Anabelian Geometry
A modern overview
Japan Europe Number Theory Exchange Seminar
Copenhagen - Nagoya
7 Dec., 2021

Benjamin Collas
RIMS - Kyoto University, Japan

Caveat emptor. Dense and not selfexplanatory slides ahead; additional examples and comments were written during the talk.
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What is arithmetic anabelian geometry
For X algebraic variety over k: does (parts of) the Arithmetic-geometric Ex. Seq.:pr

1 → π1et (X ⊗ k̄, ∗) → π1et (X, ∗) → Gal(k̄/k) → 1
reconstructs (parts of) the variety X?
Some insights.
I Tate Conjecture: A, B abelian varieties.

(AG)[relative/semi-abs./absolute]
Faltings’83

b ' HomGk [H1 (AK̄ , Zb), H1 (BK̄ , Zb)]
HomK (A, B) ⊗ Z
I Analytic/Grp. Th. input: For X smooth curve of genus g.
SGA1top
Riemann existence theorem∆X ' πb1 (X(C)an )The r-cusps inertia gives a system of generators of ∆X .
⇒ Typical context: Take X hyperbolic curve (2g-2+r>0), i.e ∆X is not abelian.
In today’s talk:
1. Anabelian principles
Nakamura‘80s;Tamagawa,Mochizuki‘90s“There exists a functorial group theoretic algorithm”
2. A new anabelian Diophantine geometry
Mochizuki,Hoshi 2010-20
Inter-universal Teichmüller geometry, beyond ring structures
3. Anabelian, A1 and motivic geometries...
(...)
Grothendieck-Teichmüller th.
Hoshi,Minamide,Mochizuki 17+;-Tsujimura 21+
c 6 Out[π1 (M)](AG) for X ∈ M = {Mg,m }g,m moduli spaces of curves: GQ 6 GT
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Bi-anabelian geometry over global fields
Let Xi be affine (r>0) hyperbolic curves over K f.g. field over Q, then:∼IsomK (X1 , X2 ) → IsomGK [ΠX1 , ΠX2 ]∼(BiK )The (g,r)-geometrical context

Smooth comp. X ,→ X̄:I g = genus(X̄),
I cusps r = #(X̄ \ X)(K̄)
For x ∈ X̄, in ∆X and ΠX :I decomp. Dx = StabΠX (x)I inertia Ix = Dx ∩ ∆X groups.
Reconstruction I [Nakamura]
Grp th. & weights
1. ΠX∆X < ΠX . As max. top. f.g. H /cl ΠX
2. ΠX(g, r). Abelianization and Deligne Frob. weights in étale homology:
0 → Q` (1) → ⊕x Q` (1) → ∆abX ⊗ Q` → T` (Jac∆ ) ⊗ Q` → 0
Distinguishes genus and inertia generators
2g and 2g − r + 1.
b(1).
3. ΠX
Ix . I is cuspidal if self-normalizing and I ' Z
For g(X)=0: Steps 1.-3. & a (K × , ) argument recover X ' P1 \ {0, 1, ∞, λ}Page 3

(Bi)-anabelian geometry: global to local fields
Reconstruction II [Tamagawa]Good red. & Rat. pts & Func. field
1. Oda’s anabelian good reduction. For v place of K, a “Serre-Tate” Out-test.
(tame)(Biv ) ∀0 v ∈ V(K): π1(Xkv ) the red. over kv finite field
Xkv ⇒ (BiK )2. Section and limits of k-points. For s : Gk → ΠX a section:∀(Im s < H) /op ΠX : YH (k) → X 6= ∅ by Lefschetz tr. on `-cohom.
x∞ = lim yH (k) ∈ X(k)Decomposition Dx =< Im s >.
Im s<H

3. The (K(Xv )× , ). Local class field theory as K(Xv )× = Ker ΨYb x× /O
bx× ×KYΨ

b × −→ ΠabKX

Then the field K(Xv ) via (techn.) valuation and functions.
Reconstruction III [Mochizuki]
Containers & Line Bdles
Let Xi be proper hyperbolic curves over K a sub-p-adic field.
(p),ab

∃

1. Hodge-Tate container for X. (∆X⊗ Cp )GK ' H 0 (X, ωX )
X ,→ P(ΓΩX)Ω
2. A p-adic Tamagawa rat. construction. A M-split
x∞ ∈ P(ΓX )(L)
3. Chern cl. of LB. A L-line bundle give a L-point (grp. th.)
I Use “only” ∆X /[∆X , [∆X , ∆X ]]“minimal reconstruction?”
T-Saïdi,Y.’20
I Use a “quotient ZX ” in the Malcev of ∆X“Lie anab. tech.?”Sawada‘20.
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Mono-anabelian geometry: “There exists a group theoretical algorithm...”Mono-anabelian Geometry. Reconstruction of X starting from ΠX only.
A. Tamagawa: “should be considered as involving all objects: One for all, all for one!
From ΠX to function & base fields K(X) and k. For X hyperbolic over k MLF.
I Inertia groups Ix . Belyi cuspidalizations
Curve of strict Belyi type.
{ΠUX → ΠX }UX ,→X
set of Ix , x ∈ X \ UX .
××
I Multiplicative groups K(X) & k . A container:κΓ(U, OU ),→H 1 (ΠU , µZb(ΠX ))×1
1k ,→ H (Gk , µZb(ΠX )) ,→ H (ΠX , µZb(ΠX ))...After cycl. sync. µZb(ΠX ) ' Ix for x ∈ X̄ \ X.

I The fields K(X) and k. Uchida (val. tech.)k MLF - From Gk to ... For Gk y Ok× < Ok. < Ok < k× as -monoids:
Local Class Field th.⇒ G
G y O× (G), O. (G), k× (G) as G-monoids
1. But Γ-orbits of Étale/Frobenius isom. of cyclotomes Λ(G) ' Λ(M ) for G y M
2. But no (Ring, , ) nor field reconstructions!
(bi-anab. for MLF)
Category of Quasi-tripods.
Hoshi’19+
Curves X st X
∃X1...
Xn = P1{0,1,∞} with ∈ {f.et, (f.et)−1 , op.im.}⇒ “Any smooth variety has an absolute anabelian basis”(Higher dim. bi-anab.)Page 5

Anabelian Diophantine Geometry - Inter-universal Teichmüller theory
Mono-anabelian results lead to a Diophantine heights inequality:
htωX (D) . (1 + )(log-diffX + log-condD )The log-theta lattice of HTs.

Log-theta latt. of Hodge-theta theatres {◦,◦ HT }:Gkv y Ok.v
(1, 1)-curves
Cat. Gl. offor
X/K, v a K-place.
Πv = π1tp (Xkv )Nb. ∃ containers & cycl. rigidity for theta-fct. ofE(kv )rig ' Gm /q Z wrt π1tp with ΠX -comp.
Mono-anabelian group theoretic reconstruction
A multi-radial algorithm
I Mono-anab. transport between Étale & Frobenius objects:
κO†×k −→ H 1 (G† k , Λ(† O))

bZ× −Ind−→∼κ−1

H 1 (G‡ k , Λ(‡ O)) −→ O‡×kI For /-monoids (Kum., theta): (Gv , Πv , ∆v , ...)

(O. (G), ∆Θ , M(Π), ...)

I Log-shell container: That contains: (O†.k , ) log→ (O‡.k , )Results
1. abc-conjecture via a Galois-Vojta reformulation.

Mochizuki’12+

2. Fermat Last Theorem via IUT-refined (Belyi & theta eval.)

FHMP’21+

3. Grothendieck Section Conj (Loc.-Glob. birat.) via IUT-Galois.

Hoshi-M
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Anabelian, Motivic & A1 -geometries...
For X non K(π, 1) and of higher dimension, replace π1et by the étale topological type
et{X}et ∈ Pro-Sp
Gives access to πN(X) for N = 1.
Unifying the three insights of Grothendieck: Arithmetic geometry, motivic theory
and Quillen model categories:
Friedlander’s pro-spaces, Morel-Voevodsky’s motivic homotopy, and Ayoub’s motives

SchQ
St(Q)

(−)A1{−}et*

4 Pro-SpoRetS

4 SH(Q)O

Σ∞/ H(Q)O{−}etNΓ)DAet (Q)(−)A1Spaces

Anab. Geo.

A1 -Geo.

A few (prospective) results:
I Higher dim.: Anabelian “polycurves” over number fields
I Higher dim.: non A1 -rigid anabelian obstruction
I Higher sym.: formalism extends from schemes to stacks

MotivesSchmidt-Stix’16...
...

... Requires further geometric foundational techniques ...
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For Further Reading...
This list of references should be used as entry points for the primary sources ([pGC],...)On anabelian geometry
Y. Hoshi, Introduction to Mono-anabelian geometry, Proc. “Fundamental Groups inArithmetic Geometry”, 2016.
S. Mochizuki, Topics in absolute anabelian geometry I, II & III, 2012, 2013, 2015.
H. Nakamura, A. Tamagawa and S. Mochizuki, The Grothendieck conjecture on thefundamental groups of algebraic curves, Sugaku Expositions (AMS), 14 (2001).
Survey with bibliographical references and notes.
On Inter-universal Teichmüller geometry
Invitation to inter-universal Teichmüller Theory & Inter-universal Teichmüller Theory
Summit, Collas, Fesenko, Hoshi, Mochizuki, Taguchi (org.), RIMS Expanding Horizonsof IUT workshop, (2021).
#IUT-WS1 - #IUT-WS2
Notes and recordings of the talks of the 2 workshops (avail. on registration).
B. Collas, Promenade in Inter-universal Teichmüller theory, RIMS-Lille seminar, 2021.
Also provides an overview of absolute mono-anabelian reconstructions.
#Text
Recent developments in Anabelian Arithmetic Geometry
Homotopic and Geometric Galois Theory, Collas, Dèbes, Nakamura & Stix (edt.),
Oberwolfach report No. 12/2021, (2021).
#Text
With references to Hoshi, Minamide, Tsujimura, Sawada’s recent works.
Thank you for your attention :: ご清聴ありがとうございました
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