Mathematisches Forschungsintitut Oberwolfach
Mini-Workshop: Arithmetic Geometry and Symmetries
around Galois and Fundamental Groups
Abstracts
Arithmetic and Homotopy of Moduli Stacks of Curves
Benjamin Collas
Let Mg,[m] be the moduli stacks of genus g curves with m-unordered marked
points, that we consider endowed with their complementary divisorial and stackstratifications. The former is a stratification at infinity and is given by the topological type (g 0 , m0 ) of curves in the Deligne-Mumford compactification of stable
curves Mg,[m] , while the later is local and is given by the flat stratification by theautomorphisms of curves.
As Q-stacks, the moduli spaces accept some Geometric Galois Representations(GGR)

ρ~s : Gal(Q̄/Q) −→ Aut[π1et (Mg,[m] ⊗ Q̄, ~s)]where ~s : Q̄{{q}} → Mg,[m] is a tangential base point associated to a chosen
Q-rational structure on Mg,[m] – tangential base points define at once some geometric base points for the fundamental group and some Q-rational π1 -sections.
They allow to bypass Falting’s limitation on rational markings on curves, and also
to benefit for the study of (GGR) from the rational Knudsen-Mumford lower dimensional (g 0 , m0 )-embeddings in terms of limit Galois representations. The stack
structure, via Hurwitz spaces, draw some connections between Geometric Galois
Representations and the Regular Inverse Problem.
By providing accessible geometries that capture key arithmetic properties, themoduli stacks of curves are fundamental spaces in arithmetic geometry, in anabelian geometry – e.g. the unordered marked M0,m are anabelian–, and in motivic theory – see the category of Mixed Tate motives.
We report on recent results on the stack arithmetic of these spaces, and on
works in progress on the use of homotopical methods: how this leads to a motivic
interpretation of these higher symmetries, and to a finer understanding of theoperadic and arithmetic properties of the divisorial stratification.
1. Stack Arithmetic of Curves (joint with S. Maugeais.)
The Deligne-Mumford stack structure of Mg,[m] is recovered through the inertia
group sheaf IM,x , which for a given point x : Spec Q → Mg,[m] geometrically
identifies with the finite automorphism group IM,x̄ ' AutC (Cx̄ ) of a Riemannsurfaces represented by x̄. By Noohi’s uniformization Theorem, it follows that
IM,x̄ ,→ π1et (Mg,[m] ⊗ Q̄, ~s),1

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i.e. the automorphisms of curves form some local ghost loops subgroups of the étale
fundamental group. This raises the question of describing the Gal(Q̄/Q)-action of(GGR) on the stack inertia groups of Mg,[m] , which is indeed of local-vs-global
nature.
Because they form the first non-trivial stack stratas, we consider in what follows
the case of the cyclic stack inertia group IM,x̄ ' hγi that we study via theirassociated special loci.
1.1. Special Loci, Irreducible Components of Hurwitz Spaces. Let thusMg,[m] (G) denote the special loci attached to a finite order group G: Mg,[m] (G) is
the Q-stacks of curves C/S that admit a faithful G-action G ,→ AutS (C). In termsof Galois action, we notice that an irreducible components of Mg,[m] (G) is a priori
defined over a number field K: this implies the stability of the π1et (Mg,[m] ⊗ Q̄)conjugacy classes of G under the action of the absolute Galois group Gal(K̄/K).
In genus 0 and for G cyclic, one proves that every such component is of the form
M0,[m]+k – with m permuted points and k fixed points, k ∈ {0, 1, 2} – thus definedover Q. This implies that the Gal(Q̄/Q)-action stabilizes the conjugacy classes of
the cyclic stack inertia in genus 0.
In higher genus, this raises the question of finding arithmetic coarse invariantsof the irreducible components of Mg,[m] (γ). The identification of the normalization
fg,[m] (G) ' Mg,[m] [G]/Aut(G) as quotient of the Hurwitz spaceof the special loci M
of G-covers Mg,[m] [G] reduces this question to the characterization of S-familiesof G-cover, which draws a first connection with the geometry of Hurwitz spaces.
For G = hγi, an answer is provided in terms of étale cohomology with thedefinition on the geometric fibers of some branching datas kr, which allows toestablish:
Theorem ([5] - Th. 4.3). The stack of γ-special loci admits a finite decompositionin irreducible components given by:aMg,[m] (γ) =Mg,[m],kr (γ),
where Mg,[m],kr (γ) denotes the Q-stack of curves inducing γ-covers with given
branching datas kr.
In a similar way to the Deligne-Mumford proof of the irreducibility of Mg,[m] ,this result relies on the existence of a Teichmüller space that parametrizes unmarked curves with given kr-datas. An arithmetic property of G-covers appears
for the general case, to ensure that the field of moduli K of certain γ-covers is
indeed a field of definition – see (Seq/Split)-condition of [8].
As in genus 0, this implies this time in every genus the conjugacy-stability of
IM = hγi under a certain local Gal(K̄/K)-action. The comparison with the global
Gal(Q̄/Q)-action of (GGR) and its complete description requires the use of thedivisorial stratification.

Mini-Workshop: AG and Symmetries around Galois and Fundamental Groups 3
1.2. Inertial Limit Galois Actions, a Tate-like Action. Let η denote thegeneric point of an irreducible component Mg,[m],kr (G) of a special loci Mg,[m] (G),and let κ(η) denote its residue field. After rigidification, one obtains an inertial
Galois action ρIη : Gal[κ(η)/κ(η)] → Aut(Iη ) on the generic stack inertia groupIη > G of the component. Since a tangential structure on Mg,[m] is a formalneighbourhood of a singular stable curve, the comparison of this local Galois action
to the global (GGR) is provided by a specialization result for Deligne-Mumford
stacks – see §3.2 and §4.2 of [6]:For any component of cyclic special loci, there exist a K-point ofMg,[m],kr (G) and a tangential base point ~s of Mg,[m] , such that theGal(Q̄/Q)-action ρ~s of (GGR) induces the inertial action ρIη .
For G cyclic, the local inertial Galois action can indeed be proven to be given
by a certain extension of the field of definition K discussed in the previous section.
From the Q-definition of the cyclic irreducible components given by the Theorem
above, one establishes more precisely:
Theorem ([5] – Th. 5.4 & [6] – Cor. 4.6, Th. 4.8). The Gal(Q̄/Q)-representationof (GGR) induces a χ-conjugacy Gal(Q̄/Q)-action on the cyclic stack inertia of
Mg,[m] . For σ ∈ Gal(Q̄/Q):(1)

χσσ.γ = h−1.hγ,σ where hγ,σ ∈ π1et (Mg,[m] ⊗ Q̄, ~s).
γ,σ .γThis result surprisingly depends on the geometry of the Hurwitz components
and of their stable compactification: if the kr-data corresponds to a class of Gcovers whose G-isotropy groups span G, the result then follows from Fried’s branchcycle argument; the general case requires a fine deformation argument of G-covers
which allows to compare G-stratas of Mg,[m],kr and Mg−1,[m]+2,kr0 of different
topological and ramification types. This comparison requires the choice of tangential Gal(Q̄/Q)-actions which are compatible at the level of the stack inertia groups.
We refer to this process under the term inertial limit Galois action.
1.3. Towards the Stack Arithmetic of Higher Stratas. By analogy with thedivisorial arithmetic of Mg,[m] , this Tate-like action motivates further studies of
the higher arithmetic of the stack stratification. The inertial limit Galois action
provides for example a mean of comparing the conjugacy factors of the Gal(Q̄/Q)action between stratas of different topological types.
We mention two immediate directions of research:(i) determines some discrete arithmetic invariants of the irreducible components of Mg,[m] (G) for non-abelian groups G;(ii) complete the description of the Gal(Q̄/Q)-action Eq. (1) by determining
relations for the conjugacy factors.
As shown in the case of the cyclic strata, progress will certainly rely on a fineunderstanding of the arithmetic of G-covers and of the Hurwitz spaces. As more
concrete examples, let us mention for (i) the use in [7] of H2 -data in terms of
mixed cohomology that complete the monodromy invariants for cyclic extensions.

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For (ii), relations could come either from the comparison of different stack inertia groups and follow either from (i), or from the use of the topological type:equations (?)-(??) of [21], and their generalization (R) in [3], are some examples
of such comparisons – in these cases of M0,[5] (Z/2Z) with respect to M0,[4] , ofM0,[6] (Z/3Z) with respect to M0,[4] , and of M1,[2] (Z/2Z) with respect to M1,1–, see also [19].
In another direction, and always by analogy with the Galois divisorial arithmetic, the Tate-like action of Eq. (1) raises the question of a motivic interpretationof this result.
2. Motivic Stack Considerations
Let k be a number field, and let M T (k) denote the category of Mixed Tate motives
over k. This is a Tannakian category of group GM T , neutralized by the canonical
Adams weight fibre functor, whose properties are tightly related to the divisorialarithmetic of the moduli schemes M0,m : it is motivically generated by the M0,m s[2], a p + 2q-motivic weight comes from a p-codimensional component of M0,m andof a q-Tate twist, relation between periods are induced by the Knudsen morphisms[22].
Motivated by the Tate-like Gal(Q̄/Q)-action on the cyclic stack inertia, we
present how Morel-Voedsky’s motivic homotopy provides a convenient context
where to develop motivic properties for the stack inertia that are Tate-compatible.
We present the main difficulties, illustrate the approach in the case of representable, and generally refer to the forthcoming [4] for the case of stacks.
2.1. Motivic Homotopy Theory and Stacks. The motivic context is givenby the unstable-stable motivic categories H(Q)  SH(Q), respectively definedas the homotopy categories of spaces Sp(Q) = sP r(SmQ ) and P1 -spectras overSp(Q) endowed with their A1 -local injective model category with respect to theétale topology. The category SH(Q) is triangulated, equivalent to Voevodsky’sDM (Q), and has the Lefschetz motive inverted at the level of morphisms as aresult of the ΣP -stabilization.
A first connection between the Gal(Q̄/Q)-stack inertia framework and the H(Q)
context is given by replacing the Galois category formalism of SGA1 by ArtinMazur and Friedlander [12] simplicial étale topological type one: we attache to thestack Mg,[m] – and similarly to its stack inertia IM – first a pro-space {Mg,[m] }et ,then by Isaksen’s étale realization functor [16] a A1 -type {Mg,[m] }A1 in sP r(SmQ ).
A more intrinsic context is indeed given by considering Mg,[m] as a specificobject of sP r(SmQ ), whose Giraud’s descent property is characterized in termsof hypercovers [10]: within this context, the (group) stack inertia identifies to a
derived loop space IM ' RHom(S1 , M) (resp. to a homotopy group sheaf).
This approach places in particular the stack motivic study of Mg,[m] withinToën-Vezzosi’s Homotopical Algebraic Geometry theory (HAG) [24]. Since P1 'Mini-Workshop: AG and Symmetries around Galois and Fundamental Groups 5
S1 ∧Gm within H(Q), and since Gm is the Lefschetz motivic divisorial monodromyof Mg,[m] , one concludes that:the motivic homotopy theory of Mg,[m] gives a favourable context where toillustrate how the S1 -loops encode the ghost 2-structure of motivic spectras.
2.2. Mixed Tate Motives and Beyond. By contrast, we now illustrate therelevance and the non-triviality of this approach on the specific examples of representable stacks M0,m ∈ SmQ . On one side, the homotopical Mixed Tate framework is given by [17] and follows Spitzweck’s representation theorem for cells modules over an Adams graded cycles algebra, as provided by Bloch-Kriz’s NBK – see
op.cit.
On the other side, the HAG context is provided by Toën’s Spec -functor of [23]4o4oSpec : AlgQ
→ sP r(Q), and by Hitchin’s Quillen equivalence AlgQ
 cdgaQ .
As a result, since the Bar complex is an homotopy colimit of diagrams, one obtains
that (the prounipotent part of) GM T is weakly equivalent to the derived loop spaceof Spec NBK . Since the prounipotency of the homotopy group sheaf characterizes
the schematic image in sP r(SchQ ), notice that a similar construction for Mg,[m]that realizes IM as motivic object requires to enlarge the aforementioned Quillen
equivalence. This lead to the DAG-context that allows to capture the S1 -motivic
inertia.
Despite the relevance of this approach, a final and fundamental difficulty is
still given by the question of a neutralizing fibre functor, which must induces
non-2-trivial geometrical de Rham-Betti comparison isomorphisms: as an étalelocally quotient stack, the rational cohomology of Mg,[m] is whose of its coarse
scheme Mg,[m] . An answer is here again provided by the HAG context, that gives
computable stack inertia periods in terms of iterated integrals that are compatible
with the Tannakian weight – via M0,m → M0,[m] and the choice of tangential
structures as involved for Galois and Mg,[m],kr (γ) see [4].
3. Arithmetic of Operads
We present how the divisorial stratification of Mg,[m] supports some fundamentalarithmetic and geometric properties. The question is two fold: from the geometricpoint of view, it is related to the fundamental question of defining on smooth objects
an operadic structure that is given by singular degeneracies; from the arithmetic
point of view it is related to Grothendieck-Teichmüller theory that is to determine
how Gal(Q̄/Q) is encoded within the geometric symmetries of Mg,[m] . More pred that contains Gal(Q̄/Q)
cisely, GT theory provides a finitely presented group GT
and factorizes the (GGR) [9, 14].
We report on how the homotopy theory of spaces and the notion of tangential
structures on M0,m provides some insight on these questions. We were recently
informed that K. Wickelgren and C. Westerland developed an independent andsimilar approach in the case of configuration spaces – see K. Wickelgren, Operad
Structure on Confn in this volume.

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3.1. Genus Zero Moduli Spaces. Motivation for this work comes from therecent operadic result of B. Fresse and G. Horel [11, 13], that interprets Drinfel’dd in terms of operad in prospaces.
definition of GT
d is isomorphic to the homotopy group of
Theorem (Fresse, Horel). The group GT∧(pro) little 2-discs operads E2 .
Here E2∧ denotes either the Sullivan rational model or the completion in prospaces.
Their fundamental groups are respectively given by the Malčev and the profinite
completion of the parenthesized braid operad in groupoids P aB ∧ . Since the Gad , this induces a Gal(Q̄/Q)-action on P aB ∧lois group Gal(Q̄/Q) is contained in GT
that is group-theoretically defined and from topological origin.
We deal with the question of recovering this result – more precisely the refinement of [1] – in terms of an arithmetic Gal(Q̄/Q)-action at the level of a Q-operadic
structure on M0,m then M0,[m] . To fix some (GGR) or Gal(Q̄/Q)-actions requires
to specify some Q-tangential structures on M0,m , i.e. to choose a formal neighbourhood Spec Q[[q]] → M0,m of some singular curves in the Deligne-Mumford
compactification of M0,m [15, 6].
In terms of operads, the choice of a tangential structure on the spaces defines thegeometric operadic composition morphisms and ensures that they are Gal(Q̄/Q)equivariant. We obtain a refinement of the Theorem above, that is from arithmetic
origin.
Theorem (conjecture). The set of Q-tangential structures {~s} over M0,m defines an operad MQ = {(M0,m , ~s)}m,~s in Q-Proschemes whose geometric étale
homotopy type is endowed with a Gal(Q̄/Q)-action – see Eq. (GGR).
The approach is based on Grothendieck’s formal-algebraization deformation
theory for curves; the operad MQ is defined in terms of Friedlander Artin-Mazurétale topological type. As a result, we obtain after completion an operad inprospaces MQ (Q̄)∧ which encodes some arithmetic a priori not distinguished by
d . More precisely, MQ (Q̄)∧ is weakly equivalent to the framed little 2-discs
GT
d ' Auth (E ∧ ) and Auth (E ∧ ) ' Auth (F E ∧ ).
operads F E2∧ , while GT222This property can already be seen at the level of configuration spaces and braids
groups – P aB is a model for E2 –, by providing a Gal(Q̄/Q)-action on an extensionof P aB ∧ that descends to the classical Gal(Q̄/Q)-action of Ihara-Matsumoto on
Braids groups [14].
Following Mac Lane’s coherence Theorem, this approach provides in particular
a computable Gal(Q̄/Q)-action which is entirely defined in 1, 2 and 3-arity. Let
us mention that Serre’s anabelian bonté of M0,m plays a key role in defining theoperadic composition on MQ .
3.2. Towards Higher Symmetries. Because this approach is close to the geometry of curves and already provides some refinement in genus zero, it motivatesand gives access to further developments in the direction of stack and higher genus
symmetries.

Mini-Workshop: AG and Symmetries around Galois and Fundamental Groups 7
In terms of stack, this motivates our work in progress on defining a similarrational operad for the genus zero moduli stack of curves with unordered marked
points M0,[m] . The ∞-model category of [20] provides the necessary context toconnect the tangential arithmetic and the 2-structure of M0,[m] . In this case, thehomotopy groups are not torsion-free – unlike the braid groups in the previous
case – but contains some stack torsion like π1orb (M0,[m] (C)) does.
Regarding the moduli spaces in higher genus, the operad MQ already comeswith additional commutativity- and associativity-like constraints at the level of
braided monoidal category. This provides additional GT-like equations, which
while already included in the pentagon-hexagon equations I, II and III defining
d , motivates in higher genus the study of a potential refinement of the originalGTgroup IΓ of [18].
(3.2.1) Define in higher genus a Grothendieck Teichmüller group IΓR given
by relations based on the tangential refined associativity and commutativity
constraints:'Gal(Q̄/Q) w

5 IΓO  u'?

) ? )
IΓR

(d
6 GT

Acknowledgments. This research was supported by he DFG programme SPP 1786
“Homotopy Theory and Algebraic Geometry”.
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