ArticlePDF Available

Drinfeld modules and torsion in the Chow groups of certain threefolds

Authors:

Abstract

Let E → B be an elliptic surface defined over the algebraic closure of a finite field of characteristic greater than 5. Let W be a resolution of singularities of E ×B E. We show that the l-adic Abel–Jacobi map from the l-power-torsion in the second Chow group of W to H3(W,Zl(2)) ⊗ Ql/Zl is an isomorphism for almost all primes l. A main tool in the proof is the assertion that certain CM-cycles in fibres of W → B are torsion, which is proven using results from the theory of Drinfeld modular curves.
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 EBbe 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 WBare 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−→ H2r1(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 :EBbe 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¯
FV¯
F, obtained by removing the singular fibers of the map V¯
FB¯
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 xcwith split multiplicative reduction at the point xis
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 zZ2(˙
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(f1(b)) ib
−−CH2(V¯
F)u
−→ CH2(Eη×ηEη)−→ 0,
where ηB¯
Fis the generic point. Our results imply that imibCH2
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:YXis 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:PQis a morphism of schemes over a field k, and kkis a
field extension, then the base changed morphism ck:PkQkwill 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, FF¯
F, with [F:F]<,GF=
Gal( ¯
F/F), H=H2r1(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(H2r1(U¯
F,Zl(r)) Ql/Zl)GFH2r1(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−→ H2r1(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 H2r1(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 VUFis the group Hr
et(V, Ql/Zl(r)). Consider the diagram
CHr(UF)tors Zl
ρF
//H2r(UF,Zl(r))
Hr1
Zar (UF,Hr(Ql/Zl(r))) γF
//
αF
OO
H2r1(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−→ H2r1(U¯
F,Ql/Zl(r)),
satisfying λrlimFαF= 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= KerfFβF:H2r1(UF,Ql/Zl(r)) H2r(U¯
F,Zl(r)),
HF= Ker(fFβFγF).
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
H2r1(U¯
F,Ql(r))GF/HGF
dF
OO
(1.5)
The map eFarises from the Hochschild–Serre spectral sequence for cohomology with μr
lm
coefficients by taking the projective limit with respect to m. The map limFeFρFis 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−→ μr
lm−→ μr
ln+m−→ μr
ln−→ 1 (1.6)
are the Bockstein (coboundary) maps
βi
F(n, m):Hi(UF
r
ln)−→ Hi+1(UF
r
lm),(1.7)
analogous maps βi
¯
F(n, m) with ¯
Freplacing Fin (1.7), and the Galois coboundary map
δ(n, m) : Ker(β2r1
¯
F(n, m))GF−→ H1(GF,Coker(β2r2
¯
F(n, m))),
TORSION CM CYCLES AND DRINFELD MODULES 549
coming from the short exact sequence of GF-modules
0−→ Coker(β2r2
¯
F(n, m)) −→ H2r1(U¯
F
r
ln+m)−→ Ker(β2r1
¯
F(n, m)) −→ 0.(1.8)
Define
L1H2r(UF
r
lm) = KerH2r(UF
r
lm)s
−→ H2r(U¯
F
r
lm),
H2r1(UF
r
ln)0= Ker(sβ2r1
F(n, m)),
Hmn = Coker(β2r2
¯
F(n, m)),
Kmn = Ker(β2r1
¯
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
r
lm)eF(n,m)
//H1(GF,H2r1(U¯
F
r
lm)) //H1(GF,Hmn)
H2r1(UF
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 :UFSpec(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 fmay 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
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 n1(cone(v)) ψn1
−−→H
n1(B·[1]) n1
−−→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=2r1,
X:= im(ψn1)→H
n1(B·[1]) im(n1) = Ker(μn)=:Y. (1.10)
The cohomology group H2r1(UF
r
ln) is identified with the hypercohomology group
R2r1fC·. The map βF(n, m) in Lemma 1.3 is induced by the map on the hypercohomology
R2r1fC·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:R2r1f(cone(u)) R2r1f(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=2r1,
Y//
κ0
Rnf(cone(u)) ν//
κu
Rnf(cone(v))
κv
fY //fHn(cone(u)) n
//fHn(cone(v))
(1.11)
where κuand κvare standard maps from the spectral sequences Y:= Ker(κvν) and
fY Ker(n). 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)) //R1fHn1(cone(v)) R1
//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)QlHQl/(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 (HQl/Zl)GF.
Remark 1.5. The cycle class map
λr:CHr(U¯
F)tors Zl−→ H2r1(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 :EBbe 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:YX, 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 π1(π(yi)). These singularities
are resolved by a single blow-up of the reduced singular locus. We denote this blow-up by
σ:WY×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 ˜
hκ1,
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:YXdenote 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:YXis smooth will be denoted j:˙
XXand 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:WXbe the tautological map.
There are only finitely many points ¯x˙
X¯
Ffor which π1x) 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(π1x)) is an order in an imaginary quadratic number field
and the N´eron–Severi group N1(f1x)) is a free Z-module of rank 4. Write Δ¯xf1x)
π1x)×π1x) for the diagonal and N1
0(f1x)) N1(f1x)) for the rank 3 submodule
spanned by {π1x)×sx),sx)×π1x),Δ¯x}, where s:XYis a section of π.
Definition 3.1. Let ¯x˙
X¯
Fbe a CM point. A 1-dimensional cycle zsupported in the fiber
f1x) is called a complex multiplication (CM) cycle if the class of zin N1(f1x)) generates
the free Z-module N1
0(f1x))of rank 1.
The subgroup of CH2(W¯
F) generated by the classes of all CM cycles in Z1(f1x)) 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(f1x)) 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 ˙
WWfor 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(f1¯
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 ˜
jto 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 zZ2
hom(W¯
F)is
supported on f1¯
F) then the class of zin CH2(W¯
F) is torsion.
Write z=ξΞ¯
Fzξwith zξZ1(f1(ξ)). Write s:XYfor 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 T3Wbe the strict transform of the diagonal in Y×XY.
Write ai,ξ for the intersection number Ti·zξ. Define
S1=T1+T2+T3,
S2=T1T2+T3,
S3=T1+T2T3,
z
ξ=2zξ
1i3
ai,ξSi·f1(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:WYfor 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(π1(ξ),Ql(1))].
Since the cohomology classes of the components of π1(ξ) together with the cohomology
class of the section sgenerate a subgroup of H2(Y¯
F,Ql(1)) which maps surjectively to
H2(π1(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·f1(ξ)=2δij .(3.1)
This proves Lemma 3.6.
Lemma 3.7. The class of z
ξin CH1(f1(ξ)) is torsion.
Proof. Recall that π1(ξ)Yis a fiber of Kodaira type Imfor some m>0. The fiber
f1(ξ) has 2m2irreducible components: m2of these correspond to irreducible components of
π1(ξ)×π1(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(f1(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(f1(ξ)),Q).
Let SW¯
Fbe a very ample non-singular hypersurface whose intersection with each component
of f1(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(g1(ξ)) //
Q
Nξ(W¯
F)
OO
CH1(f1(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=2m21.
Observe that Ti·f1(ξ)CH1(f1(ξ)) lies in the right kernel of the bottom row of (3.2) since
on W¯
F,Ti·f1(ξ) is numerically equivalent to Ti·f1(x) for any xX¯
F. As the pairing
Span{Si}1i3Span{Ti·f1(ξ)}1i3−→ Q
is non-degenerate by (3.1), rank(h)2m21 + 3. By Lemma 3.8, his surjective.
To complete the proof of Proposition 3.5 we need only show that ξΞ¯
Fai,ξSi·f1(ξ) gives
a torsion class in CH2(W¯
F). Since Ti·ξΞ¯
Fz
ξ= 0, it follows that ξΞ¯
Fai,ξ = 0. Since Fis
finite, ξΞ¯
Fai,ξξCH1(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 xXbe a closed point. Set A=H0(X−{x},OX).
Fix an integer r>0. For aA\{0}, define daZby 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 GS
and a ring homomorphism Φ : AEnd(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 aA\{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:ABis 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, λΦ, of Drinfeld A-modules over Sis a homomorphism of S-group
schemes satisfying λΦ(a)=Φ
(a)λfor all aA.
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 bA, with b= (0), be
an ideal. Any element bbdefines 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 bbwill 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)rG(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
HG[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 TSexists, and there is
a point P(G×ST)(T) such that
H×ST=
xA/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 bAis 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 mAwrite ˆ
Amfor the m-adic completion of A. Let Kdenote
the quotient field of A,Kthe completion at the place x,AKthe valuation ring,
πAa 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)/FId 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 KK(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)/FId, 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)/FId −→ Pic(A)×GL(1,A/b)/FId.
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 gGL(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 gϕ=ϕg1.
(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
{0N(b),where N(b):=(A/b)A/b
0F
qGL2(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 cAis 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)/FId. 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 zM1(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 bAas 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 bAas above. Let zM1(bc) lie over zM1(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 bAas above so that (z)/Spec(A/bc). Let zM1(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
KM0(c)K,cKc
KCH1(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(cKc
K)=0.Thuswe
may realize a positive multiple of cKc
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
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 GG/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
ΛCsatisfying (mc)1ΛΛΛ and Λ/Λ
=(mc)1A/A as A-modules. In this language
the Atkin–Lehner map wmsends a pair (Λ,Λ)to(Λ
,m1Λ). Iterating 2ntimes gives
(mnΛ,mnΛ), 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 cM0(c),γ1(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 mAwill 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 cKM(bc)K, the fiber ˜γ1
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)KM0(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 ˜γ1
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 FAAut(Φ) or
FAut(Φ) where Fis a quadratic extension of F.
(ii) For each maximal ideal mAthere 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:= FFA.
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 AA/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)/FId 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:MCbe 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 cCbe a closed point
such that Gacts with trivial stabilizers on a dense open subset of f1(c). Write ¯
f:M/G C
for the obvious map. Then the tautological dominant morphism f1(c)/G ¯
f1(c)is an
isomorphism.
Proof. Let Hbe the stabilizer of a closed point mM. Write ¯mM/H for the image of
m. Let Sbe the semi-local ring of Mat the orbit Gm. Then SGis a local ring. Let tSGbe
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 f1(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, SSpec(A/m)Spec(A).
Proposition 9.6. (i) There is a canonical map, iF:˘
M0(c)kM0(mc)k, such that the
induced map γkiFis the normalization of M0(c)k.
(ii) The composition γkwmiF:˘
M0(c)kM0(c)kis the Frobenius morphism relative to
kcomposed with the normalization.
(iii) The map
˘
M0(c)k˘
M0(c)k
(iF,wmiF)
−−−−−−M0(mc)k
is surjective.
(iv) A cusp cM0(mc)intersects the image of iFif and only if cKis not a ramification
point of γK:M0(mc)KM0(c)K.
(v) If the cusp cKM0(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→ FΦ(a)F1is 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→ FΦF1. 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)kM(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, γkiF:˘
M0(c)kM0(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 → FΦF1. 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)kM(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 γkwmiF:M0(c)kM0(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)kM(bc)kis the identity, while the composite
˜γkwmiFis purely inseparable of degree #(A/m) by (ii). As noted in the proof of (i),
˜
iFextends to a morphism ˜
iF:M(bc)kM(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:YXwith 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 cAwith x. The
complement of the support of the conductor, ˙
XX, is the locus over which πis smooth.
Write ˙
Y,˙
Wς0(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 cKM0(c)Kbe a cusp. There is a non-constant morphism of
K-curves ρK:M0(c)KYKsending 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:YLYLfor translation by minus the image
of c.NowρK:= ρ2τρ1gives 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=˙
XSupp(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
¨
XX. 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 ˙
XSupp(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:YXis a relatively
minimal, semi-stable, non-isotrivial, elliptic surface with section, and ˙
XXis the locus over
which πis smooth. Set ˙
W=˙
Y×˙
X˙
Yand write ˙
f:˙
W˙
Xfor the tautological map. Let
i¯x:f1x)W¯
Fdenote the inclusion of a fiber above a CM point ¯xX¯
F.
Theorem 0.2 will follow easily from the following result.
Theorem 11.1. The image of i¯x:CH1(f1x)) CH2(˙
W¯
F)is a torsion group.
Proof. Observe that CH1
hom(f1x)) is a torsion group since it is isomorphic to the ¯
F-
points of the abelian variety Pic0(f1x)). With notation as in §3, the N´eron–Severi group
of f1x) is generated by Ti·f1x), for 1 i3, and a complex multiplication cycle. Since
¯xgives a torsion class in CH1(˙
X¯
F), f1x)CH1(˙
W¯
F)tors and Ti·f1x)CH2(˙
W¯
F)tors.
Thus to complete the proof of Theorem 11.1, it suffices to show that a single 1-cycle supported
in f1x), whose class in the N´eron–Severi group N1(f1x)) does not lie in the subgroup
N1
0(f1x)) = Span{Ti·f1x)}1i3, is torsion in CH2(˙
W¯
F).
It is at this point that the theory of Drinfeld modular curves enters. Fix a point xX˙
X.
After replacing Fby a finite extension (also denoted F) we may assume that xis 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 mAfor the maximal ideal corresponding to xand cAfor the conductor of YK
divided by x.
Consider the composition of finite morphisms
˘
M0(c)k
iF
−−˙
M0(mc)(γ,γwm)
−−−−−˙
M0(c)×˙
X˙
M0(c)˙ρ×˙ρ
−−˙
W. (11.1)
By Proposition 9.6(i) and (ii), the image of (γ, γ wm)kiFis 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
sNand βK(˙
M0(mc))such that
div(β)=N0cwm˙c)+
s
i=1
NiCiDiv( ˙
M0(mc)).(11.3)
Proof. By the Manin–Drinfeld theorem, Theorem 7.1, there exist βK(˙
M0(mc)) and
NNsuch that, on the generic fiber,
div(β)=N(cKwmcK)Div( ˙
M0(mc)K).
Thus there is a divisor Dsupported on closed fibers of ˙
ϑ0(mc) so that
div(β)=Ncwm˙c)+DDiv( ˙
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 iLwith 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)=N0cwm˙c)+s
i=1 NiCi, where N0N. Intersecting with Cigives
0 = deg(div(β)|Ci)=N0˙c·Ci+NiCi·Ci=N0˙c·CiNiC
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 CiC
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 mNsuch that multiplication by m,mEnd( ˙
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 gto (11.3) yields
0=g(div(β)) = m2s
i=1
NieiΓCH2(˙
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(f1x)) of N1(f1x)). 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.
References
1. S. Abhyankar,Resolution of singularities of embedded algebraic surfaces (Academic Press, New York,
1966).
2. A. Beilinson, ‘Height pairing between algebraic cycles’, Current trends in arithmetical algebraic geometry,
Arcata, CA, 1985 (ed. K. A. Ribet), Contemporary Mathematics 67 (American Mathematical Society,
Providence, RI, 1987) 1–24.
3. S. Bloch, ‘Torsion algebraic cycles and a theorem of Roitman’, Compositio Math. 39 (1979) 107–127.
4. S. Bloch, ‘Algebraic cycles and values of L-functions’, J. reine angew. Math. 350 (1984) 94–108.
5. J.-L. Colliot-Th´
el`
ene, ‘Cycles alg´ebriques de torsion et K-th´eorie alg´ebrique’, Arithmetic algebraic
geometry, Trento, 1991 (ed. E. Ballico), Lecture Notes in Mathematics 1553 (Springer, New York,
1993) 1–49.
6. J.-L. Colliot-Th´
el`
ene, J.-J. Sansuc and C. Soul´
e, ‘Torsion dans le groupe de Chow de codimension
deux’, Duke Math. J. 50 (1983) 763–801.
7. P. Deligne, ‘Th´eorie de Hodge II’, Inst. Hautes ´
Etudes Sci. Publ. Math. 40 (1972) 5–57.
8. P. Deligne and D. Husemoller, ‘Survey of Drinfeld modules’, Current trends in arithmetical algebraic
geometry, Arcata, CA, 1985 (ed. K. A. Ribet), Contemporary Mathematics 67 (American Mathematical
Society, Providence, RI, 1987) 25–91.
9. M. Deschamps,‘R´eduction semi-stable’, eminaire sur les pinceaux de courbes de genre au moins deux
(ed. L. Szprio), Ast´erisque 86 (Soci´et´e Math´ematique de France, Paris, 1981) 1–34.
10. V. G. Drinfel’d, ‘Elliptic modules’, Math. USSR Sbornik 23 (1976) 561–592.
11. M. Flach, ‘A finiteness theorem for the symmetric square of an elliptic curve’, Invent. Math. 109 (1992)
307–327.
12. W. Fulton,Intersection theory (Springer, New-York, 1984).
13. O. Gabber, ‘Sur la torsion dans la cohomologie l-adique d’une varieti´e’, C. R. Acad. Sci. Paris S´er. I Math.
297 (1983) 179–183.
14. E.-U. Gekeler,‘
¨
Uber Drinfeldsche Modulkurven vom Hecke-Typ’, Compositio Math. 57 (1986) 219–236.
15. E.-U. Gekeler, ‘A note on the finiteness of certain cuspidal divisor class groups’, Israel J. Math. 118
(2000) 357–368.
16. E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel (eds), Drinfeld modules, modular
schemes and applications (World Scientific, Singapore, 1997).
17. A. Grothendieck,Revˆetements ´etales et groupe fondamental (SGA 1), Lecture Notes in Mathematics
224 (Springer, New York, 1971).
18. G.-J. van der Heiden, ‘Weil pairing and the Drinfeld modular curve’, PhD thesis, Groningen, 2003,
http://irs.ub.rug.nl/ppn/25493871X; parts published as ‘Factoring polynomials over finite fields using
Drinfeld modules’, Math. Comp. 73 (2004) 317–322, ‘Local-global problem for Drinfeld modules’,
J. Number Theory 104 (2004) 193–209, ‘Weil pairing for Drinfeld modules’, Monatsh. Math. 143 (2004)
115–143, and ‘Drinfeld modular curve and Weil pairing’, J. Algebra 299 (2006) 374–418).
566 TORSION CM CYCLES AND DRINFELD MODULES
19. U. Jannsen,Mixed motives and algebraic K-theory, Lecture Notes in Mathematics 1400 (Springer, New
York, 1990).
20. N. M. Katz and B. Mazur,Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108
(Princeton University Press, 1985).
21. T. Lehmkuhl, ‘Compactification of the Drinfeld modular surfaces’, Habilitationsschrift, G¨ottingen, 2000.
22. Q. Liu,Algebraic geometry and arithmetic curves (Oxford University Press, 2002).
23. S. Mildenhall, ‘Cycles in a product of elliptic curves and a group analogous to the class group’, Duke
Math J. 67 (1992) 387–406.
24. J. Milne,´
Etale cohomology (Princeton University Press, 1980).
25. C. Schoen, ‘On the computation of the cycle class map for nullhomologous cycles over the algebraic closure
of a finite field’, Ann. Sci. ´
Ecole Norm. Sup. (4) 28 (1995) 1–50.
26. C. Schoen, ‘On the image of the l-adic Abel–Jacobi map for a variety over the algebraic closure of a finite
field’, J. Amer. Math. Soc. 12 (1999) 795–838.
27. C. Schoen, ‘Specialization of the torsion subgroup of the Chow group’, Math. Z. 252 (2006) 11–17.
28. C. Schoen, ‘Torsion in the cohomology of fiber products of elliptic surfaces’, Preprint, Duke University,
2002.
29. C. Schoen, ‘Divisor class groups of certain normal crossing surfaces’, Preprint, Duke University, 2002.
30. J. H. Silverman,The arithmetic of elliptic curves (Springer, New York, 1986).
31. C. Soul´
e, ‘Groupes de Chow et K-th´eorie de vari´et´es sur un corps fini’, Math. Ann. 268 (1984) 317–345.
32. L. Taelman, ‘Drinfeld modular curves have many points’, Master’s Thesis, Groningen, 2002,
arXiv:math.AG/0602157.
33. J. Tate, ‘Endomorphisms of abelian varieties over finite fields’, Invent. Math. 2 (1966) 134–144.
34. J. Tate, ‘Relations between K2and Galois cohomology’, Invent. Math. 36 (1976) 257–274.
35. B. Totaro, ‘Torsion algebraic cycles and complex cobordism’, J. Amer. Math. Soc. 10 (1997) 467–493.
36. C. Weibel,An introduction to homological algebra (Cambridge University Press, 1994).
37. Y. Zarhin, ‘Endomorphisms of abelian varieties over fields of finite characteristic’, Math. USSR Izvestija
9 (1975) 255–260.
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
Book
0 Introduction.- 1. Local Theory.- 1 Terminology and preliminaries.- 2 Resolvers and principalizers.- 3 Dominant character of a normal sequence.- 4 Unramified local extensions.- 5 Main results.- 2. Global Theory.- 6 Terminology and preliminaries.- 7 Global resolvers.- 8 Global principalizers.- 9 Main results.- 3. Some Cases of Three-Dimensional Birational Resolution.- 10 Uniformization of points of small multiplicity.- 11 Three-dimensional birational resolution over a ground field of characteristic zero.- 12 Existence of projective models having only points of small multiplicity.- 13 Three-dimensional birational resolution over an algebraically closed ground field of charateristic ? 2, 3, 5.- Appendix on Analytic Desingularization in Characteristic Zero.- Additional Bibliography.- Index of Notation.- Index of Definitions.- List of Corrections.