Arithmetic in the fundamental group of a p-adic curve.
On the p-adic section conjecture for curves

Florian Pop; Jakob Stix

doi:10.1515/crelle-2014-0077

Published by De Gruyter September 26, 2014

Arithmetic in the fundamental group of a p-adic curve. On the p-adic section conjecture for curves

Florian Pop and Jakob Stix

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0077

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Abstract

We establish a valuative version of Grothendieck’s section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1101397

Funding statement: The first author was supported by the NSF Grant DMS-1101397 and also received support from MATCH of the University of Heidelberg. Both authors would like to thank the Isaac Newton Institute Cambridge, UK, for the support and the excellent working conditions during the NAG Programme in 2009 where we proved the results of this note.

A The zoo of valuations for 2-dimensional semilocal fields

Let k be a complete discrete valued field with valuation ring 𝔬 and perfect residue field κ, e.g., k is a finite extension of ℚp. Let v denote the canonical valuation on k.

A.1 The Riemann–Zariski space of 𝔬-valuations

Let K be the function field of a smooth projective geometrically connected curve X over k. In this appendix we discuss the space

Val𝔬⁡(K)=Valk⁡(K)∪Valv⁡(K)={w:valuation on ⁢K⁢ with ⁢w⁢(𝔬)≥0}

of equivalence classes of valuations w on K whose valuation ring Rw contains 𝔬, or equivalently, the restriction of w to k is either the trivial valuation or equals v.

A.1.1 Models

In this paper a model or more precisely a regular model with strict normal crossing of X over 𝔬 is a regular scheme 𝒳 which is flat and proper over Spec⁡(𝔬) together with a k-isomorphism of X with the generic fibre 𝒳k such that the reduced special fibre 𝒳κ,red is a divisor with strict normal crossings on 𝒳. In particular, unfortunately a stable model in general is not a model in the sense of this paper. By a result of Lichtenbaum, [10, Theorem 2.8], models are automatically projective over Spec⁡(𝔬). We denote the underlying topological space of 𝒳 by 𝒳top, whereas 𝒳cons denotes 𝒳top when given the finer constructible topology.

A.1.2 The center

For a valuation w∈Val𝔬⁡(K) the valuative criterion of properness implies a canonical map Spec⁡(Rw)→𝒳 which maps the closed point of Spec⁡(Rw) to the centerxw∈𝒳cons of the valuation w on the model 𝒳. Maps between different models, which are the identity on X, respect the center of a valuation. The resulting map

(A.1)center:Val𝔬⁡(K)→lim←⁡𝒳cons,

where the projective limit ranges over all models of K, identifies lim←⁡𝒳cons with Val𝔬⁡(K) which is a subspace of the Riemann–Zariski space of K/k.

A.1.3 The valuation ring

The centers of a valuation w∈Val𝔬⁡(K) determine the valuation ring Rw=lim→⁡𝒪𝒳,xw where the direct limit ranges over all models. The inverse map to (A.1) is described as follows. To a compatible system of points a𝒳∈𝒳cons on all models we associate first the ring Ra=lim→⁡𝒪𝒳⁢,⁢a𝒳. The ring Ra is the valuation ring of a valuation w of K because for every element f∈K∗ at least one of f and f-1 belongs to Ra, see [2, Chapter VI, Section 1.2]. Indeed, the indeterminacy of f, that is the set of points where neither f nor f-1 is defined, disappears on a fine enough model.

A.1.4 The patch topology

The patch topology on Val𝔬⁡(K) is defined as the topology induced from the pro-finite product topology by the injective map

sign:Val𝔬⁡(K)↪∏f∈K∗{-,0,+}

that assigns to a valuation w the collection of signs of the value w⁢(f) for each f∈K∗, where the sign of f is + if w⁢(f)>0, it is - if w⁢(f)<0 and the sign is 0 if w⁢(f)=0. The condition on a collection of signs to belong to a valuation ring, namely that the subset in K of 0 and the nonnegative elements forms a ring which contains at least one of f,f-1 for each f∈K∗, is a closed condition. Hence Val𝔬⁡(K) is a pro-finite space, in particular it is compact and Hausdorff.

The map defined in (A.1) is a homeomorphism from Val𝔬⁡(K) endowed with the patch topology to lim←⁡𝒳cons with respect to the lim←-topology. The subset

Valv⁡(K)={w∈Val𝔬⁡(K):w|k=v}⊂Val𝔬⁡(K)

is a closed subset in the patch topology described by the condition that w⁢(π)>0 for a uniformizer π of 𝔬. The set Valv⁡(K) corresponds to the subset lim←⁡𝒳κ,cons⊂lim←⁡𝒳cons, where 𝒳κ⊂𝒳 is the special fibre.

A.2 Types of valuations

We sketch the classification of the zoo of valuations and fix the terminology.

A.2.1 Type

We define the type of a valuation w∈Val𝔬⁡(K) as the well-defined number 0, 1 or 2 given by the height 0⁢p⁢t⁢(xw)=dim⁡(𝒪𝒳,xw) in the sense of scheme theory of its center xw∈𝒳 for all sufficiently fine models 𝒳 with respect to the system of all models. The unique valuation of type 0 is the trivial valuation.

A.2.2 Type 1

Valuations of type 1 are the discrete valuations associated to prime divisors on an arbitrary model fine enough such that the respective divisor appears. The corresponding prime divisor is either vertical, i.e., it maps to the closed point of Spec⁡(𝔬), or horizontal, i.e., it maps finitely to Spec⁡(𝔬). The first are called of type 1v whereas the latter valuations are called of type 1h.

Notation 1

The usual notation for a valuation of K of type 1v will be α. The corresponding prime divisor of a fine enough model will be denoted by Yα and by abuse of notation has a generic point denoted by α again. The precise meaning of α will always be clear from the context.

A.2.3 Type 2

All the remaining valuations are of type 2 and thus have all their centers at closed points of the special fibre.

Let w be a valuation of type 2. For each valuation α of type 1 we define the distance of w to α on the model 𝒳 as the infimum of the number of irreducible components in a 1-dimensional connected subscheme Z⊂𝒳 which contains the center of α and of w. If w keeps finite distance to any valuation of type 1 as we vary over the system of all models, then there is a unique valuation α of type 1 with a closed point y on the associated divisor such that w is the composition of α with the valuation vy on the residue field of α associated to y. So w=vy∘α is called of type 2v (resp. type 2h) if α is vertical (resp. horizontal).

The remaining valuations have centers which move away from any valuation of type 1 and are called of (coarse) type 2u(unbounded).

A.3 Rigid analytic viewpoint

Valuations of type 2 can be understood in terms of the associated rigid analytic space Xrig. For a model 𝒳 we get a specialisation map

sp𝒳:Xrig→𝒳κ,red

from the rigid space to the set of closed points of the special fibre. The preimage of a smooth closed point of 𝒳κ,red is an open disc, the preimage of a node of 𝒳κ,red is an annulus.

To a valuation w of type 2 we associate the system C𝒳=sp𝒳-1⁢(xw) of preimages of the centers indexed by the system of all models. The system of subsets C𝒳 is monotone decreasing with respect to inclusion when the model becomes finer. The valuation is uniquely determined by the system of the C𝒳 as

Rw=⋃𝒳𝒪𝒳⁢(C𝒳)=⋃𝒳{f∈K:f⁢ defined on ⁢C𝒳,∥f∥C𝒳,∞≤1},

where ∥f∥C𝒳,∞ is the sup-norm of f on C𝒳. The various types belong to distinctive geometric pictures of the system of the C𝒳 as follows.

A.3.1 Type 2h

For fine enough models, C𝒳 is an open disc with fixed center x∈Xrig and radius converging to 0 with finer and finer models.

Figure 1

Type 2h.

A.3.2 Type 2v

For fine enough models, C𝒳 is an annulus such that the corresponding annuli for finer and finer models share one common boundary.

Figure 2

Type 2v.

A.4 Type 2 but unbounded distance

The valuations of type 2u can be described and arranged into types in more detail as follows.

Every closed point y in the reduced special fibre carries invariants (ey,α),fy equal to the tuple (ey,α) of the multiplicities of the components on which y lies in the special fibre 𝒳κ and the residue field degree fy of y over κ. Any closed point x in the generic fibre X=𝒳k that specialises to y has to have residue field κ⁢(x) with

eκ⁢(x)/k=∑αmα⁢ey,α

with mi∈ℕ≥1 and fy|fκ⁢(x)/k. On the other hand, there is always an x with the minimal possible values of e,f.

A.4.1 Type 2usmooth

For a valuation w of type 2u the value ∑αexw,α remains bounded if and only if for fine enough models ultimately all centers xw belong to the smooth locus of the reduced special fibre. Such a valuation is called of type 2usmooth or 2usm (ultimately smooth).

A.4.2 Type 2unode

We call a valuation w of type 2unode or 2un (ultimately node) if for all fine enough models the center lies in a node of the reduced special fibre.

A.4.3 Type 2ualt

For a valuation of type 2u, if neither type 2unode nor type 2usmooth applies, then the center xw in the pro-system of models alternate between the smooth locus of the reduced special fibre and its nodes, and hence these are called of type 2ualternating or 2ualt (unbounded alternating).

$Figure 3 Type 2ualternating${{}_{\rm alternating}}$.$

Figure 3

Type 2ualternating.

A.4.4 Rigid analytic description of type 2usmooth

For a cofinal set of models, C𝒳 is an open disc without common center in Xrig. The radius of the discs converges to 0 with finer and finer models. There is a unique limit point in X⁢(kalg^)∖X⁢(kalg), where kalg^ is the completion of kalg.

A.4.5 Rigid analytic description of type 2unode

For fine enough models, C𝒳 is a p-adic annulus such that the corresponding annuli for finer and finer models share no common boundaries.

$Figure 4 Type 2unode${{}_{\rm node}}$.$

Figure 4

Type 2unode.

A.5 Algebraic structure

The information on the algebraic structure associated to a valuation w according to its type is summarized in Table 1. The rational rank of w or better its value group Γw is defined as dimℚ⁡(Γw⊗ℚ), see [2, Chapter VI, Section 10.2]. The rank of w, hauteur in [2, Chapter VI, Section 4.4], is the Krull dimension dim⁡Spec⁡(Rw) of its valuation ring Rw.

Table 1

type	value group	ℚ-rank	rank	on k	residue field
0	1	0	0	trivial	K
1h	ℤ	1	1	trivial	finite over k
1v	ℤ	1	1	v	function field over κ of transcendence degree 1
2h	ℤ⊕ℤ lex.	2	2	v	finite over κ
2v	ℤ⊕ℤ lex.	2	2	v	finite over κ
2un	ℤ⊕ℤ⁢γ⊂ℝ	2	1	v	finite over κ
2usm	ℤ	1	1	v	infinite, algebraic over κ
2ualt	⋃n1en⁢ℤ	1	1	v	algebraic over κ
	with lim⁡en=∞

The residue field for a valuation w of type 2usmooth has to be algebraic over κ of infinite degree. Indeed, otherwise the extension 𝔬≺Rw had finite residue degree f=[κ(w):κ] and finite index of value groups e=(w(K):v(k)), which implies that K as a k vector space has dimk⁡K=e⁢f, a contradiction. In particular, if we ultimately pick smooth centers xw and the residue field degree [κ⁢(xw):κ] remains finite, then we actually deal with a valuation of type 2h.

A.6 Valuations of the universal cover

From now on we fix a geometric generic point η¯:Spec⁡(Ω)→X of X as base point. Let K~ be the function field of the associated pointed universal pro-étale cover X~ of X, i.e., K~⊂Ω is the maximal algebraic extension of K which is unramified over X. We conclude that π1⁢(X,η¯) equals Gal⁡(K~/K).

A.6.1 The Riemann–Zariski space of the universal cover

Note that the prolongation Val𝔬⁡(K~) of Val𝔬⁡(K) to K~ endowed with the patch topology is a projective limit

Val𝔬⁡(K~)→∼lim←K′⁡Val𝔬⁡(K′)

of the spaces Val𝔬⁡(K′) equipped with the patch topology, where K′ ranges over all finite intermediate extensions K′/K in K~/K. As above, one has a homeomorphism

center:Val𝔬⁡(K~)→∼lim←X′⊂𝒳′⁡𝒳cons′

and the subset

Valv⁡(K~)={w~∈Val𝔬⁡(K~):w~|k=v}⊂Val⁡(K~)

is a closed subset in the patch topology described by the condition that w~⁢(π)>0 for a uniformizer π of 𝔬. Thus Valv⁡(K~) is a compact, Hausdorff, pro-finite space which furthermore is canonically a pro-finite limit

(A.2)center:Valv⁡(K~)→∼lim←K′,𝒳′⁡𝒳κ,cons′

of the pro-finite spaces 𝒳κ,cons′, where 𝒳κ,cons′ is the reduced special fibre of 𝒳′ endowed with the constructible topology.

A.6.2 Types and the universal cover

The canonical restriction map

Val𝔬⁡(K′)→Val𝔬⁡(K)

is surjective. Furthermore, for w′↦w, by the fundamental inequality, the residue field extension κ⁢(w′)/κ⁢(w) is finite and the inclusion of value groups w⁢(K)⊂w′⁢(K′) has finite index, see [2, Chapter VI, Section 8]. Hence the type of a valuation is preserved under the restriction map Val𝔬⁡(K′)→Val𝔬⁡(K), and the classification into types also applies to valuations in Val𝔬⁡(K~).

A.6.3 Notational convenience

The map Val𝔬⁡(K~)→Val𝔬⁡(K) will be denoted by

w~↦w=w~|K

which implicitly could also imply a choice of a preimage w~ of the valuation w if the latter happens to appear first.

B Unramified Hilbert Zerlegungstheorie

We keep the notation and assumptions from Appendix A.

B.1 Nearby points

For a geometric point y on a model 𝒳 we set

𝒳yh=Spec⁡(𝒪𝒳,yh)

for the scheme of nearby points and

𝒳ysh=Spec⁡(𝒪𝒳,ysh)

for the scheme of strictly nearby points. The intersection with the generic fibre we denote by

𝒰yh=Spec⁡(𝒪𝒳,yh⊗𝔬k)⊂𝒳yh

and

𝒰ysh=Spec⁡(𝒪𝒳,ysh⊗𝔬k)⊂𝒳ysh.

For y equal to the center xw of a valuation w∈Val𝔬⁡(K), more precisely, for a choice of geometric point above the closed point of the valuation ring which induces a geometric point x¯w above each center, we abbreviate

𝒰wh:=𝒰x¯wh⊆𝒳wh:=𝒳x¯wh

and

𝒰wsh:=𝒰x¯wsh⊆𝒳wsh:=𝒳x¯wsh.

In the limit over all models 𝒳 of K we get

Uwh=lim←𝒳⁡𝒰wh⊆Xwh=lim←𝒳⁡𝒳wh

and

Uwsh=lim←𝒳⁡𝒰wsh⊆Xwsh=lim←𝒳⁡𝒳wsh.

We note that Uwh (resp. Uwsh) is a limit of affine p-adic curves over k (resp. knr). In particular, the cohomological dimension of Uwsh for étale constructible sheaves is at most 2.

B.2 Hilbert decomposition and inertia group

Let us fix a choice of a geometric generic point ξ¯y of 𝒰ysh such that (𝒰ysh,ξ¯y)→(X,η¯) becomes a pointed map.

The decomposition group resp. inertia group in the sense of Hilbert at y is given by the image Dy, resp. Iy, of the natural map π1⁢(𝒰yh,ξ¯y)→π1⁢(X,η¯), resp. π1⁢(𝒰ysh,ξ¯y)→π1⁢(X,η¯), induced by the inclusions. We suppress the choice of base points in the notation for decomposition and inertia groups.

B.3 Decomposition and inertia group of a valuation

For w∈Val𝔬⁡(K) let ξ¯w be a geometric generic point of Uwsh such that (Uwsh,ξ¯w)→(X,η¯) becomes a pointed map. The compatibility of ξ¯w with η¯ describes a unique prolongation w~ of w to K~ by the property Kw~sh=K~⋅Kwsh and similarly Kw~h=K~⋅Kwh in Ω. Here Kwh (resp. Kwsh) is a (strict) henselisation of K in w, and similarly for w~. We easily observe the following lemma.

Lemma 1

For a valuation w∈Valo⁡(K) of type 2 but not of type 2h we have

Spec⁡(Kwh)=Uwh,

whereas for w of type 2h refining α of type 1h the nearby points Uwh equals the spectrum of the valuation ring the extension of α to Kwh and moreover equals Uαh=Xαh. ∎

The decomposition group (resp. inertia group) in the sense of valuation theory of w, or more precisely the prolongation w~|w to a valuation of K~, is given by the image Dw~|w of π1⁢(Uwh,ξ¯w)→π1⁢(X,η¯), resp. the image Iw~|w of π1⁢(Uwsh,ξ¯w)→π1⁢(X,η¯).

The dependence on w~ is through the choice of a path connecting the base points ξ¯w and η¯ to the effect of conjugating Dw~|w and Iw~|w within π1⁢(X,η¯). If no confusion arises, we will simplify the notation to Dw=Dw~|w (resp. Iw=Iw~|w).

B.4 Reconciliation of valuation theory and arithmetic geometry

The two viewpoints of inertia and decomposition groups are related via the compliance of π1 with affine projective limits. We may assume that ξ¯w induces ξ¯x¯w∈𝒰wsh for every model of X, and then find

(B.1)Dw~|w=lim←𝒳⁡Dxw and Iw~|w=lim←𝒳⁡Ixw,

where the limits are in fact simply intersections of closed subgroups in π1⁢(X,η¯).

Moreover, let α be a valuation of type 1 and let y be a geometric point localised in a closed point of the divisor Yα associated to α on a suitable model 𝒳. Then we have the following diagram when the corresponding geometric points are compatibly chosen.

The scheme 𝒰yNisα is the generic fibre of 𝒳yNisα which is the maximal strict étale neighbourhood in between 𝒳yh→𝒳 which is Nisnevich at α, i.e., such that the point α splits in the image of ξ¯α after an appropriate choice is fixed. With

Dy,α=im⁡(π1⁢(𝒰yNisα,ξ¯y)→π1⁢(X,η¯)),

we find

(B.2)Iα⊆Iy⊆Dy∩∩Dα⊆Dy,α⊆π1⁢(X,η¯)⁢.

Let w∈Val𝔬⁡(K) be a valuation of rational rank 2, i.e., of type 2h or 2v, and a refinement of the valuation α. Then in the limit over all models we deduce from (B.2) and (B.1) that

Iα⊆Iw⊆Dw⊆Dα

because Dα=lim←𝒳⁡Dxw,α.

Acknowledgements

We would like to thank Yves André and Shinichi Mochizuki for pointing out potential relations with analogous questions in the context of the tempered fundamental group. Especially, in light of Shinichi Mochizuki’s comments, our exposition gained in clarity and our pro-finite result was put in the right perspective. We further like to thank Jordan Ellenberg, Hélène Esnault, Kiran Kedlaya, Minhyong Kim, Mohamed Saïdi, Tamás Szamuely, Akio Tamagawa, and Olivier Wittenberg for their interest in our work.

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Received: 2011-12-20

Published Online: 2014-9-26

Published in Print: 2017-4-1