Resolution of non-singularities and absolute anabelian conjecture

Resolution of non-singularities and absolute anabelian conjecture

Emmanuel Lepage

IMJ-PRG, Sorbonne Université

2nd Kyoto-Hefei Workshop on Arithmetic Geometry

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

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 „

ÑΠY

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 Quasi-Belyi type if there are maps:

X Y

f.´et.

oo

dominant

P¹zt0,1,8u

Theorem (Mochizuki)

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

Intermediate steps

Xr “lim

ÐÝ^pS,s0qÑpX,x₀qS: universal pro-finite étale cover ofX. Natural actionΠ_X ñXr

Definition

Letx closed point ofX andx˜PXr a preimage ofx.

D_˜x “Stab_ΠXpx˜q ĂΠX: decomposition group ofx Dx “conjugacy class ofD_˜x.

An isomorphismφ: Π_X Ñ^„ Π_Y is point-theoretic ifDĂΠ_X is a decomposition group if and only ifφpΠ_Xqis a decomposition group.

Proposition

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.

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_|KpX¹ _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 (punctured) Mumford curve, then X satisfies RNS.

4 If X is of Belyi type, then X satisfies RNS.

X Y

f.´et

oo

dominant

Z :hyp.ppuncturedqMumford curve

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.semi´normsAÑRě0,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 (Ąrigid points))

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

Example: the affine line

LetX “SpecpCprTsq “A¹_Cp. IfaPC_p,r PRě0,|ř

ia_ipT ´aqⁱ|_ba,r :“max_ip|a_i|rⁱq ùb_a,r PA^1,an_Cp If r=0,b_a,r of type 1.

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

Ifr R|p|^Q,ba,r of type 3 (rkp|KpXq^ˆ|_ba,rq “2).

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

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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^anzιpG_Xq: 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^„ lim ÐÝ

X{K¹

GX

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 „

ÑGY.

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IfX satisfies RNS, one gets a natural homeomorphism Xr^anĂXr

an

:“lim ÐÝ

S

SÑ^„ lim ÐÝ

pS,sq

GS,

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

Apply isom. p´q_G to open subgps ofΠ_X andΠ_Y,ùhomeomorphism φ˜:Xr^an Ñ^„ Yr^an

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

φ^an :X^anÑ^„ Y^an presp. φ^an_Cp :X^an

C^p

Ñ„ Y^an

C^pq.

Action compatibility ùñ If˜x PXr^an,φpD˜xq “Dφp˜˜xq

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

Does every homeomorphismX^an ÑY^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_Cp,x a node ofX_s. ThenXétale loc. »SpecO_Cpru,vs{puv ´aq.

Leteedge of dual graphGXofX.

Setlengthpeq:“vpaqùmetric onGX.

X_p2q^an »inj lim_XVpGXqùnatural metricd onX_p2q^an. φ^log :X^log_s Ñ^„ Y^log_s ùñ φ^an_p2q:X^an

Cp,p2qÑY^an

Cp,p2qis an isometry.

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Letx₀PX_p2q^an. Proposition Let x PX_C^an

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

dpx₀,xq:“sup

U

zPUinf_p2qdpx₀,zq “ `8 where U goes through open neighbourhood of x in X^an

C^p. Sketch:

Metric extends toιpG_Xqùreduce to the case of a disk inX^an

CpzιpG_Xq.

In a disk, explicit description of the metric:

dpba,r,b_a¹_,r¹q “|log_pprq ´logppr¹q| if|a´a¹|ďmaxpr,r¹q

“ ´2vpa´a¹q ´log_pprq ´log_ppr¹q| if|a´a¹|ěmaxpr,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. Ş

iBpa_i,r_iq “ H, r_i Ñr ą0.

ùñ φ^an preserves 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^an preserves rigid points.

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IfX satisfies RNS andΠ_X »Π_Y, doesY satisfies RNS?

Not known in general...

Theorem (Mochizuki )

LetB(curves of Belyi type) be the smallest family of hyperbolic orbicurves over finite extensions ofQp such that:

P¹zt0,1,8ubelong toB;

If X belongs toBand Y ÑX is an open embedding, then Y belongs toB;

If X belongs toBand Y ÑX is finite étale, then Y belongs toB;

If X belongs toBand X ÑY is finite étale, then Y belongs toB.

If X belongs toBand Y ÑX is a partial coarsification, then Y belongs toB.

If X and Y are hyperbolic orbicurves such that X PM, then every isomorphismΠ_X Ñ^„ Π_Y comes from an isomorphism X Ñ^„ Y

Corollary

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

Hyperbolic (punctured) 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 „

ÑΠY comes from an isomorphism X Ñ^„ Y

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Sketch of RNS for Mumford curves

LetX{K be a proper Mumford curve.

It is enough to show that the union of images of skeletal valuation by arbitrary finite étale morphisms are dense insideX^an.

p-adic Hodge theory ùñ H¹pX,Zpp1qqÑ^pX H⁰pX,Ω¹_Xq bK Cp

Local computation in a disk:

Lemma

If cPH¹pX,Zpp1qq, let Y_n,c ^φÑ^n,c X be the finite étaleµ_pⁿ-cover corr. to c.

If x PXpCpqs.t.DcPH¹pX,Zpp1qqs.t. p_Xpcq ‰0and mult_xp_Xpcqis not of the form p^k´1for any k ,

thenDznP pYn,cqKn s.t.φn,cpznqÑⁿ x .

LetX be a hyperbolic curve overK. Letx PXpK¹q, whereK¹ is a finite extension ofK.

H¹pX,Zpp1qq{pKer_{ _} p_Xq ^{ //}

H⁰pX,Ω¹_Xq ^{evx //}

K¹ ĂCp

H¹pX¹,Zpp1qq{pKerp_X¹q ^{ //}H⁰pX¹,Ω¹_X1q

ev_x¹

88

whereX¹ is any topological finite cover andx¹ PX¹pK¹qis a preimage ofx.

AsgpX¹q Ñ 8, dim_QpH¹pX¹,Qpp1qq{pKerp_X¹q Ñ 8, but dim_QpK¹ stays finite

ùñ DpX¹,x¹qandcPH¹pX,Zpp1qqzpKerp_X¹ qs.t. p_Xpcqpx¹q ‰0.

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