COMMENTS ON THE MAIN THEOREM OF POP-STIX

Shinichi Mochizuki
Updated November 15, 2010

Let k be a ﬁnite extension of Qp , k an algebraic closure of k, and X a proper
hyperbolic curve over k. Write ΠX for the étale fundamental group of X [relativeto some basepoint], Πtp
X for the tempered fundamental group of X [relative to somebasepoint], anddef

ΠX  Gk = Gal(k/k),

Πtp
X  Gkfor the natural projections. Also, we write K for the function ﬁeld of X and Ok
for the ring of integers of k. Then:(1) During the conference on Galois-Teichmüller theory held at RIMS, Kyoto University, in October 2010, it was suggested by Yves André that the Main Theoremof [PS] should imply the following assertion:(A1) Every section of ΠX  Gk arises, up to ΠX -conjugacy, from a sectionof Πtp
X  Gk .
We refer to (7) below for a more detailed formulation. One consequence of thisassertion (A1) is the following assertion:(A2) The “Proﬁnite p-adic Section Conjecture” for hyperbolic curves may
be reduced to the “Tempered p-adic Section Conjecture” for hyperbolic
curves.
As far as I can see, (A1) and (A2) may be veriﬁed immediately.
(2) In light of the discussion of (1), it is natural to ask the question (cf. also thediscussion of (8) below):tp(Q1) If one starts with a section s : Gk → Πtp
X of ΠX  Gk , then does theMain Theorem of [PS] tell you anything new concerning the section s?

In this context, we observe that, even without applying the theory of [PS], it follows
immediately from [Semi], Theorem 5.4, (i) — cf. also the discussion of [Semi],
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2SHINICHI MOCHIZUKI

Example 5.6; [Semi], Remark 6.9.1 — that the section s determines a system ofvertices. . .  vi+1  vi  . . .
of any coﬁnal system of ﬁnite étale connected Galois coverings of X with stablereduction. . . → Xi+1 → Xi → . . .
of X — i.e., each vi is an irreducible component of the special ﬁber of the stablemodel of Xi that is ﬁxed by the natural action of the image Im(s) of the sections. Here, i ranges over the positive integers, and, after possibly passing to a coﬁnalsubsystem, the notation “” may be interpreted as the statement that one of thefollowing two [mutually exclusive] conditions holds:(C1) For each positive integer i, the irreducible component vi+1 maps quasiﬁnitely to the irreducible component vi .
(C2) For each positive integer i, the irreducible component vi+1 maps to aclosed point xi of the irreducible component vi .
In the case of (C1), the vi determine a discrete valuation of K that contains Ok ,but not k. In the case of (C2), the vi determine a system of closed points. . . → xi+1 → xi → . . .
and hence a corresponding inductive system of normal local rings. . . → Ri → Ri+1 → . . .
— each of which contains Ok , but not k — whose union R∞ is, consequently, anormal local ring that is not necessarily noetherian. That is to say, in either of thetwo cases (C1), (C2):(A3) One obtains a normal local ring R ⊆ K that contains Ok , but not k, suchtpthat the section s : Gk → Πtp
X is contained in [one of the ΠX -conjugatesof] the decomposition group associated to R.
The statement of (A3) is reminiscent of the Main Theorem of [PS] in the sense
that, like the Main Theorem of [PS], the conclusion (A3) may be thought of as a
reduction of the Section Conjecture to an essentially local problem. On the otherhand,(Q2) it is not clear, in the case of (C2), whether or not the normal local ring
R that is obtained in (A3) is necessarily a valuation ring.
Perhaps a deﬁnitive answer to (Q2) would require results along the lines of Tamagawa’s “resolution of nonsingularities” (cf. [Tama]). It would be interesting,however, if one could give a deﬁnitive answer to (Q2) via an elementary argument.
Although I have not studied the proof of the Main Theorem of [PS] in detail, atleast at a naive level, it is diﬃcult to believe that one must apply [Tama] in orderto give an aﬃrmative answer to (Q2): Indeed, at least at a naive level, it it diﬃcult

COMMENTS ON THE MAIN THEOREM OF POP-STIX3

to believe that the essential consequence of [Tama] in the present context should
be any stronger than the existence of a system of vertices {vi }i as above.
(3) Thus, in summary, the theory of [Semi] already implies the statement (A3)concerning sections s : Gk → Πtp
X of the tempered fundamental group. From thispoint of view:(A4) In the tempered case, the Main Theorem of [PS] may be regarded as astrengthening of (A3) to the eﬀect that the ring R of (A3) may be taken
to be a valuation ring (that may possibly contain k).
That is to say, (A4) may be thought of as being at least a partial answer to (Q1).
(4) Another consequence of [Semi], Theorem 5.4, (i), and [Semi], Lemma 5.5, is theconclusion that, after possibly passing to a coﬁnal subsystem, one of the following
holds:(U1) The system of vertices {vi }i obtained in (3) is “essentially” unique.
(U2) There exists an “essentially” unique system of edges {ei }i — i.e., each ei
is a node of the special ﬁber of the stable model of Xi such that ei+1 → ei— that are ﬁxed by the natural action of Im(s).
Here, the term “essentially” is used in the evident sense; we leave the routine taskof giving a precise formulation to the reader. In particular, in the case that (U1)holds, one concludes that the system of vertices {vi }i coincides with the system ofvertices determined by any valuation obtained as in the Main Theorem of [PS].
(5) In fact, unlike the case with [PS], by combining the analysis of verticial decomposition groups given in [NodNon], Proposition 3.9, (i), with a similar argument
to the argument used (in [Semi]) to prove [Semi], Theorem 5.4, (i), one concludes
that the consequence (A3) obtained above from [Semi], Theorem 5.4, (i), continues
to hold if one takes k to be an arbitrary complete discrete valuation ﬁeld of mixed
characteristic. (Here, we note in passing that one veriﬁes easily — by considering
the pro-p Kummer map — that, for any such k, the resulting absolute Galois group
Gk continues to be center-free, hence slim.) In this situation, the analysis of verticial decomposition groups given in [NodNon], Proposition 3.9, (i), also yields thefollowing version of “uniqueness”:(U3) For each i, any two possibilities for the vertex vi either coincide, areadjacent, or admit a common adjacent vertex.
Here, we note that (U3) is weaker that the version of “uniqueness” involving (U1),(U2) [cf. (4)] precisely because one does not have an analogue of [Semi], Lemma
5.5, in the case of more general k.
(6) In a similar vein, unlike the case with [PS], the consequences (A3), (U3) obtained
above continue to hold, even in the more general setting discussed in (5), if, for Σ

4SHINICHI MOCHIZUKI

a set of primes that contains a prime l = p, one replaces ΠX by the geometricallypro-Σ quotient(Σ)ΠX  ΠXof ΠX and Πtp
X by the “Σ-tempered fundamental group”
tp,(Σ)

ΠX

(Σ)

⊆ ΠX

(Σ)

given by the image of Πtp
X in ΠX .
(7) In the notation of (6), we observe that another important consequence of [Semi],
Theorem 5.4, (i), in the present context is the following:(A5) The natural map
tp,(Σ)

Sect(ΠX(Σ)/Gk ) → Sect(ΠX /Gk )tp,(Σ)

tp,(Σ)(Σ)-conjugacy classes of sections of ΠX— i.e., from ΠX→ Gk to ΠX (Σ)
conjugacy classes of sections of ΠX → Gk — is injective.
In particular, we obtain the following consequence of the Main Theorem of [PS] (cf.
(A1)):(A6) If Σ is the set of all primes, and k is a ﬁnite extension of Qp , then thenatural map
Sect(Πtp
X /Gk ) → Sect(ΠX /Gk )is bijective.
tp,(Σ)A proof of (A5) may be sketched as follows. Consider two sections s, t : ΠX→(Σ)γ
Gk such that s = t for some γ ∈ ΠX (where the superscript “γ” denotes conjugation by γ). Then, by applying (A3), we conclude that the sections s, t give rise,
respectively, to systems {vi }i , {wi }i of vertices, which (cf. (U3); [NodNon], Proposition 3.9, (i)) are, in some appropriate sense, “essentially unique”. In particular,
we obtain that, for each i, the vertices wi and γ(vi ) either coincide, are adjacent,or admit a common adjacent vertex. But this implies that the systems {vi } andtp,(Σ){γ(vi )} correspond to the same “tempered basepoint”, i.e., that γ ∈ ΠX, so stp,(Σ)
and t are ΠX-conjugate, as desired.
(8) One reason why I was interested in, for instance, the pSC (i.e., the “(proﬁnite/proΣ) p-adic Section Conjecture for hyperbolic curves”) was because I was able to show
in [AbsTopII], Corollary 2.9, thatpSC =⇒ abs pGC— i.e., that the pSC implies the “absolute (proﬁnite/pro-Σ) p-adic version of theGrothendieck Conjecture” (cf. [AbsTopII], Corollary 2.9, for more details). On theother hand, from the point of view of verifying the abs pGC, results such as (A2)

COMMENTS ON THE MAIN THEOREM OF POP-STIX

5or (A6) are not so interesting, since one already knows (cf., e.g., [Semi], Theorem
6.6) that the absolute proﬁnite p-adic version of the Grothendieck Conjecture may
be reduced to the absolute tempered p-adic version of the Grothendieck Conjecture.
It is precisely this state of aﬀairs that prompted me, upon hearing of (A2), to posethe question (Q1).

Bibliography[NodNon] Y. Hoshi, S. Mochizuki, On the Combinatorial Anabelian Geometry of NodallyNondegenerate Outer Representations, to appear in Hiroshima Math. J.
[Semi] S. Mochizuki, Semi-graphs of Anabelioids, Publ. Res. Inst. Math. Sci. 42(2006), pp. 221-322.
[AbsTopII] S. Mochizuki, Topics in Absolute Anabelian Geometry II: Decomposition Groups,
RIMS Preprint 1625 (March 2008).
[PS] F. Pop and J. Stix, Arithmetic in the fundamental group of a p-adic curve —On the p-adic section conjecture for curves, preprint, Philadelphia-HeidelbergCambridge, August 2010.
[Tama] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. Res.
Inst. Math. Sci. 40 (2004), pp. 1291-1336.