Non-abelian Chabauty study group

74 Pages·2017·0.51 MB·English

Non-abelian Chabauty study group
Giulio Bresciani, Julian L. Demeio, Guido Lido, Davide Lombardo
2017–2018
2
Contents
1 Colemanintegration 5
1.1 IntegratingonJacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
(cid:82)
1.2 Constructionof intherigidsetting . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Affinerigidgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Monsky-Washnitzercohomology . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Specialization;thealgebraoflocallyanalyticfunctions . . . . . . . 14
1.2.5 ConstructionoftheColemanintegral . . . . . . . . . . . . . . . . . 15
2 ThemethodofChabauty(viaColemanintegration) 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Chabauty-Coleman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Boundsonthenumberofrationalpoints . . . . . . . . . . . . . . . . . . . 20
3 Gruppofondamentaleétale 25
3.1 Rivestimentiétalefiniti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Ilgruppofondamentaleétale . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Alcuneproprietà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Schemiingruppi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 TheDeRhamunipotentfundamentalgroup 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Definitionof π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1,DR
4.3 Thecaseofthethrice-puncturedprojectiveline . . . . . . . . . . . . . . . . 47
4.4 Furtherconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 ThecanonicalDeRhaminvariantpath . . . . . . . . . . . . . . . . . . . . . 50
Q
5 ProofofSiegel’stheoremover 53
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 DeRhamUnipotentFundamentalGroup . . . . . . . . . . . . . . . . . . . 53
5.3 Thefundamentaldiagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 TheUnipotentÉtaleFundamentalgroup . . . . . . . . . . . . . . . . . . . 55
5.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5 Themorphism D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3
4 CONTENTS
5.6 ProofofSiegel’stheorembyKim’smethod . . . . . . . . . . . . . . . . . . 58
5.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Periodrings 63
6.1 Wittvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1.3 Teichmüllerrepresentatives . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.4 PropertiesoftheWittvectors . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Periodrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.1 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.2 Periodsforthecyclotomiccharacter . . . . . . . . . . . . . . . . . . 66
6.2.3 Hodge-Taterepresentations . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.4 ConstructionofBdR . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B
6.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
cris
Chapter 1
Coleman integration
1.1 Integrating on Jacobians
Q
Let K be a p-adic field, that is, a finite extension of , and let J be the Jacobian of a
p p
smoothcurve C/K . Thepurposeofthischapteristoconstructanintegrationmap
p
H0(J,Ω1)× J(K ) → K
J p p
(cid:82)P
(ω,P) (cid:55)→ ω
thatsatisfies:
1. K -linearityin ω;
p
(cid:82)P+Q (cid:82)P (cid:82)Q
2. additivityin P: ω = ω+ ω
3. non-degeneracymodulotorsion: (cid:82)Pω = 0forallω ∈ H0(J,Ω1)ifandonlyif Pis
J
torsion.
Remark 1.1.1. In fact, the same construction goes through for any abelian variety over
a p-adicfield.
Lemma1.1.2. LetCot(J) bethecotangentspaceto J attheorigin. Theevaluationmap
ev : H0(J,Ω1) → Cot(J)
J
ω (cid:55)→ ω(0)
isanisomorphism.
Proof. The map P (cid:55)→ τ∗ω (from J to Cot(J)) is algebraic. Since J is complete and Cot(J)
P
is affine, it must be constant. This proves that any differential form is translation in-
variant, so ev is injective. It is also surjective, for example because the two spaces have
the same dimension, or because any differential form defined at 0 can be extended to a
translation-invariantdifferentialform.
5
6 CHAPTER1. COLEMANINTEGRATION
Lemma 1.1.3. For any ω ∈ H0(J,Ω1) there exists a unique analytic map λ : J(K ) → K
J ω p p
suchthat dλ = ω, λ (0) = 0,and λ : J(K ) → K isagrouphomomorphism.
ω ω p p
Proof. Writeω(0) = ∑Fdz forsome F ∈ K [[z ,...,z ]]. Byanobviousanaloguetothe
i i i p 1 g
Poincaré lemma, there exists G ∈ K [[z ,...,z ]] such that dG = ω, and G converges
p 1 g
on some open ball B. Without loss of generality1, we can assume that B is an (open)
(cid:0) (cid:1)
subgroupof J(K ). Bycompactness, J(K ) : B =: N isfinite2,andwecanset
p p
1
λ (P) = G(NP).
ω
N
Onechecksthatdλ = ω: thisisclearon B,andbytranslation-invarianceitisthentrue
ω
on all of J. To show that λ is a homomorphism, notice that (at least when restricted
ω
(cid:82)b
to B) the map ω := λ (b)−λ (a) has all the formal properties of integration (by
a ω ω
definition), hence for a,b ∈ B such that a+b ∈ B (which is automatic, because B is a
subgroup)wehave
(cid:90) a+b (cid:90) a (cid:90) a+b
λ (a+b) = ω = ω+ ω
ω
0 0 a
(cid:90) a (cid:90) b
= ω+ τ∗ω = λ (a)+λ (b).
a ω ω
0 0
This easily implies that λ is a homomorphism. Finally, uniqueness follows from local
ω
uniquenessatzerocombinedwiththerequirementthat λ beahomomorphism.
ω
(cid:82)P
Definition1.1.4. Weset ω = λ (P).
ω
Proposition 1.1.5. This definition satisfies conditions (1)-(3) in the definition of the Coleman
integral.
Proof. 1. Let ω ,ω be two elements of H0(J,Ω1) and λ ,λ be the two associated
1 2 J ω1 ω2
homomorphisms. Set λ := λω +λω : we want to prove that λ = λω +ω . This
1 2 1 2
follows by uniqueness of λω +ω and linearity of d, since λ(0) = 0, dλ = d(λω +
1 2 1
λ ) = ω +ω ,and λ isahomomorphism.
ω2 1 2
2. Directconsequenceofthepreviouslemma.
(cid:82)P
3. Wehave ω = 0forall ω ifandonlyif λ (P) = 0forall ω. DenotebyTanJ the
ω
tangentspaceattheoriginto J(K ) andconsiderthemap
p
ι : J(K ) → Tan(J)
p
x (cid:55)→ (ω (cid:55)→ λ (x)).
ω
1atheoremofMattuck[Mat55]saysthatJ(Kp)containsanopensubgroupisomorphic(asatopological
group) to OKgp. In particular, the images in J(Kp) of the sets (pjOKp)g form a basis of neghbourhood
around0consistingofopensubgroups.
2this number should be thought of as a large power of p; however, due to the possible presence of
torsion,itisnottrueingeneralthatNisexactlyapowerof p.
(cid:82)
1.2. CONSTRUCTIONOF INTHERIGIDSETTING 7
We claim that this is a locally injective homomorphism. Linearity is clear, and
local injectivity follows from the fact that the differential at 0 of this map is the
identity (indeed dλ | = ω). In particular, kerι is finite, and hence a subgroup of
ω 0
thegroupoftorsionpoints. OntheotherhandTan(J)istorsion-free,soTorsJ(K )
p
issentto0by ι. Thisprovesthat ι inducesaninjectivemap
J(K )/TorsJ(K ) (cid:44)→ Tan(J)
p p
orequivalentlyapairing
J(K )/TorsJ(K )×Tan(J)∨ → K
p p p
which is non-degenerate on the left. Since we have a canonical identification of
(cid:16) (cid:17)
Tan(J)∨ with H0 J(K ),Ω1 wearedone.
p J
(cid:82)
1.2 Construction of in the rigid setting
Theconstructionoftheintegrationmapgivenintheprevioussectionreliesonthegroup
structure of the Jacobian; in this section we shall sketch a much more general construc-
tion of the Coleman integral, which however relies on deeper results (specifically, the
studyoftheeigenvaluesofFrobeniusontheso-calledMonsky-Washnitzercohomology
ofthevarietyforwhichwewanttoconstructtheintegral).
1.2.1 Notation
Q
Let K be a p-adic field (finitely generated extension of ) with ring of integers R, uni-
p
formizer π, and quotient field k. For later use, we fix an automorphism σ of K that
reducestothe p-powermapon k andextenditto K.
1.2.2 Affine rigid geometry
TheTatealgebrais
(cid:40) (cid:41)
T := K(cid:104)t ,...,t (cid:105) := ∑a tI : lim |a | → 0 .
n 1 n I I
|I|→∞
Itisaverynicering! Forexample,itsatisfiesWeierstrasspreparationanddivision:
Definition1.2.1. AWeierstrasspolynomialis
f(t ,...,t ) = tm +tm−1g (t ,...,t )+···+g (t ,...,t ),
1 n 1 1 m−1 2 m 0 2 m
whereeach g isanelementof T and g (0,...,0) = 0.
i n−1 i
8 CHAPTER1. COLEMANINTEGRATION
Theorem1.2.2. Thefollowinghold:
• let f ∈ T be such that f(0,...,0) = 0. Suppose that the power series f(t ,...,t ) has
n 1 n
at least one term involving only one variable3 (say t ): then we have f = uW with u a
1
unitandW aWeierstrasspolynomial.
• given f ∈ T and a Weierstrass polynomial g, there exist q ∈ T and a Weierstrass
n n
polynomial r suchthat f = qg+r.
We also have a version of Noether’s normalization, and therefore the weak Null-
stellensatz: every maximal ideal is a Galois conjugacy class of geometric points. More
formally,
Spm(T ) = {K−homomorphism ψ : T → K}/Gal(K/K)
n n
ThemaximalspectrumoftheTatealgebraistherefore
(cid:110) (cid:111)
n
Spm(T ) = (z ,...,z ) ∈ K : |z | ≤ 1 /Gal(K/K).
n 1 n i
×
To see that points need to have integral coordinates, notice that if z ∈ K is non-
i
integral,then t −z = −z (1−z −1t ) isaunitin T ,soitcannotmaptozero.
i i i i i n
n
The unit polydisk is B := {(z ,...,z ) ∈ K : |z | ≤ 1}. An affinoid algebra is
n 1 n i
a K-algebra with a surjective map T → A for some n. The prototypical example is
n
A = T /(t t −1) (whichisa p-adicanalogueofS1):
2 1 2
(cid:110) (cid:111)
Spm(A) = {(z ,z ) ∈ B : z z = 1} = x ∈ K : |x| = 1 /Gal(K/K).
1 2 2 1 2
Remark 1.2.3. T is a Banach algebra with respect to the so-called Gauss norm: for a
n
series f = ∑a xI, its Gauss norm is (cid:107)f(cid:107) = max |a |. Every affinoid algebra inherits a
I I I
structureofBanachalgebra(theGaussnormpassestothequotientbecauseanyidealin
theTatealgebraistopologicallyclosed).
Remark 1.2.4. Proofs for many useful facts concerning Tate algebras can be found in
http://www.mat.uc.cl/~rmenares/TateAlgebras.pdf.
1.2.3 Monsky-Washnitzer cohomology
Weshallworkwiththeso-calledwcfgalgebras:
Definition1.2.5. 1. The standard weakly complete finitely generated (wcfg) algebra
is
(cid:40) (cid:41)
T † = ∑a tI : a ∈ R,∃r > 1suchthat lim |a |r|I| = 0 .
n I I I
|I|→∞
2. Awcfgalgebraisan R-algebra A† withasurjectivehomomorphism T † → A†.
n
3Noticethatonecanalwaysreducetothissituationbyanappropriatechangeofvariables.
(cid:82)
1.2. CONSTRUCTIONOF INTHERIGIDSETTING 9
3. Givenawcfgalgebra A†,its(π-adic)completionis A(cid:98) := lim A†/πnA†.
←−n
Notice that the Gauss norm on the Tate algebra T restricts to a norm on T † (with
n n
respecttothisnormT † isnotcomplete). Weshallneedthefollowingtheorem,whichis
n
aconsequenceoftheso-calledArtinapproximationproperty:
Theorem1.2.6. (see[vdP86]andthereferencestherein)Thefollowinghold:
• Let ε > 0. Givenadiagramofwcfgalgebras
A
f
(cid:15)(cid:15)
(cid:111)(cid:111)
B/J B
g
and a morphism u(cid:98) : A(cid:98) → B(cid:98) such that f(cid:98) = g(cid:98)◦u(cid:98), there exists a morphism u : A → B
suchthat f = g◦u and |u−u| ≤ ε.
(cid:98)
• Let ε > 0. Givenadiagramofwcfgalgebras
A(cid:79)(cid:79)
f
g (cid:47)(cid:47)
C B
and a morphism u(cid:98) : A(cid:98) → B(cid:98) such that g(cid:98) = f(cid:98)◦u(cid:98), there exists a morphism u : A → B
suchthat f = g◦u and |u−u| ≤ ε.
(cid:98)
Differentials
Themodulesofdifferentialsassociatedwith A† = T †/(cid:104)f ,..., f (cid:105) are
n 1 m
(cid:76)A†dt
Ω1 := i
A† (cid:28) (cid:12) (cid:29)
∂fjdt (cid:12) j = 1,...,m
∂t i (cid:12)
i
A†
and
Ωk := ΛkΩ1 .
A† A†
Theyareprojective A†-modules. ThedeRhamcomplex Ω• is
A†
0 → Ω0 → Ω1 → ··· → Ωk .
A† A† A†
10 CHAPTER1. COLEMANINTEGRATION
Cohomology
Ifif A† isawcfgalgebrathen A = A†/π isanhonestfinitelygenerated k-algebra.
Definition 1.2.7. The Monsky-Washnitzer cohomology of A, denoted by H (A/K), is the
MW
cohomologyofthedeRhamcomplex Ω• ⊗K.
A†
It is a theorem of Berthelot [Ber97] that Hi (A/K) is a finite-dimensional K-vector
MW
spaceforall i.
Theorem1.2.8. (Elkik[Elk73],vanderPorten[vdP86])
1. Let A be a smooth finitely generated k-algebra. There exists a flat wcfg algebra A† such
that A = A†/π.
2. Anytwosuchliftsareisomorphic.
3. Anymorphism f : A → B canbeliftedtoamorphism f† : A† → B†.
4. Any two maps f , f with the same reduction modulo π induce homotopic maps Ω• ⊗
0 1 A†
K → Ω• ⊗K.
B†
Proof. 1. It suffices to show that there is a smooth lift, and then weakly-complete it.
Existenceofthesmoothliftisshownin[Elk73].
2. By flatness and the fact that A is smooth, one obtains that A/πnA and B/πnB
are smooth over R/πnR for all n. In particular, there is a projective system of
morphisms A(cid:98) → B(cid:98). By Artin approximation, there is a morphism i : A → B
which is an isomorphism modulo π. We want to show that it is an isomorphism.
By flatness, A/πA ∼= πnA/πn+1A and B/πB ∼= πnB/πn+1B, hence in particular
keri ⊆ (cid:84) πnA = (0). Asforsurjectiveness,seeLemma1.2.9below.
n
(cid:127) In [vdP86] it is claimed that surjectivity modulo π is “a consequence of the
Weierstrass theorems", but no details are given. Any errors in the previous proof
(orinLemma1.2.9below)areentirelydowntotheauthorofthesenotes.
3. Similarto2.
4. We only give a sketch. The idea is to mimic the proof of Poincaré’s lemma: we
look for maps α ,α : B(cid:104)T(cid:105)† → B† and f : B(cid:104)T(cid:105)† → B† such that f = α ◦ f. We
0 1 i i
define α bysending T to i. Now
i
Ωq+1(B(cid:104)T(cid:105)†/K) = B(cid:104)T(cid:105)† ⊗ Ωq+1(B/K)⊕B(cid:104)T(cid:105)†dT⊗Ωq(B(cid:104)T(cid:105)†/K)
B
andonedefines δ tobe0onthefirstsummandand
q
(cid:32) (cid:33)
(cid:90) 1
g⊗ω (cid:55)→ g(t)dt ω
0