R ESEARCH I NSTITUTEFOR M ATHEMATICAL S CIENCESKYOTOUNIVERSITY,Kyoto,Japan ByYuYANGOctober2015 DegenerationofPeriodMatricesofStableCurves RIMS-1835

RIMS-1835

Degeneration of Period Matrices of Stable Curves

By

Yu YANG

October 2015

R _ESEARCH I NSTITUTE FOR M ATHEMATICAL S _CIENCES

Degeneration of Period Matrices of Stable Curves

Yu Yang

Abstract

In the present paper, we study the extent to which linear combina- tions of period matrices arising from stable curves are degenerate (i.e., as bilinear forms). We give a criterion to determine whether a stable curve admits such a degenerate linear combination of period matrices. In particular, This criterion can be interpreted as a certain analogue of the Weight Monodromy Conjecture for non-degenerate elements of pro-ℓlog

´

etale fundamental groups of certain log points associated to the log moduli stackM^logg .

Keywords: period matirces, stable curves, log ´etale fundamental groups.

Mathematics Subject Classification: Primary 14H30; Secondary 14H10.

Contents

1 Review of log ´etale fundamental groups of stable curves 5 1.1 Log structures on stable curves . . . 5 1.2 Log ´etale fundamental groups . . . 6 2 Degeneration of period matrices of stable curves 9 2.1 Pro-ℓperiod matrices of stable curves and their functorial properties 9 2.2 Degeneration of pro-ℓperiod matrices . . . 18 2.3 Relationship with the Weight Monodromy Conjecture . . . 25

Introduction

The anabelian geometry of hyperbolic curves concerns the problem of recon- structing hyperbolic curves from their fundamental groups. In order to un- derstand these fundamental groups, many techniques of algebraic geometry are applied. On the other hand, in the case of stable curves over algebraically closed fields, the introduction of some ideas of a combinatorial nature allows one to prove some results in much greater generality under very weak hypotheses (cf.

[14], [15], [6], [7]). By applying this point of view, we are able to discuss not only phenomena that arise scheme-theoretically but also phenomena that arise purely group-theoretically. Before we explain the main question that motivated the theory developed in the present paper, let us recall some basic facts con- cerning period matrices.

LetXbe a stable curve of genusgover an algebraically closed fieldkand ΓX

the dual graph ofX. Then one has a natural exact sequence of freeZℓ-modules (cf. [14] Definition 1.1 (ii) and Remark 1.1.3.)

0−→M_X^ver−→M_X −→M_X^top−→0, whereM_X :=π₁^ℓ-adm(X)^ab, M_X^top :=π^ℓ₁(Γ_X)^ab, M_X^ver := Im(⊕

v∈v(ΓX)π^ℓ₁(X_v− Node(X))^ab −→ MX) (cf. Notations and Conventions of the present paper), where Node(X) denotes the set of nodes ofX. The stable curveX determines a morphism froms := Speck to the moduli stackMg, and the pull-back log structure of the natural log structure on Mg determines a natural log struc- ture on Speck; denote the resulting log scheme by s^log which admits a chart (Speck,⊕

e∈e(ΓX)N). The pro-ℓ log ´etale fundamental groupπ^ℓ₁(s^log) is natu- rally isomorphic to⊕

e∈e(ΓX)Zℓ(1). Therefore, we obtain a natural action of

⊕

e∈e(ΓX)Zℓ(1) on the extension 0 −→ M_X^ver −→ M_X −→ M_X^top −→ 0. This extension determines an extension class [MX], which may be regarded as a ho- momorphism, which we refer to as thepro-ℓ period matrix morphism ofX (cf.

Proposition 2.3, Definition 2.4, and the surrounding discussion) f_X :π₁^ℓ(s^log)∼= ⊕

e∈e(Γ_X)

Zℓ(1)−→Hom(M_X^top⊗M_X^top,Zℓ(1)).

For each element a ∈ ⊕

e∈e(Γ_X)Zℓ(1), we refer to f_X(a) as the pro-ℓ period matrixassociated to a.

If a = (a_e)_e ∈ ⊕

e∈e(ΓX)Zℓ(1)_e is a positive definite element (cf. Defini- tion 2.5), then the subgroup generated bya can be regard as the image of the pro-ℓ completion of the inertia group of a p-adic local field. Thus, by apply- ing Faltings-Chai’s theory (or the Weight Monodromy Conjecture for curves), we know that the pro-ℓ period matrix fX(a) is positive definite, hence also non-degenerate. This non-degeneracy property of pro-ℓ period matrices is the most non-trivial part in S. Mochizuki’s proof of the combinatorial version of the Grothendieck conjecture (=ComGC) for semi-graphs of anabelioids in the case of outer representations of IPSC-type (cf. [14] Corollary 2.8). More precisely, Mochizuki proved that the pro-ℓperiod matrix associated to a positive definite element of any finite admissible covering X^′ −→ X of X is non-degenerate.

Moreover, Mochizuki gave a criterion to determine whether or not an isomor- phism between fundamental groups of semi-graphs of anabelioids that is compat- ible with the respective outer Galois actions by inertia groups is graphic(i.e., the isomorphism preserves verticial subgroups and edge-like subgroups). By considering the pro-ℓlog ´etale fundamental groups which arise from acuspand applying the ComGC in the IPSC-type case, Mochizuki gave an algebraic alter- native proof of the injectivity theorem in the affine case due to M. Matsumoto (cf. [15]). But if one wants to extend Matsumoto’s theorem to the projective case, it is natural to attempt to prove the ComGC in the case of outer represen- tations of NN-type case (i.e., the out Galois action arising from a non-degenerate (= all the coordinates of the element are nonzero)a= (ae)e∈⊕

e∈e(Γ_X)Zℓ(1)

(cf. [6] Definition 2.4 (iii))). On the other hand, if one attempts to imitate the proof of the ComGC in the IPSC-type case, one has to consider whether or not the pro-ℓ period matrix arising from a node is non-degenerate. Y. Hoshi and S. Mochizuki proved a version of the ComGC in the NN-type case under cer- tain assumptions, and by applying this version of the ComGC, they successfully extended the injectivity theorem to the projective case (cf. [6]).

More generally, in the theory of combinatorial anabelian geometry, in order to extend results (e.g., the ComGC) in the IPSC-type case to the NN-type case, one has to consider whether or not the pro-ℓperiod matrix arising from anon- degenerate element of π₁^ℓ(s^log) ∼= ⊕

e∈e(ΓX)Zℓ(1) is degenerate. It is difficult to determine in general whether or not the pro-ℓ period matrix associated to a givennon-degenerate element is degenerate. But at least we can ask which stable curves admit a non-degenerate element that gives rise to a degenerate pro-ℓperiod matrix. This question may be formulated as follows:

Question 0.1. Does there exist a criterion to determine whether or not the stable curveX admits an elementa= (ae)e∈⊕

e∈e(Γ_X)Zℓ(1)such that ae̸= 0 for eacheand, moreover, the pro-ℓ period matrixfX(a)is degenerate?

In present paper, our main theorem is a criterion as follows (cf. Theorem 2.9):

Theorem 0.2. Let X be a stable curve over an algebraically closed field k, Γ_X the dual graph of X. Then X is a pro-ℓperiod matrix degenerate curve (cf.

Definition 2.5) if and only if the maximal untangled subgraphΓ^ø_X(cf. Definition 2.7) ofΓX is not a tree (i.e.,r(Γ^ø_X) := rank(H¹(Γ^ø_X,Z))̸= 0).

The Weight Monodromy Conjecture for curves holds if and only if the period matrix associated to an element of the inertia group is non-degenerate. Thus, our main theorem may also be interpreted as asserting that a certain analogue of the Weight Monodromy Conjecture for non-degenerate elements ofπ^ℓ₁(s^log) (cf. Corollary 2.11).

In Section 1, we recall some basic facts concerning log structures and log

´etale fundamental groups of stable curves.

In Section 2, we discuss the topic of degeneracy of pro-ℓ period matrices of stable curves and prove Theorem 0.2. Finally, we explain the relationship between Theorem 0.2 and the Weight Monodromy Conjecture.

Acknowledgements

I would like thank my advisor Professor Shinichi Mochizuki for suggesting the topic of the present paper and carefully reading preliminary versions of the present paper. Also, I would like to thank Yuichiro Hoshi for many helpful discussions. I would like to express my deepest gratitude toS, for giving me constant support, warm encouragements during the most painful period, 2013.

Without her, this paper could not be written.

Notations and Conventions

Numbers:

Ifk is a field, we shall write (char(k), n) = 1 if char(k) and nare relatively prime or char(k) = 0. The notationZwill be used to denote the ring of rational integers. We always use the notation ℓ to denote a prime number such that ℓ̸= char(k). The notationsZℓ and Qℓ denote theℓ-adic completions ofZand Q, respectively.

Curves and their moduli stacks:

By a curve over a field, we mean a finite type, separated, connected, one dimensional reduced scheme over a field.

Anr-pointed stable curve(X, D_X) of type (g, r) over a schemeSconsists of a flat, proper morphismX −→ S, together with a closed subschemeD_X ⊆X such that for each geometric pointsofS:

(i) The geometric fiberX_sis a reduced and connected curve of genusgwith at most ordinary double points (i.e., nodes).

(ii)Xs is smooth at the points ofDX.

(iii) The composite morphismDX ⊆X−→S is finite ´etale of degreer.

(iv) Let E be an irreducible component of Xs of genusgE. Then the sum of the degree of the restriction ofDX to E and the number of points whereE meets the closure of the complement ofE inXsis≥3−2gE.

(v) dim(H¹(Xs,OX_s)) =g.

In this situation, one verifies easily that 2g−2 +r is≥1.

We shall say that anS-schemeX is astable curveof genusgoverS if (X,∅) is a 0-pointed stable curve of genusg overS.

We shall say that a pointed stable curve (X, D_X) over a schemeSissmooth if the morphism of schemesX −→S is smooth.

We denote (X, D_X) a pointed stable curve over S with divisor of marked pointsD_X and underlying schemeX. For simplicity we also use the notation X to denote the pointed stable curve (X, D_X) when there is no confusion.

Let Mg,r be the moduli stack of stable curves of type (g, r) over SpecZ (where we regard the marked points as ordered), Mg,r the open substack of Mg,r parametrizing pointed smooth curves. ThenM^logg,r is the log moduli stack obtained by equipping Mg with the natural log structure associated to the divisor with normal crossings Mg,r \ Mg,r ⊂ Mg,r relative to SpecZ. Let Xg,r −→ Mg,r be the universal stable curve over Mg, and Dg ⊂ Xg,r the divisor given by the inverse image inXg,r of the divisor Mg,r\ Mg,r ⊂ Mg,r. Dg,r determines a log structure onXg,r; denote the resulting log stack byX^logg,r. Thus, we obtain a morphism of log stacksX^logg,r−→ M^logg,r. In particular, ifr= 0 (i.e., stable curve), we use notationMg (resp. M^logg , Xg, X^logg ) to denote the stackMg,0(resp. M^logg,0,Xg,0, X^logg,0).

For more details on stable curves, pointed stable curves and their moduli stacks, see [3], [9].

Galois categories and their fundamental groups:

We denote the categories of finite ´etale, finite Kummer log ´etale, finite tame, and finite admissible coverings of “(−)” by Cov(−), Cov((−)^log), Covtame(−), Covadm(−), respectively, and the categories of finite ℓ-´etale, finite ℓ-Kummer log ´etale, finiteℓ-tame, and finiteℓ-admissible coverings of “(−)” by Cov^ℓ(−), Cov^ℓ((−)^log), Cov_tame^ℓ (−), Cov^ℓ_adm(−) respectively.

The notations π₁(−), π₁((−)^log), π₁^tame(−), π^adm₁ (−) will be used to denote the ´etale, Kummer log ´etale, tame, and admissible fundamental groups of “(−)”, respectively; the notationsπ^ℓ₁(−), π^ℓ₁((−)^log), π^ℓ-tame₁ (−), π₁^ℓ-adm(−) will be used to denote the pro-ℓ´etale, pro-ℓKummer log ´etale, pro-ℓtame, and pro-ℓadmis- sible fundamental groups, respectively; the notation (−)^abdenotes the abelian- ization of the group (−)

For more details on Kummer log ´etale coverings, admissible coverings, log admissible coverings and their fundamental groups for (pointed) stable curves, see [10], [13].

1 Review of log ´ etale fundamental groups of sta- ble curves

In this section, we recall some basic facts concerning log structures and log ´etale fundamental groups of stable curves.

1.1 Log structures on stable curves

In this subsection, we will recall some basic facts concerning log structures of stable curves; for generalities on log schemes, see [8].

Let X be a generically smooth stable curve over a complete DVR (R,m_R) with algebraically closed residue fieldk:=R/m_R,πa uniformizer ofm_R. Write K for the quotient field of R and Xs (resp. Xη) for the special fiber (resp.

generic fiber) of X over R. Thus, the stable curve X −→ SpecR induces a morphismϕX : SpecR−→ Mg. The completion of the local ring ofMg at the pointϕX_s :s:= Speck−→ Mgis isomorphic toOJt1, ..., t3g−3K, where we write O for k (resp. the ring of Witt vectors with coefficients in k) if char(k) = 0 (resp. if char(k) =p >0), and thet1, . . . , t3g−3 are indeterminates.

If we denote the number of nodes ofXsbymand assign labelsi= 1, . . . , m to each of the nodes, then the completion of the local ring of Xs at the node labelediis isomorphic to RJxi, yiK/(xiyi−πⁿⁱ), and the indeterminateti may be chosen so as to correspond to the deformations of the node of X_s labeled i. Then the log structure on SpecOJt₁, ..., t_m, t_m+1, ..., t_3g−3K induced by the log structure ofM^logg may be described as the log structure associated to the following chart:

N^d−→ OJt1, ..., tm, tm+1, ..., t3g−3K, where (ai)i7→∏

i≤mt^a_iⁱ. We denote this log scheme by S₁^log:= (SpecOJt1, ..., t3g−3K,N^m).

Moreover, we also obtain a log structure on the closed point ofS1 by restrict- ing the log structure of S₁^log; we denote the resulting log scheme by s^log₁ :=

(Speck,N^m). On the other hand, the closed point of SpecR determines a log structure on SpecR, which admits a chart

N −→ R

1 7→ π.

We denote the resulting log scheme by S₂^log := (SpecR,N). Write s^log₂ :=

(Speck,N) for the log scheme obtained by restricting the log structure ofS₂^log to the closed point ofS2. Thus, we obtain a cartesian commutative diagram

X₂^log −−−−→ X₁^log −−−−→ X^logg



y y y S₂^log −−−−→ S₁^log −−−−→ M^logg

— whereX₁^log (resp. X₂^log) is defined so as to render the right-hand (resp. left- hand) square in the diagram cartesian; the underlying scheme of X₁^log (resp.

X₂^log) may be identified withXg×_MgSpecOJt₁, ..., t_3g−3K(resp. X); for suitable choices of the indeterminatest1, . . . , tm, the lower horizontal arrow in the left- hand square of the diagram may be described, fori= 1, . . . , m, as follows:

S₂^log:= (SpecR,N) −→ S₁^log:= (SpecOJt1, ..., t3g−3K,N^m) R ←− OJt1, ..., t3g−3K

πⁿⁱ ←−p ti

N ←− N^m

∑

iaini ←−p (ai).

1.2 Log ´ etale fundamental groups

For more details on the definition of the notion of a finite Kummer log ´etale covering, see [10] Section 3. Let Y^log be a connected fs log scheme. Choose a strict log geometric point ey^log −→ Y^log (i.e., a log geometric point (cf. [10]

4.2) over a strict geometric point (cf. [5] Section 2, Definition 1)y^log−→Y^log).

Then this choice of a strict log geometric point determines an associated log

´etale fundamental groupπ1(Y^log).

Let ℓ be a prime number that is ̸= char(k). For a proof of the following specialization theorem for log ´etale fundamental groups, see [18] Theorem 2.2.

Proposition 1.1. Suppose thatX₂^log is as above. Letη := SpecK−→SpecK be a geometric point of SpecK. Write K^t for the maximal tamely ramified extension ofK in K,RK^t for the integral closure of R in K^t, η^t := SpecK^t, (SpecRK^t)^log for the log scheme obtained by equipping SpecRK^t with the log structure determined by the sheaf of nonzero regular functions, andes^log₂ for the log scheme

Speck×SpecR_Kt (SpecR_Kt)^log

— where we identify the residue field ofR_Kt with k. Thus, we obtain a natural strict log geometric point es^log₂ −→ S₂^log induced by η. Then there is a natural isomorphism between the pro-ℓ log ´etale fundamental groups at the respective fibers ofX₂^log overη and es^log₂ , which is well-defined up to composition with an inner automorphism, as follows:

π₁^ℓ((X₂^log)η)∼=π^ℓ₁((X₂^log)η^t)−→π^ℓ₁((X₂^log)_eslog 2 ).

Next, let us recall that if C −→ U is a family of hyperbolic curves over a regular scheme U, and, for n a positive integer, we write Cn for the n-th configuration space associated toC−→U, then there is an associated homotopy exact sequence as follows (cf. [16] Proposition 2.2 (iii)):

1−→π^ℓ₁((C_n)_u)−→π₁^ℓ(C_n)−→π₁^ℓ(U)−→1,

where u is a geometric point of U. Since, for i = 1,2, the interior of S_i^log is a regular scheme, by applying the theorem of log purity and the deformation theory of log schemes (cf. [5] Section 4, Corollary 1), we obtain a homotopy exact sequence as follows (for the definition of stable log curves, see [7] Section 0):

Corollary 1.2. Suppose thatX_i^log−→S_i^log, wherei∈ {1,2}, is the morphism discussed above. Letsi−→Si be a geometric point ofSi. Writes^log_i for the log scheme obtained by equippings_i with the log structure determined by restricting the log structure ofS_i^log tos_i. Letes^log_i −→S_i^log be a strict log geometric point of S_i^logthat factors through the natural morphisms^log_i −→S^log_i . Then the following sequence is exact:

1−→π^ℓ₁((X_i^log)_eslog i

)−→π^ℓ₁((X_i^log)_slog i

)−→π₁^ℓ(s^log_i )−→1

On the other hand, there is a classical scheme-theoretic description of the groupπ₁^ℓ((X_i^log)_eslog

i ) that does not require one to apply the theory of log schemes, namely, by means of thepro-ℓ admissible fundamental group. We use the nota- tionπ₁^ℓ-adm(Xs) to denote the pro-ℓadmissible fundamental group of the special fiberXs. We have a proposition as follows.

Proposition 1.3. Let i ∈ {1,2}. Suppose that Xs, X_i^log, and es^log_i are as in Corollary 1.2 and the following discussion. Fix a strict geometric point e

x^log_i −→ (X_i^log)_slog

i whose image is a smooth point of the underlying scheme

(X_i^log)s_i. Then there is a natural isomorphism of fundamental groups, which is well-defined up to composition with an inner automorphism, as follows:

π^ℓ-adm₁ (Xs)∼=π^ℓ₁((X_i^log)_eslog i )

— whereπ^ℓ₁(−)is taken with respect to the base point determined by the strict geometric pointex^log_i −→(X_i^log)_slog

i

;π^ℓ-adm₁ (−) is taken with respect to the base point determined by the underlying morphism of schemes ofxe^log_i −→(X_i^log)_si. Proof. Write (s₁)^log_n (resp. (s₂)^log_n ) for the log scheme determined by the mor- phism of monoids

1

n·N^m −→ k a 7→ 0, (resp.

1

n ·N −→ k a 7→ 0),

wherenis a positive integer such that (n,char(k)) = 1. Ifn^′andn^′′are positive integers such thatn^′ dividesn^′′, then we consider the morphism of log schemes (s1)^log_n′′ −→ (s1)^log_n′ (resp. (s2)^log_n′′ −→ (s2)^log_n′ ) determined by the morphism of monoids

1

n^′ ·N^m −→ 1 n^′′ ·N^m

a 7→ a.

(resp.

1

n^′ ·N −→ 1 n^′′ ·N a 7→ a).

If we allow n^′ and n^′′ to vary, then these morphisms determine an inductive system, whose inductive limit is easily seen to be isomorphic tose^log₁ (resp. es^log₂ ).

In the following, we shall fix one such isomorphism, which we shall use toidentify this inductive limit withes^log₁ (resp. es^log₂ ).

To complete the proof of the Proposition, it suffices to construct, in a natural way, an equivalence between the Galois categories Cov^ℓ_adm(X_s) and Cov^ℓ((X₁^log)_eslog

1

) (resp. Cov^ℓ((X₂^log)_eslog 2

)). Here, we note that Cov^ℓ((X₁^log)_eslog 1

) (resp. Cov^ℓ((X₂^log)_eslog

2

) ) may be identified with lim−→ⁿCov^ℓ((X₁^log)_(s

1)^logn ) (resp.

lim−→ⁿCov^ℓ((X₂^log)_(s

2)^log_n )). Since any finite Kummer log ´etale covering of (X1)^log

(s₁)^log_n

(resp. (X2)^log

(s₂)^log_n ) determines a multi-log admissible covering (i.e., a disjoint union of log admissible coverings) after base-change to (s1)^log_m (resp. (s2)^log_m) for some positive integer m >> 0, the Proposition follows immediately from [12]

Proposition 3.11.

Remark 1.3.1. The isomorphism π₁^ℓ-adm((X2)s)=∼π₁^ℓ((X₂^log)_eslog

2 ) can be also deduced by applying the log purity theorem, the specialization theorem for Kummer log ´etale fundamental groups, and the specialization theorem for ad- missible fundamental groups.

2 Degeneration of period matrices of stable curves

In this section, we assume thatkis an algebraically closed field.

2.1 Pro-ℓ period matrices of stable curves and their func- torial properties

In this subsection, we give the definition of the pro-ℓperiod matrix morphism associated to a stable curve overk.

Let X be a stable curve of genus g over k. Write Γ_X for the dual graph of X, v(ΓX) for the set of vertices of ΓX, e(ΓX) for the set of edges of ΓX, and ΠX := π₁^ℓ-adm(X) for the pro-ℓ admissible fundamental group of X. We use the notation Xv to denote the irreducible component of X corresponding tov ∈v(ΓX). Thus, Uv :=Xv\Node(X) is an open subscheme ofXv, where Node(X) denotes the set of nodes of X; the pro-ℓ´etale fundamental group of Uv, which we denote by Πv :=π₁^ℓ(Uv), may be regarded as the decomposition group⊆ΠX (which is well-defined up to ΠX-conjugation) associated to v. For e ∈ e(ΓX), write Πe (∼= Zℓ(1)) for the decomposition group ⊆ ΠX (which is well-defined up to ΠX-conjugation) associated to e. Write π^ℓ₁(ΓX) for the pro-ℓ completion of the topological fundamental group of the dual graph ΓX. Finally, we use the notation M_X (resp. M_X^top, M_X^ver, M_X^edge) to denote the abelianization of Π_X(resp. the abelianization ofπ₁^ℓ(Γ_X), Im(⊕

v∈v(Γ_X)Π^ab_v −→

MX), Im(⊕

e∈e(Γ_X)Π^ab_e −→MX)).

By the definitions given above, we obtain a filtration as follows:

0⊆M_X^edge⊆M_X^ver⊆M_X. Moreover, there are two natural exact sequences:

0−→M_X^ver−→M_X −→M_X^top−→0, 0−→M_X^edge−→M_X^ver−→M_X^ver/M_X^edge−→0.

For more details on the first exact sequence, see [14] Definition 1.1 and [14]

Remark 1.1.4. Furthermore, we have the following proposition which can be proved by using the structure of Picard schemes of stable curves (cf. [1] Section 9.2, Example 8) and the theory of Raynaud extensions (cf. [4] Chapter II, Section 1). On the other hand, for a purely group-theoretic proof, see [6] Lemma 1.4.

Proposition 2.1. Forv∈v(ΓX), writeX_v^′ for the normalization ofXv,J(X_v^′) for the Jacobian ofX_v^′, and (∆^cpt_v )^ab for the pro-ℓ ´etale fundamental group of J(X_v^′)(i.e., theℓ-adic Tate module associated toJ(X_v^′)). Then, we have

M_X^ver/M_X^edge∼=⊕

v

(∆^cpt_v )^ab.

The stable curveX−→Speckdetermines a classifying morphism Speck−→

Mg to the moduli stackMg. Thus, we obtain a log structure on Speck, natu- rally associated to the stable curveX, by restricting the log structure ofM^logg ; denote the resulting log scheme by s^log_X . We also obtain a stable log curve X^log := X^logg ×_M^log

g

s^log_X over s^log_X whose underlying scheme is X. Thus, we have an isomorphismI_slog

X

:=π^ℓ₁(s^log_X )∼=⊕

e∈e(ΓX)Zℓ(1)_e. Furthermore, there are natural actions of I_slog

X

on the exact sequences 0 −→ M_X^ver −→ MX −→

M_X^top −→ 0 and 0 −→ M_X^edge −→ M_X^ver −→ M_X^ver/M_X^edge −→ 0. Denote the extension class corresponding toM_X by

[MX]∈Ext¹_I

slog X

(M_X^top, M_X^ver).

By [11] Example 0.8, there is a spectral sequence converging to Ext^p+q_I

slog X

(M_X^top, M_X^ver).

whoseE2-term is given by H^p(I_slog X

,Ext^q_Z(M_X^top, M_X^ver)). In particular, we obtain a long exact sequence as follows:

0−→H¹(I_slog X

,Hom_Z(M_X^top, M_X^ver))−→Ext¹_I

slog X

(M_X^top, M_X^ver)

−→H⁰(I_slog X

,Ext¹_Z(M_X^top, M_X^ver)).

Since M_X, M_X^top, M_X^ver, M_X^edge are free Zℓ-modules of finite rank, we thus con- clude that the morphism H¹(I_slog

X

,Hom_Z(M_X^top, M_X^ver))−→Ext¹_I

slog X

(M_X^top, M_X^ver) is an isomorphism. Thus, the extension class [MX] may be regarded as an ele- ment of H¹(I_slog

X ,Hom_Z(M_X^top, M_X^ver)).

Here, we observe that, for any two finitely generated freeZℓ-modulesM, N, we have natural isomorphisms

Hom_Z(M, N)∼= lim←−_n Hom_Z/ℓnZ(M/ℓⁿM, N/ℓⁿN)∼= Hom_Zℓ(M, N).

Thus, we shall use the notation Hom(−,−) to denote Hom_Zℓ(−,−).

Proposition 2.2. In the notation of the above discussion, the actions ofI_slog X

onM_X^top,M_X^ver,M_X^edge, andMX/M_X^edge are trivial.

Proof. First, we have two exact sequences as follows:

0−→M_X^edge−→MX−→MX/M_X^edge−→0 and

0−→M_X^ver−→MX −→M_X^top−→0.

By Poincar´e duality (cf. [14] Proposition 1.3), we have natural isomorphisms M_X^edge ∼= Hom(M_X^top,Zℓ(1))

and

M_X^ver∼= Hom(MX/M_X^edge,Zℓ(1)).

Thus, to complete the proof of our claim, it suffices to show (sinceM_X^edge⊆M_X^ver, andI_slog

X

acts trivially onZℓ(1)) that the action ofI_slog X

onM_X^ver(orM_X/M_X^edge) is trivial. Next, let us write X1 −→ S1 for the restriction of the tautological curveXg over the moduli stackMg to the spectrum of the completion of the local ring at the point ofMg corresponding to X. For each vertexv ofv(ΓX), writeUv:=Xv\Node(X),Mvfor the image inM_X^verof the decomposition group associated tov. Then every open subgroup of Mv corresponds to an abelian

´etale covering of the curve Uv, and every ´etale covering of Uv lifts uniquely (up to unique isomorphism), without base change, to an ´etale covering of the formal neighborhood ofU_vin X₁, the claim follows immediately. Alternatively, the claim may be verified by observing that every open subgroup ofM_X/M_X^edge corresponds to an abelian ´etale covering of the stable curveX, and every ´etale covering ofX lifts uniquely (up to unique isomorphism) to an ´etale covering of X₁ without base change.

This completes the proof of our proposition.

By using Proposition 2.2, we can prove a proposition as follows:

Proposition 2.3. In the notation of the above discussion, then the natural map H¹(I_slog

X

,Hom(M_X^top, M_X^edge))−→ H¹(I_slog X

,Hom(M_X^top, M_X^ver)) is injective, and (if, by abuse of notation, we identify the domain of this injection with its image via the injection, then) the extension class

[M_X]∈H¹(I_slog X

,Hom(M_X^top, M_X^edge)).

Proof. The short exact sequence 0−→M_X^edge −→M_X^ver−→M_X^ver/M_X^edge−→0 ofI_slog

X

-modules determines a long exact sequence

0−→Hom(M_X^top, M_X^edge)^IsXlog −→Hom(M_X^top, M_X^ver)^IslogX

−→Hom(M_X^top, M_X^ver/M_X^edge)^IslogX −→H¹(I_slog X

,Hom(M_X^top, M_X^edge))

−→H¹(I_slog

X ,Hom(M_X^top, M_X^ver))−→H¹(I_slog

X ,Hom(M_X^top, M_X^ver/M_X^edge))−→. . .

— where the superscript “I_slog X

” denotes the submodule ofI_slog X

-invariants. Since the functor Hom(M_X^top,−) is exact, and the actions ofI_slog

X onM_X^top, M_X^ver, and M_X^ver/M_X^edge are trivial, the morphism

Hom(M_X^top, M_X^ver)^IslogX −→Hom(M_X^top, M_X^ver/M_X^edge)^IslogX is a surjection. Thus, the morphism

H¹(I_slog X

,Hom(M_X^top, M_X^edge))−→H¹(I_slog X

,Hom(M_X^top, M_X^ver)) is an injection.

Since the action of I_slog X

on M_X/M_X^edge is trivial (cf. Proposition 2.2), it follows formally that the image of the extension class [MX] via the morphism H¹(I_slog

X

,Hom(M_X^top, M_X^ver))−→ H¹(I_slog X

,Hom(M_X^top, M_X^ver/M_X^edge)) is 0. This implies that

[MX]∈H¹(I_slog X

,Hom(M_X^top, M_X^edge)).

This completes the proof of the proposition.

Remark 2.3.1. LetY^•:= (Y, D) be a pointed stable curve over Speck. Then just as in the non-pointed case, we have a filtration as follows:

0⊆M_Y^cusp_• ⊆M_Y^edge_• ⊆M_Y^ver• ⊆M_Y• ↠M_Y^top_• :=M_Y•/M_Y^ver•,

whereM_Y• denotes the abelianization ofπ₁^ℓ-adm(Y^•);M_Y^ver_• (resp.M_Y^edge_• ,M_Y^cusp_• ) denotes the subgroup ofM_Y• generated by the subgroups that arise from the irreducible components (resp. nodes and cusps, cusps). Similar arguments to the arguments given in the proofs of Proposition 2.2 and 2.3 imply that the actions of I_slog

Y• on M_Y^top_•, M_Y^ver•, M_Y^edge_• , MY•/M_Y^edge_• are trivial, and, moreover, that we obtain a corresponding extension class

[MY^•]∈H¹(I_slog

Y•,Hom(M_Y^top•, M_Y^edge• )).

Since H¹(I_slog X

,Hom(M_X^top, M_X^edge))∼= Hom(I_slog X

,Hom(M_X^top, M_X^edge)), by Poincar´e duality (cf. [14] Proposition 1.3), the extension class [MX] corresponds to a con- tinuous group homomorphism

f_X :I_slog

X −→Hom(M_X^top⊗M_X^top,Zℓ(1)).

Definition 2.4. We shall refer to the morphismf_Xdiscussed above as thepro-ℓ period matrix morphism associated toX. For an elementa∈I_slog

X

, we shall refer to the quadratic formfX(a) onM_X^top as thepro-ℓperiod matrix associated toa.

Note thatf_X(a) is a symmetric quadratic form on M_X^top for eacha∈I_slog X

(cf.

[4] Chapter III Section 8).

In the next two remarks, we will explain the functorial properties of period matrices.

Remark 2.4.1. We discuss a certain functorial property that relates the pro- ℓ period matrix morphisms associated to a stable curve to the corresponding morphism associated to a stable “sub-curve”.

Let X be a stable curve over s := Speck which is sturdy (i.e., the genus of the normalization of each irreducible component of Y is ≥ 2), ΓX the dual graph of X, V a subset of v(ΓX)∪

e(ΓX). Suppose that UV := X \ ((∪

v∈V Xv)∪ (∪

e∈V e)) is aconnectedcurve. Write (gV, rV) for the type ofUV; XV for the compactification of UV (i.e., the closure of UV in the scheme ob- tained by normalizingXV at the nodes ofX\UV). Thus, the pair (XV, XV\UV) determines a pointed stable curveX_V^•, which may be regarded as associated to V. Ifv ∈v(ΓX), then by applying these conventions in the case where “V” is taken to be [v] := (v(ΓX)\ {v})∪

Node(Xv), we obtain a pointed stable curve X_[v]^• of type (g_v, r_v), whereg_v is the genus ofX_[v], andr_v is the cardinality of

the set {

X_v ∩

( ∪

v̸=w∈v(Γ_X)

X_w)} ∪

Node(X_v).

Thus, if we writes^log_X (resp. s^log_V ; (s^U_V)^log) for the log scheme whose underlying scheme iss, and whose log structure is obtained by pulling back the log structure of the log moduli stackM^logg (resp. M^loggV;M^loggV,rV) via the classifying morphism σ(resp. σV; σ_V^U) associated toX −→s(resp. XV −→s;X_V^• −→s, i.e., for a suitable choice of ordering of the cusps), then we obtain a stable log curve

X^log −→s^log_X (resp. X_V^log−→s_V^log;X_V^•log−→(s^U_V)^log)

by pulling back the morphism of log stacksX^logg −→ M^logg (resp.X^loggV −→ M^loggV; X^loggV,rV −→ M^loggV,rV). IfSis a Deligne-Mumford stack over SpecZ, writeSsfor the stackS ×SpecZsovers. Then the geometry of the stable curveX, together with the original choice of a subsetV ofv(ΓX), determine a clutching morphism of moduli stacks (i.e., for a suitable choice of ordering of the cusps):

ψ:N := (MgV,rV)_s×s

∏

v∈V

(Mgv,rv)_s−→(Mg)_s

LetN^logbe the log stack whose underlying stack isN, and whose log struc- ture is the pull-back of the log structure of (Mg)^log_s byψ. On the other hand, we also have a log structure determined by the divisor given by the union of pull-backs toN of the divisors at infinity of each of the factors (MgV,rV)_sand (Mg_v,r_v)s, for v ∈V; write NV^log for the resulting log stack, which, as is eas- ily verified, is isomorphic to the log stack (Mg_V,r_V)^log_s ×s

∏

v∈V(Mg_v,r_v)^log_s . We have a natural morphism between the two log stacks N^log and (Mg_V)^log_s obtained by composing the following three morphisms:

N^log −→ NV^log −→(Mg_V,r_V)^log_s −→(Mg_V)^log_s .