The Absolute Anabelian Conjecture for Curves with resolution of Non-Singularities

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The Absolute Anabelian Conjecture for Curves with resolution of Non-Singularities

Emmanuel Lepage

IMJ-PRG, Sorbonne Université

Foundations and Perspectives of Anabelian Geometry

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Outline

X,Y: hyperbolic curves over finite extension ofQp

Absolute anabelian conjectureAACpX,Yq:

Isomorphisms of étale fundamental groupΠ_X Π_Y come from isomorphismsX Y

(absolute: not given with an augmentation map toG_K) Resolution of non-singularities (RNS_X):

Every semistable model ofX is dominated by the stable model of some finite étale cover ofX

Main result of this talk:

RNS_X & RNS_Y ùñ AACpX,Yq

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relative Grothendieck Anab. Conj. (Isom. version)

Theorem (S. Mochizuki)

X{K , Y{L : two hyperbolic curves over sub-p-adic fields.

Π_X,Π_Y: étale fundamental groups of X and Y . G_K,G_L: absolute Galois groups of L and K . Assume following commutative diagram:

Π_X ^//

Π_Y

G_K ^//G_L

such that G_K ÑG_Lis induced by an isomorphism K Ñ L.

ThenΠ_X ÑΠ_Y is induced by an isomorphism X Ñ Y

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absolute Anab. Conj. (Isom version)

Conjecture (AAC(X,Y), S. Mochizuki)

X{K , Y{L : two hyperbolic curves over p-adic fields.

ΠX,ΠY: étale fundamental groups of X and Y . Assume we have an isomorphism:

ϕ: ΠX Ñ ΠY

ThenΠX ÑΠY is induced by an isomorphism X Ñ Y Proposition

Under the same assumptions,ϕinduces an isomorphism G_K Ñ G_L (but not known to be geometric in general)

Remark

Neukirch-Uchida + Rel. AC ùñ AAC over Number Fields

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Curves of Quasi-Belyi type

Definition

A hyp. curveX is of pseudo-Belyi type if there are maps:

X Y

f.´et.

oo

dominant

P¹zt0,1,8u

Theorem (Mochizuki)

If X,Y are curves of pseudo-Belyi type, then AACpX,Yqis true.

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Intermediate steps

Xr limÐÝ^p^S,s⁰^qÑp^X^,x⁰^qS: universal pro-finite étale cover ofX. Natural actionΠ_X ñ rX

Definition

Letx closed point ofX andx˜P rX a preimage ofx.

D_˜_x StabΠXpx˜q ΠX: decomposition group ofx Dx conjugacy class ofD˜x.

An isomorphismϕ: ΠX Ñ ΠY is point-theoretic ifDΠX is a decomposition group if and only ifϕpΠ_Xqis a decomposition group.

Theorem (Mochizuki)

Letϕ: Π_X Ñ Π_Y be point-theoretic, thenϕis induced by an isomorphism X Ñ Y

Characterization of decomposition groups viacurspidalization techniques.

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Resolution of Non-Singularities

Definition

LetX be a hyperbolic curve over an algebraic closureQpofQp. X satisfies resolution of non-singularities (RNS_X) if for every semi-stable modelXofX, there exists a finite étale cover f :Y ÑX such thatf extends to a morphismYÑXwhereYis the stable model ofY.

Y ^st.model ^//

f. ét

Y X^semi^st.model^//X

LetX be a hyperbolic curve over a finite extensionKofQp.X satisfies resolution of non-singularities (RNS_X) if its pullback to an algebraic closure ofK does.

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Valuative version of RNS

Definition

A valuationv onKpXqis of type 2 if it extends the valuation ofQp and its residue fieldk˜_v is of transcendance degree 1 overFp.

Example

IfXis a normal model ofX andZ is a irreducible component of the special fiberX_s, thenvz mult_Z is a valuation of type 2 onKpXq. A valuation of this form whereXis the stable model (if it exists) is calledskeletal

Proposition

X satisfies resolution of non-singularities if and only if for every valuation v of type2on KpXq, there exists a finite étale cover Y ÑX and a skeletal valuation v¹ on KpYqsuch that v v_|¹_K_p_X_q.

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Example of Curves with RNS

A smooth curve overQpis a Mumford curve if every normalized irreducible component of its stable model is isomorphic toP¹. Theorem

Let X,Y be two hyperbolic curves overQpand assume that Y satisfies RNS.

1 If there is a dominant map f :X ÑY , then X satisfies RNS.

2 If there is a finite étale cover f :Y ÑX , then X satisfies RNS.

3 If X is a Mumford curve, then X satisfies RNS.

4 ùIf X is of pseudo-Belyi type, then X satisfies RNS.

X Y

f.´et

oo

dominant

Z :hyp.ppuncturedqMumford curve

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Main Result

Theorem

Let X and Y be two hyperbolic curves over finite extensions ofQp

satisfying RNS. Then every isomorphism of fundamental groups ϕ: Π_X Ñ Π_Y is induced by an isomorphism X Y .

Remark

Includes some proper curves, contrary to the quasi-Belyi type result.

Sketch of the proof:

Step 1: One just needs to show thatϕispoint-theoretic.

Step 2: Recovery of the topological Berkovich space.

Step 3: Characterization of rigid points.

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Berkovich spaces

LetX be an alg. variety{K non-archimedean field.

IfX SpecA,

X^an tmult.seminormsAÑR_¥0,extending norm ofKu topology: coarsest s.th. @f PA,x :|pxq|PX ÞÑ|fpxq|PRcont.

X ÞÑX^anfunctorial, maps open coverings to open coverings.

ùñ glues together for generalX.

set theor.,X^an tpx,||q;x PX,||:mult.norm onkpxqu Example of points:

ùñ X_cl ãÑX^an(rigid points)

XpKˆq ÑX^an (type 1 points (rigid points))

IfX is a smooth curve, valuations of type 2 onKpXqinduce points inX^an(type 2 points)

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Example: the affine line

LetX SpecpCprTsq A¹_C_p. IfaPCp,r PR_¥0,|°

ia_ipT aqⁱ|_b_a,r :max_ip|a_i|rⁱq ùba,r PA^1,an_C_p If r=0,b_a,r of type 1 (Cp-points).

Ifr P|p|^Q,b_a,r of type 2.

Ifr R|p|^Q,b_a,r of type 3 (rkp|KpXq|_b_a,rq 2).

+ points of type 4 corr. to decreasing sequences of balls with empty intersection, completion ofKpXqis an immediate extension ofCp.

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Berkovich curves

X{K: smooth curve over non-archimedean field X: smooth compactification ofX

X{O_K: semi-stable model ofX{K

GX: dual graph of the semi-stable curveX_s

There is a natural topological embeddingιand a strong deformation retractionπ:

GX ^ι //X^an

xx π

X^anzιpGXq: disjoint union of potential open disks (becomes a disk after finite extension of the base field).

By taking the inverse limit over all potential semi-stable models, they induce a homeomorphism

X^anÑ limÐÝ

X{K¹

GX

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Sketch of the proof

Step 2: Recovery of the topological Berkovich space.

Theorem (Mochizuki)

Let X{K and Y{L be two hyperbolic curves over finite extensions ofQp

that admit stable modelsX{O_K andY{O_L. They are naturally enriched as log-schemesX^log andY^log. Then every isomorphismΠ_X Ñ Π_Y induces an isomorphism of log-special fibersϕ^log :X^log_s Ñ Y^log_s .

In particular, it induces an isomorphism of dual graphs of the stable reduction:ϕ_G :GX Ñ GY.

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IfX satisfies RNS, one gets a natural homeomorphism Xr^an rX

an

:limÐÝ

S

SÑ limÐÝ

pS,sq

GS,

whereS goes through pointed finite étale covers ofX admitting stable reduction over their constant field (S: smooth compactification ofS).

Apply isom. pq_G to open subgps ofΠX andΠY,ùhomeomorphism ϕ˜:Xr^anÑ r Y^an

(compatible with the actions ofΠ_X andΠ_Y andϕ). Quotient by actions of the fundamental groups (resp. geom. fund. groups)ù

ϕân:XânÑ Yân presp. ϕân_C_p :Xân

C^p

Ñ Y^an

C^pq. Action compatibility ùñ If˜x P rX^an,ϕpDx˜q Dϕ˜p˜xq

ùTo show point-theoreticity, it is enough to show thatϕ^anmaps rigid points to rigid points.

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Does every homeomorphismX^anÑY^anpreserves rigid points?

No (cannot distinguish between type 1 and type 4 points).

Need of a stronger property about this homeomorphism.

Step 3: Metric characterization ofCp-points.

LetXbe a semi-stable model ofX_C_p,x a node ofX_s. ThenXétale loc. SpecO_C_pru,vs{puvaq.

Leteedge of dual graphGXofX.

Setlengthpeq:vpaq ùmetric onGX.

X_pân₂_qinj lim_XVpGXq ùnatural metricd onX_pân₂_q. ϕ^log :X^log_s Ñ Y^log_s ùñ GXÑ GYis an isometry apply to open subgroupsùϕân_p₂_q :Xân

Cp,p2q ÑY^an

Cp,p2qis an isometry.

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Letx₀PX_p^an₂_q. Proposition Let x PX_C^an

p, then x is aCp-point (is of type 1) if and only if:

dpx₀,xq:sup

U

zPinfU_p₂_qdpx₀,zq 8

where U goes through open neighbourhood of x in X^an

Cp. Sketch:

Metric extends toιpGXq ùreduce to the case of a disk inX^an

CpzιpGXq. In a disk, explicit description of the metric:

dpb_a,r,b_a1,r¹q |log_pprq log_ppr¹q| if|aa¹|¤maxpr,r¹q 2vpaa¹q logpprq logppr¹q if|aa¹|¥maxpr,r¹q Ifb_a1,r¹ Ñx of type 1,r¹ Ñ0 sodpba,r,b_a1,r¹q Ñ 8.

IfpBpa_i,r_iqqiPNis a decreasing seq of balls s.t.

iBpa_i,r_iq H, r_i Ñr ¡0.

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ùñ ϕ^anpreserves points of type 1.

Ifx is a point of type 1, thenx is rigid, if and only if the imageppDxqby the augmentation mapp: Π_X ÑG_K is open.

ùñ ϕ^anpreserves rigid points.

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IfX satisfies RNS andΠ_X Π_Y, doesY satisfies RNS?

Not known in general...

Theorem (Mochizuki)

LetQT (quasi-tripods) be the smallest family of hyperbolic orbicurves over finite extensions ofQp such that:

P¹zt0,1,8ubelong toB;

If X belongs toQT and Y ÑX is an open embedding, then Y belongs toQT;

If X belongs toQT and Y ÑX is finite étale, then Y belongs to QT;

If X belongs toQT and X ÑY is finite étale, then Y belongs to QT.

If X belongs toQT and Y ÑX is a partial coarsification, then Y belongs toQT.

If X and Y are hyperbolic orbicurves such that X PQT, then every isomorphismΠX Ñ ΠY comes from an isomorphism X Ñ Y

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Corollary

LetMbe the smallest family of hyperbolic orbicurves over finite extensions ofQpsuch that:

Hyperbolic Mumford curves belong toM;

If X belongs toMand Y ÑX is an open embedding, then Y belongs toM;

If X belongs toMand Y ÑX is finite étale, then Y belongs toM;

If X belongs toMand X ÑY is finite étale, then Y belongs toM.

If X belongs toMand Y ÑX is a partial coarsification, then Y belongs toM.

If X and Y are hyperbolic orbicurves such that X PM, then every isomorphismΠX Ñ ΠY comes from an isomorphism X Ñ Y