subscheme.

(ii) The group action of Theorem 5.2(iv) extends to a biregular action on M(b).

(iii) The map ϑ(b)extends to an equivariant morphism

ϑ(b):M(b)−→ M1(b).

(iv) The ﬁbers of ϑ(b)are connected.

(v) The structure morphism ς(b)is smooth on the complement of ς(b)−1(Spec(A/b)).

(vi) The scheme M(b)−M(b)with its reduced scheme structure consists of sections of ϑ(b).

(vii) The group action on M(b)−M(b)induced by Theorem 5.2(iv) permutes the irreducible

components transitively. The stabilizer of one component may be identiﬁed with

{0}×N(b),where N(b):=(A/b)∗A/b

0F∗

q⊂GL2(A/b).

Proof. (i) See [21, 5.3.5].

(ii) See [10, 9.2].

(iii) See [10, 9.3; 21, 5.3.5].

(iv) By Zariski’s connectedness principle [22, 8.3.6(ii)], it suﬃces to show that the generic

ﬁber is geometrically connected. In other words, it suﬃces to show that M(b)×Spec(A)Spec(C)

has [K(b):K] connected components. This is known from the analytic theory of Drinfeld

modules [16, Lecture 8, pp. 116, 117].

(v) See [10, 5C; 21, 5.3.5].

(vi) See [18, 5.9] or [21, 5.3.5].

(vii) See [21, 4.4, 5.1.10 and 5.3.5] (and also [18, 5.9.2]).

The irreducible components of M(b)−M(b) are called cusps, as are their images in quotients

of M(b) by subgroups of the group described in Theorem 5.3(vii).

TORSION CM CYCLES AND DRINFELD MODULES 557

6. Modular varieties for Γ0(c)-level structures

References for this section are [14] and [32, 4.3].

If c⊂Ais divisible by two distinct primes, we deﬁne M0(c) and M0(c) to be the quotients of

M(c) and M(c), respectively, by the subgroup B(c)⊂GL(2,A/c) consisting of matrices with a

zero in the lower left-hand corner. If cfails to be divisible by two distinct primes (for example,

c= (1)), let bbe an ideal prime to csuch that bc is divisible by two distinct primes. Deﬁne

M0(c) to be the quotient of M(bc) by GL(2,A/b)B(c) (viewed as a subgroup of GL(2,A/bc)by

the Chinese remainder theorem) and deﬁne M0(c) analogously. This deﬁnition is independent

of the choice of auxiliary ideal b.

Write M0(c) for the functor which assigns to each A-scheme Sthe set of isomorphism classes

of Drinfeld modules over Swith a Γ0(c)-level structure. Finally, write M1(1) for the quotient

of M1(b) by the action of GL(1,A/b)/F∗Id. This is independent of the choice of b.

Theorem 6.1. (i) The quotient M0(c)is a coarse moduli scheme for the functor M0(c).

(ii) The scheme M1(1) is a coarse moduli scheme for rank 1Drinfeld modules without level

structure.

(iii) The quotient M0(c)is irreducible and normal. It is proper and ﬂat over Spec(A).It

contains M0(c)as an open dense subset.

(iv) The structure morphism factors:

ς0(c):M0(c)ϑ0(c)

−−−→M1(1)

−→ Spec(A).

(v) The ﬁbers of ϑ0(c)are connected.

(vi) For z∈M1(1) with (z)/∈Spec(A/c), the ﬁber ϑ0(c)−1(z)is irreducible.

(vii) The connected components of M0(c)\M0(c)are sections of ϑ0(c). They are in bijective

correspondence with elements of the set Pic(A)×B(c)\GL(2,A/c)/N (c).

Proof. (i) and (ii) See [32, 5.3.5].

(iii) This follows from the analogous properties in the case of full level structure by taking

quotients.

(iv) Choose b⊂Aas above. Now (iv) follows from Theorem 5.3(iii) and Theorem 5.2(vi)

by taking the quotient of M(bc) by the action of GL(2,A/b)B(c).

(v) Choose b⊂Aas above. Let z∈M1(bc) lie over z∈M1(1). By Theorem 5.3(iv) the ﬁber

ϑ(bc)−1(z) is connected. Since det : GL(2,A/b)B(c)→GL(1,A/bc) is surjective, the quotient

map M(bc)→M0(c) sends ϑ(bc)−1(z)ontoϑ0(c)−1(z).

(vi) Choose b⊂Aas above so that (z)/∈Spec(A/bc). Let z∈M1(bc) lie over z.By

Theorem 5.3(iv) and (v) the ﬁber ϑ(bc)−1(z) is irreducible. The result follows as in (v).

(vii) The ﬁrst assertion follows from Theorem 5.3(vi). The second assertion follows from

Theorem 5.3(vii) by taking quotients.

7. The Manin–Drinfeld theorem

An important result for classical modular curves is the Manin–Drinfeld theorem. Gekeler

[15] has proved an analog for Drinfeld modular curves. We need this result in the following

form.

Theorem 7.1. For any two cusps cK,c

K∈M0(c)K,cK−c

K∈CH1(M0(c)K)tors.

Proof. Gekeler’s theorem says that, for any irreducible component of M0(c)C, any degree

zero divisor supported on the cusps is torsion in the Chow group. Since the cusps are algebraic

over K, the statement remains true when Cis replaced by a suitable ﬁnite extension Kof K.

558 CHAD SCHOEN AND JAAP TOP

By Theorem 6.1(iii), M0(c)Kis irreducible. By Theorem 6.1(vii), deg(cK−c

K)=0.Thuswe

may realize a positive multiple of cK−c

Kas the norm of a degree zero divisor supported on

the cusps of an irreducible component of M0(c)K. The result follows from [12, 1.4].

8. Atkin–Lehner automorphisms

We need a notion for Drinfeld modular curves which is analogous to Atkin–Lehner involutions

of elliptic modular curves. The treatment here extends somewhat that in [14].

Henceforth we assume that m,band care pairwise coprime ideals in A. We say that a rank 2

Drinfeld module has a Γ0(m)Γ(bc)-level structure when it has both a Γ0(m)- and a Γ(bc)-level

structure. Suppose that bc is divisible by two distinct primes (note that here and elsewhere

we do not exclude the case c= 1). The functor rank 2 Drinfeld A-modules with a Γ0(m)Γ(bc)-

level structure is representable [32, 5.3.3]. We denote the corresponding ﬁne moduli space by

M(m0,bc) and compactify it, embedding it in M(m0,bc):=M(mbc)/B(m).

Suppose given a rank 2 Drinfeld A-module (G,Φ) equipped with a Γ0(m)-structure Hand

aΓ

0(c)-structure H. Then H=H+H gives a Γ0(mc)-structure.

Theorem 8.1. (i) The association

(G,H)−→ (G/H,H/H+G[m]/H)

gives rise to a natural transformation of functors

wm:M0(mc)−→ M 0(mc).

(ii) The map wminduces a morphism wm:M0(mc)→M0(mc)of schemes over Spec(A).

(iii) The map wmis an automorphism whose order divides 2n, where nis the order of m

in Pic(A).

(iv) The map wmextends to an automorphism of M0(mc).

Proof. (i) Given a Drinfeld module (G,Φ) with Γ0(mc)-level structure, taking the quotient

by the ﬁnite ﬂat group scheme associated to the Γ0(m)-level structure gives a group scheme

which is locally isomorphic to Ga[32, 2.3.2]. Each Φ(a)∈End(G) induces an endomorphism

of the quotient. This gives rise to a Drinfeld module on the quotient [32, 3.2.4] which acquires

a level Γ0(c)-structure from the image of the Γ0(c)-structure on (G,Φ) and a Γ0(m)-structure

from the image of G[m].

(ii) Fix an ideal bwhich is prime to mc with the property that bc is divisible by two distinct

primes. By an argument analogous to the proof of (i), the association G→G/Hgives a

natural transformation from the functor Drinfeld A-modules with Γ0(m)Γ(bc)-structure to

itself. As this functor is representable, we obtain a morphism of ﬁne moduli spaces which is

GL(2,A/b)B(c)-equivariant. Passing to the quotient gives the desired map wm.

(iii) It suﬃces to show that w2n

m=Id on M0(mc)(C). Since M0(mc) is a coarse moduli

scheme, an element of M0(mc)(C) corresponds to a Drinfeld module with Γ0(mc)-level structure

over C. By the analytic theory of Drinfeld modules, a Drinfeld module over Cis given by

C/Λ, where Λ ⊂Ca discrete subgroup, which is a rank 2 projective A-submodule (cf. Lopez’s

article [16, p. 37]). A Γ0(mc)-structure on C/Λ is given by a second projective A-submodule

Λ⊂Csatisfying (mc)−1Λ⊃Λ⊃Λ and Λ/Λ∼

=(mc)−1A/A as A-modules. In this language

the Atkin–Lehner map wmsends a pair (Λ,Λ)to(Λ

,m−1Λ). Iterating 2ntimes gives

(m−nΛ,m−nΛ), which is isomorphic to the original pair since mnis principal.

(iv) See [10, 9.2].

Forgetting the Γ0(m)-level structure gives a natural transformation γ:M0(mc)→M

0(c).

TORSION CM CYCLES AND DRINFELD MODULES 559

Lemma 8.2. (i) The natural transformation γgives rise to a ﬁnite surjective morphism

γ:M0(mc)→M0(c). For each cusp c⊂M0(c),γ−1(c)is supported on cusps.

(ii) When mis a maximal ideal, deg(γ)=1+#(A/m).

Proof. (i) Forgetting the Γ0(m)-level structure gives a natural transformation from the

functor rank 2 Drinfeld A-modules with Γ0(m)Γ(bc)-level structure to rank 2 Drinfeld

A-modules with Γ(bc)-level structure. This gives a morphism of ﬁne moduli spaces,

˜γ:M(m0,bc)−→ M(bc),

which induces γ:M0(mc)→M0(c) by taking the quotient by the natural GL(2,b)B(c)-action.

One deduces from [21, 3.4.1(ii)] that both ˜γand γare ﬁnite and surjective. They extend to

ﬁnite morphisms of the compactiﬁcations by [10, 9.2]. The ﬁnal assertion in (i) is clear.

(ii) We need only count the number of points in the inverse image of a suﬃciently

general geometric point. The m-torsion in a Drinfeld A-module over Cis isomorphic to a

2-dimensional A/m-vector space. Now Γ0(m)-structures correspond to 1-dimensional subspaces.

There are 1 + #(A/m) of these.

9. Geometry of cusps and special ﬁbers

Henceforth m⊂Awill denote a maximal ideal prime to c. This section contains the

technical results about the cusps of M0(mc) and the ﬁber ς0(mc)−1(m) which are needed for

the construction of torsion cycles in the proof of the main theorem. As in the previous section

we make frequent use of an auxillary ideal bprime to mc and such that bc is divisible by two

distinct primes.

For an integral A-scheme M,MKdenotes the generic ﬁber and K(M) denotes the ﬁeld of

rational functions.

Lemma 9.1. For any cusp cK∈M(bc)K, the ﬁber ˜γ−1

K(cK)⊂M(m0,bc)Kconsists of two

cusps. The map ˜γKis unramiﬁed at one cusp and has ramiﬁcation degree #(A/m)at the other.

The same statements hold for γK:M0(mc)K→M0(c)K.

Proof. By Theorem 5.3(vii) the cusps of M(bc)Kare parametrized by cosets

Pic(A)×GL(2,A/mbc)/N (mbc).

Thus cusps of M(m0,bc)Kare parametrized by double cosets

Pic(A)×B(m)\GL(2,A/mbc)/N (mbc).

By the Chinese remainder theorem cusps of M(m0,bc) lying over a given cusp of M(bc) are

parametrized by

B(m)\GL(2,A/m)/N (m)=B(m)\GL(2,A/m)/B(m).

This is the classical Bruhat decomposition of GL(2,A/m) and consists of two elements.

The B(m)-orbit, B(m)/N (m)⊂GL(2,A/m)/N (m), corresponds to cusps of M(bcm)Kover

cKwhose decomposition ﬁeld in the extension K(M(bc)) ⊂K(M(mbc)) is K(M(mbc))B(m).

The corresponding cusp of M(m0,bc)Kis unramiﬁed. The same argument as was used in

Lemma 8.2(ii) shows that ˜γKhas degree 1 + #(A/m). Thus the second cusp in ˜γ−1

K(cK) has

ramiﬁcation degree #(A/m). The corresponding result for γKfollows by taking the quotient

by GL(2,A/b)B(c).

We next study the ﬁber ς0(mc)−1(m) (cf. [14,§5]). In preparation we recall a few basic facts

about automorphisms of Drinfeld modules.

560 CHAD SCHOEN AND JAAP TOP

Lemma 9.2. (i) Let Φbe a rank 2Drinfeld A-module. Then F∗A∗Aut(Φ) or

F∗Aut(Φ) where Fis a quadratic extension of F.

(ii) For each maximal ideal m⊂Athere exist only ﬁnitely many isomorphism classes of

rank 2Drinfeld A-modules Φover the algebraic closure of A/mfor which Aut(Φ) F∗.

Proof. (i) See [16, 2.11, p. 73].

(ii) With the notation used in (i), suppose that Aut(Φ) = F∗ F∗=A∗. Put B:= F⊗FA.

The Drinfeld A-module Φ naturally yields a Drinfeld B-module of rank 1. Vice versa, any

Drinfeld B-module of rank 1 yields by restriction to Aa rank 2 Drinfeld A-module with

automorphism group F∗. Since there are only ﬁnitely many isomorphism classes of such Drinfeld

B-modules, the result follows.

Set k=A/mand use the subscript kto denote base change with respect to A→A/m

(for example, M0(c)k:= M0(c)⊗AA/m). Then M0(c)kis a ﬁber of the quotient variety

M(bc)/GL(2,A/b)B(c). The next lemma compares this with the quotient of the ﬁber M(bc)k.

Write ˘

M0(c)kfor the normalization of M0(c)kat the cusps. A recent result of Taelman [32,

Theorem 5.5.2] implies that the normal variety M0(c) is regular along the cusps. It follows that

˘

M0(c)kM0(c)k. However, we have not used this here; instead, we apply Lemma 9.4 below

which seems to be of independent interest.

Lemma 9.3. ˘

M0(c)kM(bc)k/GL(2,A/b)B(c).

Proof. Note that GL(2,A/b)B(c)/F∗Id acts on M(bc) and M(bc)k, by Lemma 9.2(i). There

are open dense subsets in each of these varieties where the stabilizer at any closed point is trivial,

by Lemma 9.2(ii). Since M(bc) is smooth over Spec(A)−Spec(A/bc) (by Theorem 5.3(v)),

M(bc)kis a smooth curve and so is M(bc)k/GL(2,A/b)B(c). Thus it suﬃces to show that the

natural morphism M(bc)k/GL(2,A/b)B(c)→M0(c)kis an isomorphism.

Observe that the stabilizer of GL(2,A/b)B(c) at every closed point of M(bc) has order prime

to pby Lemma 9.2(i). Thus the lemma is a consequence of the following result.

Lemma 9.4. Let f:M→Cbe a smooth morphism of varieties in characteristic pwith

target a smooth curve. Let Gbe a ﬁnite group which acts with stabilizers of order prime to p

on M. Assume that fis G-equivariant when Gacts trivially on C. Let c∈Cbe a closed point

such that Gacts with trivial stabilizers on a dense open subset of f−1(c). Write ¯

f:M/G →C

for the obvious map. Then the tautological dominant morphism f−1(c)/G →¯

f−1(c)is an

isomorphism.

Proof. Let Hbe the stabilizer of a closed point m∈M. Write ¯m∈M/H for the image of

m. Let Sbe the semi-local ring of Mat the orbit Gm. Then SGis a local ring. Let t∈SGbe

the image of a uniformizing parameter on Cat c. Consider the following commutative diagram

with injective vertical maps:

S/tS SH/tSH

oo

a

SG/tSG

oo

b

S/tS ⊃(S/tS)H⊃(S/tS)G

We must show that bis an isomorphism. The map ais an isomorphism since p|H|,so

H1(H, tS) = 0. Smoothness of fimplies that S/tS is non-singular, so (S/tS)Hand (S/tS)G

are normal. Hence SH/tSHis also normal. The localization of SHat ¯mis etale over SG[17,

V.2.2]. Since etaleness is preserved by base change, the localization of SH/tSHat ¯mis etale

TORSION CM CYCLES AND DRINFELD MODULES 561

over SG/tSG.NowSG/tSGis normal since SH/tSHis (see [24, I.3.17]). Since Sis integral

over SG,S/tS is integral over SG/tSGand bis an integral extension. Since (S/tS)Gis normal

and local, it is an integral domain.

To show that bis an isomorphism, it suﬃces to show that the map induced by bon fraction

ﬁelds is an isomorphism. This calculation may be done at any nearby point on f−1(c)/G.

Thus we may assume that H=1. Now S(respectively S/tS) is ﬁnite and etale over SG

(respectively (S/tS)G). Hence S(respectively S/tS) is free of rank |G|as an SG(respectively

(S/tS)G) module. On the other hand, applying SGSG/tSGshows that S/tS is free of rank

|G|as an SG/tSG-module. Now the inclusion of fraction ﬁelds induced by bis an isomorphism

since the total quotient ring of S/tS has dimension |G|when viewed as a vector space over

either ﬁeld.

This completes the proof of Lemma 9.3.

Corollary 9.5. The map

ς0(c):M0(c)−→ Spec(A)

is smooth on the complement of ς0(c)−1(A/c).

Proof. We note that ς0(c) is ﬂat by Theorem 6.1(iii), the closed ﬁbers are non-singular

curves by Lemma 9.3, and the residue ﬁelds of closed points in Spec(A) are perfect.

Deﬁne dmby qdm=#k. A Drinfeld A-module over a scheme Sis said to have characteristic

mif the structure map factors, S→Spec(A/m)→Spec(A).

Proposition 9.6. (i) There is a canonical map, iF:˘

M0(c)k→M0(mc)k, such that the

induced map γk◦iFis the normalization of M0(c)k.

(ii) The composition γk◦wm◦iF:˘

M0(c)k→M0(c)kis the Frobenius morphism relative to

kcomposed with the normalization.

(iii) The map

˘

M0(c)k˘

M0(c)k

(iF,wm◦iF)

−−−−−−−→M0(mc)k

is surjective.

(iv) A cusp c⊂M0(mc)intersects the image of iFif and only if cKis not a ramiﬁcation

point of γK:M0(mc)K→M0(c)K.

(v) If the cusp cK∈M0(mc)Kis unramiﬁed under γK, then the cusp wm(cK)is ramiﬁed.

Proof. (i) Let bbe an auxiliary ideal as above. For a Drinfeld A-module over a scheme Sin

characteristic mwith full level bc-structure (G,Φ), the line bundle underlying Gis trivialized

by the choice of a section βx0of the bc-torsion subgroup. Such a Drinfeld module may be viewed

as a homomorphism Φ : A→O

S(S){τ}, where multiplication in the skew polynomial ring is

determined by bqτ=τb. Conjugation in this ring by F=τdmamounts to raising the coeﬃcients

of polynomials in τto the qdmth power. Now a→ F◦Φ(a)◦F−1is a Drinfeld A-module over

S, since the constant term in each skew polynomial Φ(a) is ﬁxed by conjugating by F.ThusF

gives a morphism of Drinfeld A-modules over S,Φ→ FΦF−1. Clearly Ker(F)isaΓ

0(m)-level

structure on Φ. Thus we have a natural transformation from the functor Drinfeld A-modules in

characteristic mwith Γ(bc)-level structure to the functor Drinfeld A-modules in characteristic

mwith Γ0(m)Γ(bc)-level structure. Both of these functors are representable, so the natural

transformation corresponds to a morphism between ﬁne moduli spaces. On the smooth curve

M(bc)kthis map separates points and tangent vectors and is thus an embedding. It extends

to a morphism ˜

iF:M(bc)k→M(m0,bc)k, since M(bc), and hence M(bc)k, is non-singular.

562 CHAD SCHOEN AND JAAP TOP

Taking the quotient by GL(2,A/b)B(c) gives a morphism

iF:˘

M0(c)k−→ M(m0,bc)k/GL(2,A/b)B(c)−→ M0(mc)k

with the property that the composition, γk◦iF:˘

M0(c)k→M0(c)k, induces the identity on

the open dense subscheme M0(c)k.

(ii) With notation as in part (i), consider the natural transformation Φ → FΦF−1. The

second Drinfeld A-module acquires a Γ(bc)-structure by applying Fto the Γ(bc)-structure

on Φ. The ﬁne moduli space M(bc)khas a standard description as a subspace of the aﬃne space

whose coordinates correspond to coeﬃcients of skew polynomials Φ(a) and to bc-torsion sections

[21, 2.5]. The endomorphism M(bc)k→M(bc)k, corresponding to the natural transformation

just described, raises all these coordinates to the qdm-power and is thus Frobenius relative

to k. This endomorphism descends to γk◦wm◦iF:M0(c)k→M0(c)k.

(iii) It suﬃces to show that the image is Zariski dense. Away from the ﬁnite set of

isomorphism classes of supersingular Drinfeld A-modules in characteristic m[16, p. 81], a

Drinfeld A-module of characteristic mover an algebraically closed ﬁeld has exactly two

Γ0(m)-structures [32, 4.3.2].

(iv) The composite ˜γk◦˜

iF:M(bc)k→M(bc)kis the identity, while the composite

˜γk◦wm◦iFis purely inseparable of degree #(A/m) by (ii). As noted in the proof of (i),

˜

iFextends to a morphism ˜

iF:M(bc)k→M(m0,bc)kwhich must be an embedding since

˜γk◦˜

iF= Id. It is clear that ˜

iF(M(bc)k) cannot meet any cusp of M(m0,bc) where ˜γis ramiﬁed.

Similarly, the embedded curve wm◦˜

iF(M(bc)k)⊂M(m0,bc) cannot meet any cusp where ˜γis

unramiﬁed. Now the result follows by taking the quotient by GL(2,A/b)B(c) and applying (iii).

(v) This follows from the proof of (iv).

10. Modular parametrizations of elliptic surfaces

Let YKbe an elliptic curve over Kwith neutral element e. The curve YKmay be regarded

as the generic ﬁber of a unique relatively minimal elliptic surface π:Y→Xwith section ¯e

containing e. Assume that YKhas split multiplicative reduction at x∞. The conductor of YK

is a divisor on Xwhich is uniquely expressible as a product of an ideal c⊂Awith x∞. The

complement of the support of the conductor, ˙

X⊂X, is the locus over which πis smooth.

Write ˙

Y,˙

W,˙ς0(c), ˙

M0(c), ˙

M1(1) and ˙

ϑ0(c) for the base changes of Y,W,ς0(c), M0(c), M1(1)

and ϑ0(c) with respect to this inclusion.

Theorem 10.1. (i) Let cK∈M0(c)Kbe a cusp. There is a non-constant morphism of

K-curves ρK:M0(c)K→YKsending cto e.

(ii) The morphism ρKextends to a ﬁnite morphism ˙ρ:˙

M0(c)→˙

Y.

Proof. (i) Let l=pbe a prime. Drinfeld’s great theorem yields a Gal( ¯

K/K)-submodule

of H1(M0(c)¯

K,Ql) which is isomorphic to H1(Y¯

K,Ql)[10,§11]. The Tate conjecture for

M0(c)K×YKgives a divisor on the product which induces an isomorphism on Galois modules

[33, Theorem 4; 37]. By adding and subtracting divisors of the form M0(c)K×eand m×YK

and by moving within a rational equivalence class, one may arrange that the divisor is a graph

of a morphism ρ0. Set L=K(M1(1)). From the commutative diagram

M0(c)K

ρ0

//

YK

Spec(L)//Spec(K)

(10.1)

TORSION CM CYCLES AND DRINFELD MODULES 563

we deduce that ρ0factors:

M0(c)K

ρ1

−→ YL

ρ2

−→ YK.

The image of cis a degree 1 point on YL. Write τ:YL→YLfor translation by minus the image

of c.NowρK:= ρ2◦τ◦ρ1gives a map with the required properties.

(ii) Choose an ideal bprime to csuch that bc is divisible by two distinct primes.

Set Y=Y×XM1(1). Set ¨

X=˙

X−Supp(b) and write ¨

M(bc), ¨

M0(c), ¨

M1(1) and ¨

Yfor,

respectively, the base changes of M(bc), M0(c), M1(1) and Ywith respect to the inclusion

¨

X→X. Since ¨

M0(c) is a quotient of ¨

M(bc), by (i) there is a rational map

˜ρ:¨

M(bc)−→ ¨

M0(c)ρ1

¨

Y(10.2)

of schemes over ¨

M1(1). Now ˜ρmay be made into a morphism by blowing up points of

indeterminancy. Since ¨

M(bc) is non-singular, by Theorem 5.3(i) any point of indeterminancy

would lead to a rational curve in a ﬁber of ¨

Y→¨

M1(1). Since no such rational curves exist, ˜ρis a

morphism. The ﬁber components of the natural map ¨

M(bc)→¨

M1(1) are permuted transitively

by GL(2,A/b)B(c), since ¨

ϑ0(c): ¨

M0(c)→¨

M1(1) has irreducible ﬁbers by Theorem 6.1(vi). By

(10.2), ˜ρis GL(2,A/b)B(c)-equivariant. Thus it does not contract any ﬁber component of

¨

M(bc) to a point. Since ˜ρis proper, it is ﬁnite.

The veriﬁcation that the rational map ρ1is a ﬁnite morphism may be done locally on an

open aﬃne space, Spec(R)⊂¨

Ywith inverse image Spec(S)⊂¨

M(bc): since Ris contained in

both Sand the GL(2,A/b)B(c)-invariants of the rational functions on ¨

M(bc), it is contained

in the GL(2,A/b)B(c)-invariants of S, which gives an aﬃne open subscheme of ¨

M0(bc). Since

Sis a Noetherian ﬁnite R-module, so are the invariants. This shows that ρ1in (10.2) is a ﬁnite

morphism.

Since ˙

Xis covered by open subsets of the form ˙

X−Supp(b) with bas above, the rational

map ρ1extends to a ﬁnite morphism ˙

M0(c)→˙

Y. Composing with ˙

Y→˙

Ygives a ﬁnite

morphism ˙ρ:˙

M0(c)→˙

Y.

11. Proof of the main theorem

We keep the notation of previous sections. In particular, Fis a ﬁnite ﬁeld of characteristic

p,Xis a smooth, projective, geometrically irreducible curve over F,π:Y→Xis a relatively

minimal, semi-stable, non-isotrivial, elliptic surface with section, and ˙

X⊂Xis the locus over

which πis smooth. Set ˙

W=˙

Y×˙

X˙

Yand write ˙

f:˙

W→˙

Xfor the tautological map. Let

i¯x:f−1(¯x)→W¯

Fdenote the inclusion of a ﬁber above a CM point ¯x∈X¯

F.

Theorem 0.2 will follow easily from the following result.

Theorem 11.1. The image of i¯x∗:CH1(f−1(¯x)) →CH2(˙

W¯

F)is a torsion group.

Proof. Observe that CH1

hom(f−1(¯x)) is a torsion group since it is isomorphic to the ¯

F-

points of the abelian variety Pic0(f−1(¯x)). With notation as in §3, the N´eron–Severi group

of f−1(¯x) is generated by Ti·f−1(¯x), for 1 i3, and a complex multiplication cycle. Since

¯xgives a torsion class in CH1(˙

X¯

F), f−1(¯x)∈CH1(˙

W¯

F)tors and Ti·f−1(¯x)∈CH2(˙

W¯

F)tors.

Thus to complete the proof of Theorem 11.1, it suﬃces to show that a single 1-cycle supported

in f−1(¯x), whose class in the N´eron–Severi group N1(f−1(¯x)) does not lie in the subgroup

N1

0(f−1(¯x)) = Span{Ti·f−1(¯x)}1i3, is torsion in CH2(˙

W¯

F).

It is at this point that the theory of Drinfeld modular curves enters. Fix a point x∞∈X−˙

X.

After replacing Fby a ﬁnite extension (also denoted F) we may assume that x∞is a degree

1 point and that Yhas split multiplicative reduction at x∞. We may similarly arrange that

the point x∈˙

X, which is the image of ¯x, is a degree 1 point. Let A=H0(X−{x∞},OX),

564 CHAD SCHOEN AND JAAP TOP

and write m⊂Afor the maximal ideal corresponding to xand c⊂Afor the conductor of YK

divided by x∞.

Consider the composition of ﬁnite morphisms

˘

M0(c)k

iF

−−−→˙

M0(mc)(γ,γ◦wm)

−−−−−−→˙

M0(c)×˙

X˙

M0(c)˙ρ×˙ρ

−−−→˙

W. (11.1)

By Proposition 9.6(i) and (ii), the image of (γ, γ ◦wm)k◦iFis the graph of the k-linear

Frobenius morphism in ˙

M0(c)2

k. Applying ( ˙ρ, ˙ρ)∗to this graph gives deg( ˙ρ)Γ, where Γ ⊂Y2

kis

the graph of Frobenius on the elliptic curve Yk. By Theorem 6.1(iv), the irreducible components

of ˙

M0(c)kare the ﬁbers ˙

ϑ0(c)−1(zi), where {z1,...,z

s}=−1(m)⊂˙

M1(1). Let Cibe the

closure of iF(ϑ0(c)−1(zi)). Then

(˙ρ, ˙ρ)∗◦(γ, γ ◦wm)∗Ci=eiΓ with ei>0.(11.2)

The connected components of ˙

M0(mc)kare the ﬁbers ˙

ϑ0(mc)−1(zi). Besides Cithis ﬁber has a

second irreducible component C

i, by Proposition 9.6(iii).

Fix a cusp ˙c⊂˙

M0(mc) which is not contained in the ramiﬁcation locus of

˙γ:˙

M0(mc)−→ ˙

M0(c).

Such cusps exist by Lemma 9.1. Write K(˙

M0(mc)) and L=K(M1(1)) for the ﬁelds of rational

functions on ˙

M0(mc) and M1(1).

Lemma 11.2. There are N0,...,N

s∈Nand β∈K(˙

M0(mc))∗such that

div(β)=N0(˙c−wm˙c)+

s

i=1

NiCi∈Div( ˙

M0(mc)).(11.3)

Proof. By the Manin–Drinfeld theorem, Theorem 7.1, there exist β∈K(˙

M0(mc)) and

N∈Nsuch that, on the generic ﬁber,

div(β)=N(cK−wmcK)∈Div( ˙

M0(mc)K).

Thus there is a divisor Dsupported on closed ﬁbers of ˙

ϑ0(mc) so that

div(β)=N(˙c−wm˙c)+D∈Div( ˙

M0(mc)).(11.4)

Every closed ﬁber of ˙

ϑ0(mc) may be regarded as a torsion element of CH1(˙

M0(mc)), since every

closed point of the aﬃne curve ˙

M1(1) may be regarded as a torsion element of CH1(˙

M1(1)).

At the expense of replacing Nby a multiple and replacing βby a power of itself times an

element of L∗, we may arrange that Dis supported on the reducible ﬁbers of ˙

ϑ0(mc).

The only reducible ﬁbers of ˙

ϑ0(mc) are those that lie above m, by Theorem 6.1(vi). Thus we

may write

D=

s

i=1

niCi+n

iC

i.(11.5)

Let νidenote the order of ziin Pic(M1(1)). Choose i∈L∗with div(i)=−νizi.By(11.4)

and (11.5), C

iappears with zero multiplicity in the prime decomposition of the divisor

div(n

i

i(β)νi). By iterating this process, we produce a function βof the form s

i=1 Mi

i(β)M,

with div(β)=N0(˙c−wm˙c)+s

i=1 NiCi, where N0∈N. Intersecting with Cigives

0 = deg(div(β)|Ci)=N0˙c·Ci+NiCi·Ci=N0˙c·Ci−NiC

i·Ci,

since wm˙c·Ci= 0 by Proposition 9.6(iv) and (v), Cj·Ci= 0 for i=jand (Ci+C

i)·Ci=0

because Ci+C

iis the ﬁber ˙

ϑ0(mc)−1(zi). The intersection theory being used here is that for

a quotient variety as described in [12, 8.3.12 and 16.1.13]. This is permissible since ˙

M0(mc)is

the quotient of a non-singular variety by the action of a ﬁnite group (cf. Section 6). Note that

TORSION CM CYCLES AND DRINFELD MODULES 565

˙

M0(mc) may be singular at points of Ci∩C

i(see [14, 5.8]). As ˙c·Ci>0 and C

i·Ci>0, we

have Ni>0.

Since ˙ρsends one cusp of ˙

M0(c) to the zero section of ˙

Y, it sends other cusps to torsion

sections of ˙

Y. Choose m∈Nsuch that multiplication by m,m∈End( ˙

Y/ ˙

X), sends each of

these torsion sections to the zero section. Observe that

g:= (m◦˙ρ, m◦˙ρ)◦(γ, γ ◦wm): ˙

M0(mc)−→ ˙

W

maps both cusps ˙cand wm˙cto the zero section of ˙

W. Thus applying g∗to (11.3) yields

0=g∗(div(β)) = m2s

i=1

NieiΓ∈CH2(˙

W).(11.6)

Now Γ, and hence the positive multiple of Γ which appears in (11.6), does not lie in the

subspace N1

0(f−1(¯x)) of N1(f−1(¯x)). Applying the pullback map CH2(˙

W)→CH2(˙

W¯

F) proves

Theorem 11.1.

Proof of Theorem 0.2.In the notation of Proposition 3.5, Theorem 11.1 implies that

˜

j∗(CH2

CM(W¯

F)) is a torsion group. Since CH2

CM(W¯

F)⊂CH2

hom(W¯

F), Proposition 3.5 implies

that CH2

CM(W¯

F) is a torsion group. Now Theorem 0.2 follows from Propositions 3.3 and 3.4,

Lemma 2.1, and Proposition 1.1.

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Chad Schoen

Department of Mathematics

Duke University

Box 90320

Durham, NC 27708-0320

USA

schoen@math·duke·edu

Jaap Top

IWI

University of Groningen

P.O. Box 800

9700 AV Groningen

The Netherlands

top@math·rug·nl