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LMS Symposia 2011
University of Durham
18 – 28 July 2011
ANABELIAN GEOMETRY
— A short survey —
Florian Pop, University of Pennsylvania
LECTURE 1: GT versus I/OM
I) Motivation/Introduction
Non-tautological description of
GalQ = Aut(Q)
The toy example: GalR via πtop1
- Var R category of R-varieties X.
- GalR ∼= {±1} acts on Xan := X(C) and πtop1 (Xan)
• Get a representation:
ρX : GalR → Out(πtop
1 (X))
- Consider πtop1 : Var R → Groups (outer hom’s)...
• (ρX)X gives rise to repr ıR : GalR → Aut(πtop1 )
FACT: ıR is an isomorphism.
Comment: Got non-tautological description of Gal R...
1
•What about other subfields of C ?
- Λ ⊆ C base field, e.g. Q, Q(x1, . . . , xn), R, Qp
- VarΛ geometrically integral Λ-varieties X.
Comment/Note: In general, Gal doesn’t act on πtop1
∗ How to do it: Replace πtop1 by
• πalg1 := (profinite compl. of πtop
1 ) is a functor
πalg1 : VarΛ → prof.Groups (outer hom’s)
- There exists canonical exact sequence:
1→ πalg1 (X, ∗)→ πet
1 (X, ∗)→ GalΛ → 1
- Represent ρX : GalΛ → Out(πalg
1 (X))
• Finally (ρX)X gives rise to a morphism:
ıΛ : GalΛ → Aut(πalg1 )
2
II) Studying GalQ via ıQ
Question/Problem:
1) Find categories V for which Aut(πalgV ) has
“nice” topological/combinatorial description.
2) Find such categories V for which
ıV : GalK → Aut(πalgV ) is an isomorphism.
• This would give a new description of GalΛ!!!
Example: Teichmuller modular tower
- Mg,n moduli space of curves
(genus g, with n marked pts.)
- “Connecting” morphisms
(“boundary” embeddings, gluings, etc.)
- T = {Mg,n | g, n} category of varieties
over Λ = Q, the Teichmuller modular tower.
- Note: M0,4∼= P1
Q \ {0, 1,∞} = P01∞.
M0,n∼= (M0,4)n−3 \ {fat diagonal}
3
FACTS (to Question/Problem 1):
- Harbater-Schneps: Let V0 := {M0,4,M0,5}.Then Aut(πalg
V0) = GT is the famous
Grothendieck–Teichmuller group.
- [MANY]: The variants IGT , IIGT , IVGT , newGT
are all of the form VGT := Aut(πalgV )...
• Conclusion: Question/Problem 1 has quite
satisfactory answer(s) for V ⊆ T .
FACTS (to Question/Problem 2):
Injectivity: Λ number field. Then
ıV : GalK → Aut(πalgV ) injective, provided:
- Drinfel’d (using Belyi Thm): P01∞ ∈ V
- Voevodsky: C ∈ V, C ⊂ E hyperbolic curve...
- Matsumoto: C ∈ V, C affine hyperbolic curve
- Hoshi-Mochizuki: C ∈ V, C hyperbolic curve
4
Surjectivity:
• Ihara/Oda–Matsumoto Conj (I/OM).
ıQ : GalQ → Aut(πalgVarQ
) is isomorphism.
FACTS:
- I/OM has positive answer.
- Actually much stronger assertions hold, e.g...
• Given Λ ⊂ C and X ∈ VarΛ, dim(X) > 1, set:
- VX = {Ui ⊂ X}i ∪ {P01∞}, with morphisms:
inclusions Uj ↪→ Ui, projections Ui → P01∞.
- (P): ıVX: GalQ → Aut(πalg
VX) is isomorphism.
Note: With X = P2Q, this gives (in principle)
a pure topol./combin. construction of GalQ!
5
III) Variants
Replacing πalg1 by variants
(compatible with the Galois action of GalΛ)
A) The pro-` variant
∗ Replace πalg1 by its pro-` quotient, hence:
- πalg1 (X) by its pro-` quotient πalg,p
1 (X)
- πV by the corresponding π`V- Aut(πalg
V ) by Aut(π`V)...
Questions:
1) Describe the image ı`V(GalΛ)...
2) Consider the pro-` I/OM...
- Lot of intensive research here concerning 1),
e.g., by Matsumoto, Hain–M, Nakamura, etc.
Comment: This generalizes Serre’s question/result about/on...
- The pro-` I/OM is true in stronger form...
6
B) The pro-` abelian-by-central I/OM
• Bogomolov’s Program
Comment:
- The “Yoga” of Grothendieck’s anabelian geometry...
Bogomolov (1990). Consider:
- K|k function field, td.deg(K|k) > 1, k = k.
- ΠcK → ΠK abelian-by-central/abelian
pro-` quotients of GalK , ` 6= char.
Conjecture (Bogomolov’s Program, 1990):K|k can be recovered from Πc
K functorially.
Comments:- tr.deg(K|k) > 1 is necessary, because...
- This goes far beyond Grothendieck’s anabelian idea...
FACT (State of the Art):Bogomolov’s Progr OK for tr.deg > dim(k) + 1.
Comments:- dim(k) = 0: tr.deg > dim(k) + 1 is equiv to...
- The case tr.deg > dim(k) + 1 follows by...
7
Back to: Pro-` abelian-by-central I/OM:
∗ Replace the profinite groups by the theirpro-` abelian-by-central quotient, hence:
- πalg1 (X) by its quotient Πc(X)→ Π(X)
- πV by the corresponding ΠcV → ΠV
- Aut(πalgV ) by Autc(ΠV) := im
(Aut(Πc
V)→ Aut(ΠV))...
• Get the pro-` abelian-by-central I/OM...
FACT (P): ıcVX: GalΛ → Autc(ΠVX
) is isom.
Comments:- Actually one proves the “birational variant” which is stronger.
- This is a special case of Bogomolov’s Program.
- Bogomolov’s Program would imply stronger assertions, like:
∗ Let dim(X) > dim(Λ) + 1 be geom rigid...
- Note: If Λ ⊂ Q, then dim(X) > 2 is enough.
- Consider UX = {Ui}i subcategory of VX .
FACTS (P): ıcUX: GalΛ → Autc(ΠUX
) and
ıUX: GalΛ → Aut(πUX
) are isomorphisms.
8
C) The tempered GT and I/OM (Andre)
∗ Replace C by Cp and πalg1 by the the corresp.
tempered (alg.) fundam. group πtemp1 hence:
- πV by the corresponding πtempV
- Aut(πalgV ) by Aut(πtemp
V )...
• Get the tempered variant GTtemp
of GT .
• Get the tempered I/OM for Λ|Qp finite.
FACTS (Andre):
1) ıtempΛ : GalΛ → Aut(πtemp
VarΛ) is isom.
2) ıQ(GalQ) ∩ GTtemp
= ıtempQp
(GalQp).
Comments:- Actually πtemp
1 (X) ⊂ πalg1 (X) for all X .
- There are “tempered variants” of other
aspects involving fundamental groups too.
9
IV) Final Question/Comments
• Arithmetical I/OM (question by David Burns)
• Pro-linear/pro-unipotent I/OM (question by Minhyong Kim)
• (Generalized) Drinfel’d upper half plane
10
Supplement 1: On GT
• Exact sequence:
1→ F2 → π1(P01∞)→ GalQ → 1,
where P01∞ := P1Q\{0, 1,∞} tripod...
and F2 = 〈τ0, τ1, τ∞ | τ0τ1τ∞ = 1〉 = Fτ0,τ1 ...
• GT = {(λ, f) | λ ∈ Z×, f ∈ [F2, F2], rel.I,II,III}the famous Grothendieck–Teichmuller group.
• ∃ can embedd GalQ ↪→ Aut(F2).
- Intensively studied by Drinfel’d, Ihara, Deligne,
Schneps, Sch.–Lochak, Sch.–Nakamura,
Sch.–Harbater, I–Matsumoto, Furusho, etc...
- Several variants IGT , IIGT , IVGT , etc. of GT .
- Actually: rel. I, II, III, are not independent
(Schneps, Sch.–Lochak; Furusho: III suffices)
- Boggi–Lochak (to be thoroughly checked):
Some variant newGT equals Aut(πalgT ).
Supplement 2): Belyi’s Theorem
- Recall RET: There exists equiv of categories
Topology&Geometry: Compl. algebraic curves:
compact Riemann projective smoothsurfaces X complex curves X
Function fields:
function fields F inone variable over C
X ←→ M(C) = F = C(X) ←→ X
• Basic Question (Grothendieck):
Which X , hence which X, hence which F ,
are defined over Q ⊂ C, hence number fields?
Theorem (Grothendieck/Belyi).
X is defined over Q ⇔ ∃ X → P01∞ etale.
Proof: “⇒” by Belyi (nice and tricky!)
“⇐” by Grothendieck (etale fundam. groups)
1
Comments:- This is the origin of Grothendieck’s “Designs d’enfants”.
- A cover X → P1C as in Theorem is a Belyi map.
- Study the action of GalQ on the space of “Designs”
(many many people: Malle, Kluners–M., Schneps,
Lochak–Sch., Zapponi, math-physicists, etc. etc. etc...)
Interesting open Question/Problem:
Higher dim extensions of Theorem above.
Two possible ways:a) Describe all X → P1
C with at most n branch points.
b) Replace curves by higher dim varieties, e.g., surfaces (?!?).
- Several partial results to b), but...
Theorem (Ronkine 2004; unpublished).
The birat. class of a complex proj. surfaceX0 of general type is defined over Q iff
∃ smooth fibration X0→→P1C\{0, 1,∞}.
2
Supplement 3) Bogomolov’s Program
Given: ΠcK → ΠK . Reconstruct K|k functorially.
Strategy of proof (P):
Main Idea: Consider P(K,+) := K×/k×
the “projectivization” of the k-v.s. (K,+).
• (K,+, ·) can be recovered from
P(K,+) endowed with its collineations,
via Artin’s Fundam. Thm. Proj. Geometries
NOW:
- Kummer Theory: K× = Homcont(ΠK ,Z`).
- And P(K,+) = K×/k× ↪→ K×.
Hence to do list: Given ΠcK→→ΠK ,
1) Recover K×/k× ↪→ K×.
2) Recover the collineations inside K×/k×.
3) Check compatibility with Galois Theory.
1
PLAN:
• Local Theory, i.e., recover:
- primes of K|k; divisorial sets DX of primes.
• Global Theory, i.e., recover:
- Div(X), then K×/k×, then collineations;
and finally check Galois compatibility.
THE GENERAL “NONSENSE”
Case k = Fp:- Pic0(X) is torsion group
- Valuations of k are trivial
Case tr.deg(K|k) > dim k:
- Specialization (Deuring, Roquette, Mumford)
- 1-motives techniques
- Reduce to the case k = Fp- Recover the “nature” of k
2
Local Theory (few words):
- primes of K|k: DVR Rv with k ⊂ Rv ⊂ Ksuch that tr.deg(Kv |k) = tr.deg(K|k)− 1.
- D = {vi}i geometric, if ∃ normal model X → k
such thatD = DX := {v |Weil prime div. of X}.
- Quasi prime divisors
• Recovering the (quasi) primes:
- 1st Method: Use B.-Tsch. “commuting pairs”...
- 2nd Method: Use techniques developed byWare, Koenigsmann, Mıac et al, Topaz...
Comment: This is very very technical stuff...
• Recovering (quasi) decomposition graphs
• Recovering rational quotients
• · · ·
LECTURE 2: Section conjectures
I) Motivation/Introduction
Effective version of the Mordell Cojecture...
(Grothendieck: Letter to Faltings, June 1983)
• Evidence: Minhyong Kim’s work...
Setting:
- k arbitrary base field; ki, ksep ⊂ k- X0 geom. integral k-variety, d = dim(X0)
- X ⊂ X0 open k-subvariety
• Canonical exact sequence:
1→ πalg1 (X, ∗)→ πet
1 (X, ∗)→ Galk → 1
• For x ∈ X0 regular ∃ vx on k(X) with:
vx(K×) = Zd and κ(vx) = κ(x).
- vx|vx prolong to k(X), get split exact seq
1→ Tvx → Zvx → Galκ(x) → 1.
Comments (about the splitting; tangential...)
1
Conclude: κ(x) ⊂ kins ⇒ Galκ(x) = Galk andprk(X) : Galk(X) → Galk has sections svx .
- X → X pro-etale universal cover
- X0 → X0 normaliz of X0 in k(X) ↪→ k(X)
- Galk(X) → πet1 (X) can projection
- x 7→ x centers of vx on X0 → X0
- Functoriality: Zvx → Zx and Tvx → Tx.
Conclude: κ(x) ⊂ kins ⇒ Galκ(x) = Galk andprX : π1(X)→ Galk has sections sx.
Cases:
a) x ∈ X: Then Tx = {1} and Zx = Galκ(x)
• x ∈ X with κ(x) ⊂ kins defines conj class of sx.
b) x ∈ X0\X: Then in general one has
Tx 6= {1} and Zx 6= Galκ(x).
• x ∈ X0\X with κ(x) ⊂ kins defines a
“bouquet” of sections ≈ H1cont(Galk, Tx).
2
Special case: X ⊆ X0 are smooth curves. Then:
- x ∈ X0\X iff Tx 6= 1.
- char(k) = 0 ⇒ Tx ∼= Z(1) as Gk-modules.Get H1
cont(Galk, Tx) ∼= k×...
Curve SC (Section Conjecture / Grothendieck)
k fin gen infinite, X → k hyperbolic non-isotrivial.Then all sections of prX are of the form sx.
Birational SC: k, X as above.
Then sections of prk(X) are of the form svx .
• Variants...
- p-adic Curve/Birat SC: Replace k fromCurve/Birat SC by a finite extension k|Qp.
- Geom pro-C Curve/Birat SC: Replace πalg1
from Curve/Birat SC by its pro-C completion...
- Etc. e.g., [ p-adic] (pro-C) Curve/Birat SC
•Note: ...Curve SC’s⇒ ...Birat SC’s [Comments]
3
II) Evidence/Facts SC:
A) “No sections” results...
- Stix, using the following local result:
• k|Qp finite, X hyperbolic. Then existenceof sections of prX ⇒ I(X) is a p-power.
- Harari–Szamuely(using the Manin-Brauer obstruction)
- Hain: k = Q(Mg), Xg gen curve of genus g.Curve SC true for Xg for g ≥ 5...
• Geom pro-p Curve SC:
- Hoshi: ...does not hold over k = Q[ζp]for Xp
0 +Xp1 +Xp
2 ⊂ P2 and p regular prime.
•Finally: Curve/Birational SC are widely open.
• Conditional result on Birational SC: k # field.
- Esnault–Wittenberg: X proj hyperbolic, with
qqqq(Jac(X)
)finite. If pr : Galgeom.ab
k(X) → Galkhas sections, then X has index 1.
4
B) Unconditional results:
• Koenigsmann: The p-adic Birat SC holds.
Comment: Actually a much stronger fact holds:
• k|Qp finite, and K|k regular extension. Then sections of
GalK → Galk originate from k-rational places of K .
- In particular, if K = k(X0) with X0 proj smooth curve,
then the sections originate from X0(k)...
Refinement: Minimalistic p-adic Birational SC
Context: k|Qp finite, µp ⊂ k, K = k(X0).
- k′′|k′ ↪→ K ′′|K ′ max Z/p metabelian ext’s.
- pr : Gal(K ′|K)→ Gal(k′|k) can projection.
- Liftable section s of pr is one coming from a
section of pr : Gal(K ′′|K)→ Gal(k′′|k)
• (P): “Bouquets” of liftable sections ↔ X0(k).
Comments: The less Galois theory, the better...
5
C) Partial results:
Tamagawa:
- k finite field, X ⊆ X0 hyperbolic.
- X → X univ pro-etale cover, K = k(X).
- Given s section of prX , consider all
Y → X finite sub-cover of X → X
such that im(s) ⊂ πet1 (Y ).
- |Y0(k)| = ∑2i=0
(−1)iTr(ϕk)∣∣Hi(Y 0,Z`(1)
)• A section s comes from a point x ∈ X0(k) iff
Y0(k) non-empty for all Y → X as above.
Nakamura:
- k number field, X ⊂ X0 affine hyperbolic.
• The points x ∈ X0\X are in bijection with
conjugacy classes of max subgroups ∆ ∼= Zof πalg
1 (X) of pure weight −2.
6
For the next three results:
- k|Qp fin, X ⊂ X0 hyperbolic, X → X univ cover
- prX : πet1 (X)→ Galk can projection
Mochizuki: Suppose X0 defined over Q.
• Then X0\X 3 x↔ maximal ∆ ∼= Z ⊂ πalg1 (X)
on which Galk acts via the cycl character.
Comment: ...the “absolute form” of the anab conj for curves.
Saidi: Defines “good sections” of prX ...
• The good sections ↔ X0(k).
Comment: Proof relies on:
- Mochizuki’s p-adic cuspidalization methods...
- Pop’s methods developed for the “minimalistic” result...
P–Stix:
• For every section s of prX ∃ valuation
w on k(X) such that im(s) ⊂ Zw.
Comments: ...relation to Mochizuki’s “combinatorial SC
- Equivalently: Every section comes from Berkovich points.
7
D) Section conjectures over R
Here: k = R, X ⊆ X0 hyperbolic, etc...
Wojtkowiak, Mochizuki, Wickelgren, etc...
πet1 (X)→ GalR has sections iff X0(R) 6= ∅
• One cannot expect a Curve SC over R...
Wickelgren:
- The geometrically pro-2 curve version holds.
- The pro-2 birational SC holds.
8
III) Final Comments
- Initial motivation of Grothendieck......Mordell Conjecture, now Faltings’ Theorem...)
- No relation between Curve SC and(an effective) Mordell Conjecture yet...
Minhyong Kim:
Using pro-unipotent completions of πalg1 designs
an algorithm (of p -adic nature) which —under
the conjectural properties of his “Selmer vari-
eties”— produces the rational points of the curve
in discussion; and more impressively, the effec-
tiveness of the algorithm is guaranteed by the va-
lidity of the Curve SC. This — I would claim —
sheds the right light on the relation and the role
of the Curve SC to an effective Mordell.
9
IV) Short list of open Problems:
1) Prove/disprove:
ıV : GalQ → Aut(πV) is not onto for V ⊆ T
2) Clarify/prove the pro-unipotent I/OM
3) Clarify the relation between the global and
the p-adic Curve/Birational SC
4) Prove/disprove the global/p-adic (birational)
section Conjecture.
5) Relation between the representations ıV and
linear represent of GalQ, respectively GalQp
10