inter-universal teichmuller theory ii

8/13/2019 Inter-Universal Teichmuller Theory II

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INTER-UNIVERSAL TEICHMULLER THEORY II:HODGE-ARAKELOV-THEORETIC EVALUATION

Shinichi Mochizuki

August 2012

Abstract. In the present paper, which is the second in a seriesof four papers, we study the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation i.e., evaluation in the style of thescheme-theoretic Hodge-Arakelov theory established by the author in previous papers of the [reciprocal ofthe l-th root of the] theta functionat l-torsion points, for l 5 a prime number.In the first paper of the series, we studied miniature models of conventional schemetheory, which we referred to as ellNF-Hodge theaters, that were associated tocertain data, called initial-data, that includes an elliptic curveEF over a number

fieldF, together with a prime number l 5. These ellNF-Hodge theaters weregluedto one another by means -links, that identify the [reciprocal of the l-th rootof the]theta functionat primes of bad reduction ofEFin one

ellNF-Hodge theaterwith [2l-th roots of] the q-parameterat primes of bad reduction ofEF in anotherellNF-Hodge theater. The theory developed in the present paper allows one to

construct certain new versions of this -link. One such new version is the gau-

link, which is similar to the -link, but involves the theta values at l-torsion points,

rather than the theta function itself. One important aspect of the constructionsthat underlie the

gau-link is the study ofmultiradialityproperties, i.e., properties

of the arithmetic holomorphic structure or, more concretely, the ring/schemestructure arising from one ellNF-Hodge theater that may be formulated insuch a way as to make sense from the point of the arithmetic holomorphic structureofanother ellNF-Hodge theater which is related to the original ellNF-Hodge

theater by means of the [non-scheme-theoretic!] gau-link. For instance, certain of

the various rigidityproperties of the etale theta functionstudied in an earlier paperby the author may be intepreted as multiradiality properties in the context of thetheory of the present series of papers. Another important aspect of the constructions

that underlie the gau-link is the study ofconjugate synchronizationvia the

F

l -symmetryof a ellNF-Hodge theater. Conjugate synchronization refers to a

certain system of isomorphisms which are freeof any conjugacy indeterminacies! between copies of local absolute Galois groups at the various l-torsion points atwhich the theta function is evaluated. Conjugate synchronization plays an importantrole in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study ofcoricity properties of [i.e., the study of

objects leftinvariantby] the gau-link. Global aspects of conjugate synchronization

require the resolution, via results obtained in the first paper of the series, of certaintechnicalities involving profinite conjugates of tempered cuspidal inertia groups.

Contents:

Introduction1. Multiradial Mono-theta Environments

Typeset by AMS-TEX

1


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2 SHINICHI MOCHIZUKI

2. Galois-theoretic Theta Evaluation3. Tempered Gaussian Frobenioids4. Global Gaussian Frobenioids

Introduction

In the following discussion, we shall continue to use the notation of the In-troduction to the first paper of the present series of papers [cf. [IUTchI],I1]. Inparticular, we assume that are given an elliptic curveEF over a number fieldF,together with a prime number l 5. In the present paper, which forms the sec-ond paper of the series, we study the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation [cf. Fig. I.1 below] i.e., evaluation in thestyle of the scheme-theoretic Hodge-Arakelov theoryof [HASurI], [HASurII] ofthe reciprocal of the l-th root of thetheta function

v

def=

q 18

v nZ

(1)n q12 (n+ 12 )2

v Un+12

v

1l

[cf. [EtTh], Proposition 1.4; [IUTchI], Example 3.2, (ii)] atl-torsion pointsin thecontext of the theory of ellNF-Hodge theaters developed in [IUTchI]. Here,

relative to the notation of [IUTchI],I1, v Vbad; qv denotes the q-parameteratv of the given elliptic curve EF over a number field F; Uv denotes the standardmultiplicative coordinate on the Tate curve obtained by localizing EF at v . Let q

v

be a 2l-th root ofqv. Then these theta values at l-torsion points will, up to afactor given by a 2l-th root of unity, turn out to be of the form [cf. Remark 2.5.1,(i)]

qj2

v

where j {0, 1, . . . , l def= (l 1)/2}, soj is uniquely determined by its imagej |Fl| def= Fl/{1}={0}

Fl [cf. the notation of [IUTchI]].

(Frobenius-like!)Frobenioid-theoretic

theta function

Kummer

. . . . . . . . .

(etale-like!)Galois-theoretic etale

theta function

evaluation evaluation

(Frobenius-like!)Frobenioid-theoretic

theta values

Kummer

. . . . . . . . .

(etale-like!)Galois-theoretic

theta values

Fig. I.1: The Kummer theory surrounding Hodge-Arakelov-theoretic evaluation


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INTER-UNIVERSAL TEICHMULLER THEORY II 3

In order to understand the significance of Kummer theory in the contextof Hodge-Arakelov-theoretic evaluation, it is important to recall the notions ofFrobenius-like and etale-like mathematical structures [cf. the discussion of

[IUTchI],I1]. In the present series of papers, the Frobenius-like structures consti-tuted by [the monoidal portions of]Frobenioids i.e., more concretely, by variousmonoids play the important role of allowing one to construct gluing isomor-phisms such as the -link which lie outside the framework of conventionalscheme/ring theory [cf. the discussion of [IUTchI],I2]. Such gluing isomor-phisms give rise to Frobenius-pictures [cf. the discussion of [IUTchI],I1]. Onthe other hand, the etale-like structures constituted by various Galoisand arith-metic fundamental groupsgive rise to the canonical splittingsof such Frobenius-pictures furnished by corresponding etale-pictures [cf. the discussion of [IUTchI],I1]. In [IUTchIII],absolute anabelian geometrywill be applied to these Galoisand arithmetic fundamental groups to obtain descriptions ofalien arithmeticholomorphic structures, i.e., arithmetic holomorphic structures that lie on theopposite side of a -link from a given arithmetic holomorphic structure [cf. thediscussion of [IUTchI],I3]. Thus, in light of the equally crucialbut substantiallydifferentroles played by Frobenius-like and etale-like structures in the present seriesof papers, it is of crucial importance to be able

to relate corresponding Frobenius-likeand etale-like versions of vari-ous objects to one another.

This is the role played by Kummer theory. In particular, in the present paper,we shall study in detail the Kummer theory that relates Frobenius-like and etale-like versions of thetheta functionand itstheta valuesat l-torsion points to oneanother [cf. Fig. I.1].

One important notion in the theory of the present paper is the notion ofmul-tiradiality. To understand this notion, let us recall the etale-picture discussedin [IUTchI],I1 [cf., [IUTchI], Fig. I1.6]. In the context of the present paper, weshall be especially interested in the etale-like versionof the theta functionand its

theta valuesconstructed in eachD-ellNF-Hodge theater ()HTD-ellNF; thus,one can think of the etale-picture under consideration as consisting of the diagram

given in Fig. I.2 below. As discussed earlier, we shall ultimately be interested inapplying various absolute anabelian reconstruction algorithmsto the various arith-metic fundamental groups that [implicitly] appear in such etale-pictures in orderto obtain descriptions of alien holomorphic structures, i.e., descriptions of objectsthat arise on onespokethat make sense from the point of view ofanother spoke. Inthis context, it is natural to classify the variousalgorithmsapplied to the arithmeticfundamental groups lying in a given spokeas follows [cf. Example 1.7]:

We shall refer to an algorithm as coric if it in fact only depends oninput data arising from the mono-analytic core of the etale-picture, i.e.,

the data that is common to all spokes.

We shall refer to an algorithm as uniradial if it expresses the objectsconstructed from the given spoke in terms that only make sense withinthe given spoke.


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We shall refer to an algorithm as multiradialif it expresses the objectsconstructed from the given spoke in terms ofcorically constructedobjects,i.e., objects that make sense from the point of view ofother spokes.

Thus, multiradial algorithms are compatible with simultaneous execution atmultiple spokes[cf. Example 1.7, (v); Remark 1.9.1], whileuniradialalgorithms mayonly be consistently executed at a singlespoke. Ultimately, in the present series ofpapers, we shall be interested relative to the goal of obtaining descriptions ofalien holomorphic structures in the establishment ofmultiradialalgorithms forconstructing the objects of interest, e.g., [in the context of the present paper] theetale-like versions of the theta functions and the corresponding theta valuesdiscussed above. Typically, in order to obtain such multiradial algorithms, i.e.,algorithms that make sense from the point of view of other spokes, it is necessaryto allow for some sort ofindeterminacy in the descriptions that appear in thealgorithms of the objects constructed from the given spoke.

etale-likeversion of

v

,{qj2

v}

. . .

|. . .


v

,{qj2

v}

. . .

()D>

|


v

,{qj2

v}

. . .


v

,{qj2

v}

Fig. I.2: Etale-picture of etale-like versions of theta functions, theta values

Relative to the analogy between the inter-universal Teichmuller theory of thepresent series of papers and the classical theory of holomorphic structures onRiemann surfaces [cf. the discussion of [IUTchI],

I4], one may think of coric

algorithms as corresponding to constructions that depend only on the underlyingreal analytic structureon the Riemann surface. Then uniradial algorithms cor-respond to constructions that depend, in an essential way, on the holomorphicstructureof the given Riemann surface, while multiradial algorithms correspond


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to constructions ofholomorphicobjects associated to the Riemann surface whichare expressed [perhaps by allowing for certain indeterminacies!] solely in terms ofthe underlying real analytic structure of the Riemann surface cf. Fig. I.3

below; the discussion of Remark 1.9.2. Perhaps the most fundamental motivat-ing examplein this context is the description of alien holomorphic structures bymeans of the Teichmuller deformations reviewed at the beginning of [IUTchI],I4, relative to unspecified/indeterminate deformation data[i.e., consistingof a nonzero square differential and a dilation factor]. Indeed, for instance, in thecase of once-punctured elliptic curves, by applying well-known facts concerning Te-ichmuller mappings [cf., e.g., [Lehto], Chapter V, Theorem 6.3], it is not difficultto formulate the classical result that

the homotopy class of every orientation-preserving homeomorphism be-tween pointed compact Riemann surfaces of genus one lifts to a uniqueTeichmuller mapping

in terms of themultiradial formalismdiscussed in the present paper [cf. Example1.7]. [We leave the routine details to the reader.]

abstract inter-universal classical complex algorithms Teichm uller theory Teichm uller theory

uniradial arithmetic holomorphic holomorphicalgorithms structures structures

arithmetic holomorphic holomorphic structuresmultiradial structures described in structures described in

algorithms terms of underlying terms of underlyingmono-analytic structures real analytic structures

coric underlying mono-analytic underlying real analyticalgorithms structures structures

Fig. I.3: Uniradiality, Multiradiality, and Coricity

One interesting aspect of the theory of the present series of papers may be seenin the set-theoretic function arising from the theta values considered above

j

qj2

v

a function that is reminiscent of the Gaussian distribution (R ) x ex

2

on the real line. From this point of view, the passage from the Frobenius-pictureto the canonical splittings of theetale-picture[cf. the discussion of [IUTchI],


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I1], i.e., in effect, the computation of the -links that occur in the Frobenius-picture by means of the various multiradial algorithms that will be established inthe present series of papers, may be thought of [cf. the diagram of Fig. I.2!] as a

sort of global arithmetic/Galois-theoretic analogue of the computation of theclassical Gaussian integral

ex2

=

via the passage from cartesian coordinates, i.e., which correspond to the Frobenius-picture, to polar coordinates, i.e., which correspond to the etale-picture cf.the discussion of Remark 1.12.5.

One way to understand the difference between coricity, multiradiality, and

uniradiality at a purely combinatorial level is by considering the Fl - and F

l -

symmetries discussed in [IUTchI],I1 [cf. the discussion of Remark 4.7.4 of thepresent paper]. Indeed, at a purely combinatorial level, the Fl -symmetrymay be

thought of as consisting of the natural action of Fl on the set of labels|Fl| ={0} Fl [cf. the discussion of [IUTchI],I1]. Here, the label0 corresponds tothe [mono-analytic] core. Thus, the corresponding etale-picture consists of variouscopies of|Fl| glued together along thecoric label0 [cf. Fig. I.4 below]. In particular,the various actions of copies ofFl on corresponding copies of|Fl| are compatiblewith simultaneous execution in the sense that they commute with one another.That is to say, at least at the level of labels, the Fl -symmetry is multiradial.

. . .

. . .

|

0

Fig. I.4: Etale-picture ofFl -symmetries

. . .

. . .

0

Fig. I.5: Etale-picture ofFl -symmetries


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In a similar vein, at a purely combinatorial level, the Fl -symmetrymay be thoughtof as consisting of the natural action ofFl on the set oflabelsFl[cf. the discussionof [IUTchI],I1]. Here again, the label0 corresponds to the [mono-analytic] core.Thus, the corresponding etale-picture consists of variouscopies ofFl glued togetheralong the coric label 0 [cf. Fig. I.5 above]. In particular, the various actions ofcopies of Fl on corresponding copies of Fl are incompatible with simultaneousexecution in the sense that they clearly fail to commute with one another. That isto say, at least at the level of labels, the Fl -symmetry is uniradial.

Since, ultimately, in the present series of papers, we shall be interested in theestablishment of multiradial algorithms, special care will be necessary in orderto obtain multiradial algorithms for constructing objects related to the a prioriuniradial Fl -symmetry[cf. the discussion of Remark 4.7.3 of the present paper;[IUTchIII], Remark 3.11.2, (i), (ii)]. The multiradiality of such algorithms will be

closely related to the fact that Fl -symmetry is applied to relate the variouscopiesof local units modulo torsion, i.e., O [cf. the notation of [IUTchI],1] atvarious labels Fl that lie in various spokes of the etale-picture [cf. the discussionof Remark 4.7.3, (ii)]. This contrasts with the way in which the a priorimultira-dial Fl -symmetrywill be applied, namely to treat various weighted volumescorresponding to the local value groupsand global realified Frobenioidsat variouslabels Fl that lie in various spokes of the etale-picture [cf. the discussion of Re-mark 4.7.3, (iii)]. Relative to the analogy between the theory of the present seriesof papers and p-adic Teichmuller theory [cf. [IUTchI],I4], various aspects of theFl -symmetry are reminiscent of the additive monodromy over theordinary

locus of the canonical curves that occur in p-adic Teichmuller theory; in a similarvein, various aspects of the Fl -symmetry may be thought of as correspondingto the multiplicative monodromy at thesupersingular points of the canonicalcurves that occur in p-adic Teichmuller theory cf. the discussion of Remark4.11.4, (iii); Fig. I.7 below.

Before discussing the theory ofmultiradiality in the context of the theory ofHodge-Arakleov-theoretic evaluation theorydeveloped in the present paper, we pauseto review the theory of mono-theta environments developed in [EtTh]. Onestarts with a Tate curve over a mixed-characteristic nonarchimedean local field.The mono-theta environmentassociated to such a curve is, roughly speaking, the

Kummer-theoretic datathat arises by extracting N-th roots of thetheta trivial-ization of the ample line bundle associated to the origin over appropriate tem-pered coverings of the curve [cf. [EtTh], Definition 2.13, (ii)]. Such mono-thetaenvironments may be constructed purely group-theoretically from the [arithmetic]tempered fundamental groupof the once-punctured elliptic curve determined by thegiven Tate curve [cf. [EtTh], Corollary 2.18], or, alternatively, purely category-theoreticallyfrom the tempered Frobenioiddetermined by the theory of line bundlesand divisors over tempered coverings of the Tate curve [cf. [EtTh], Theorem 5.10,(iii)]. Indeed, the isomorphism of mono-theta environments between the mono-theta environments arising from these two constructions of mono-theta environ-ments i.e., from tempered fundamental groups, on the one hand, and from tem-pered Frobenioids, on the other [cf. Proposition 1.2 of the present paper] may bethought of as a sort of Kummer isomorphism for mono-theta environments[cf. Proposition 3.4 of the present paper, as well as [IUTchIII], Proposition 2.1,(iii)]. One important consequence of the theory of [EtTh] asserts that mono-theta


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environments satisfy the following three rigidity properties:

(a) cyclotomic rigidity,

(b) discrete rigidity, and(c) constant multiple rigidity

cf. the Introduction to [EtTh].

Discrete rigidity assures one that one may work with Z-translates [where wewrite Z for the copy of Z that acts as a group of covering transformations on the

tempered coverings involved], as opposed toZ-translates [i.e., whereZ =Z denotesthe profinite completion of Z], of the theta function, i.e., one need not contend

withZ-powers of canonical multiplicative coordinates [i.e., U], or q-parameters[cf. Remark 3.6.5, (iii); [IUTchIII], Remark 2.1.1, (v)]. Although we will certainlyuse this discrete rigidity throughout the theory of the present series of papers,this property of mono-theta environments will not play a particularly prominent

role in the theory of the present series of papers. TheZ-powers of U and q thatwould occur if one does not have discrete rigidity may be compared to the PD-formal series that are obtained,a priori, if one attempts to construct thecanonicalparameters ofp-adic Teichmuller theoryvia formal integration. Indeed, PD-formalpower series become necessary if one attempts to treat such canonical parameters

as objects which admitarbitrary

O-powers, where

O denotes the completion of the

local ring to which the canonical parameter belongs [cf. the discussion of Remark

3.6.5, (iii); Fig. I.6 below].

Constant multiple rigidity plays a somewhat more central role in thepresent series of papers, in particular in relation to the theory of thelog-link, whichwe shall discuss in [IUTchIII] [cf. the discussion of Remark 1.12.2 of the presentpaper; [IUTchIII], Remark 1.2.3, (i); [IUTchIII], Proposition 3.5, (ii); [IUTchIII],Remark 3.11.2, (iii)]. Constant multiple rigidity asserts that the multiplicativemonoid

OFv

v

which we shall refer to as the theta monoid generated by the reciprocalof the l-th root of the theta function and the group of unitsof the ring of inte-gers of the base field Fv [cf. the notation of [IUTchI],I1] admits a canonicalsplitting, up to 2l-th roots of unity, that arises from evaluationat the[2-]torsionpoint corresponding to the label 0 Fl [cf. Corollary 1.12, (ii); Proposition 3.1,(i); Proposition 3.3, (i)]. Put another way, this canonical splitting is the splittingdetermined, up to 2l-th roots of unity, by

v O

Fv

v. The theta monoid of

the above display, as well as the associated canonical splitting, may be constructedalgorithmically from the mono-theta environment [cf. Proposition 3.1, (i)]. Rela-tive to the analogy between the theory of the present series of papers and p-adicTeichmuller theory, these canonical splittings may be thought of as correspondingto the canonical coordinates of p-adic Teichmuller theory, i.e., more precisely,to the fact that such canonical coordinates are also completely determined withoutany constant multiple indeterminacies cf. Fig. I.6 below; Remark 3.6.5, (iii);[IUTchIII], Remark 3.12.4, (i).


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Mono-theta-theoretic rigidity property Corresponding phenomenon inin inter-universal Teichmuller theory p-adic Teichmuller theory

mono-theta-theoretic lack of constant multiple constant indeterminacy ofmultiple canonical coordinates

rigidity on canonical curves

lack ofZ-power indeterminacymono-theta-theoretic of canonical coordinatescyclotomic on canonical curves,

rigidity Kodaira-Spencerisomorphism

multiradialityofmono-theta-theoretic Frobenius-invariantconstant multiple, nature of

cyclotomic canonical coordinates

rigidity

mono-theta-theoretic formal = non-PD-formaldiscrete nature of canonical coordinatesrigidity on canonical curves

Fig. I.6: Mono-theta-theoretic rigidity properties in inter-universal Teichmuller

theory and corresponding phenomena in p-adic Teichmuller theory

Cyclotomic rigidity consists of a rigidity isomorphism, which may be con-structed algorithmicallyfrom the mono-theta environment, between

the portion of the mono-theta environment which we refer to as theexterior cyclotome that arises from the roots of unity of the basefieldand

a certain copy of the once-Tate-twisted Galois module

Z(1) which

we refer to as the interior cyclotome that appears as a subquotientof the geometrictempered fundamental group

[cf. Definition 1.1, (ii); Proposition 1.3, (i)]. This rigidity is remarkable as weshall see in our discussion below of the corresponding multiradiality property


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in that unlike the conventional construction of such cyclotomic rigidity isomor-phisms via local class field theory [cf. Proposition 1.3, (ii)], which requires one touse the entiremonoid with Galois action Gv OFv , the only portion of themonoidOFv that appears in this construction is the portion [i.e., the exteriorcyclotome] corresponding to the torsion subgroupO

Fv O

Fv[cf. the notation

of [IUTchI],I1]. This construction depends, in an essential way, on the com-mutator structure of theta groups, but constitutes a somewhat different approachto utilizing this commutator structure from the classical approach involvingirre-ducibilityof representations of theta groups [cf. Remark 3.6.5, (ii); the Introductionto [EtTh]]. One important aspect of this dependence on the commutator structureof the theta group is that the theory of cyclotomic rigidity yields an explanationfor the importance of the special role played by the first power of [the reciprocal

of the l-th root of] thetheta function in the present series of papers [cf. Remark3.6.4, (iii), (iv), (v); the Introduction to [EtTh]]. Relative to the analogy betweenthe theory of the present series of papers and p-adic Teichmuller theory, mono-theta-theoretic cyclotomic rigidity may be thought of as corresponding either tothe fact that the canonical coordinates ofp-adic Teichmuller theoryare completely

determined without anyZ-power indeterminaciesor [roughly equivalently] to theKodaira-Spencer isomorphismof the canonical indigenous bundle cf. Fig.I.6; Remark 3.6.5, (iii); Remark 4.11.4, (iii), (b).

The theta monoidO

Fv

v

discussed above admits both etale-like and Frobenius-like [i.e.,Frobenioid-theoretic]versions, which may be related to one another via a Kummer isomorphism [cf.Proposition 3.3, (i)]. The unit portion, together with its natural Galois action, ofthe Frobenioid-theoretic version of the theta monoid

Gv OFv

forms the portion at v Vbad of theF-prime-strip Fmod that is preserved,up to isomorphism, by the -link [cf. the discussion of [IUTchI],I1; [IUTchI],Theorem A, (ii)]. In the theory of the present paper, we shall introducemodifiedversions of the -link of [IUTchI] [cf. the discussion of the -, gau-linksbelow], which, unlike the -link of [IUTchI], only preserve[up to isomorphism] theF-prime-strips i.e., which consist of the data

Gv OFv = O

Fv/O

Fv

[cf. the notation of [IUTchI],I1] at v Vbad associated to theF-prime-strips preserved [up to isomorphism] by the -link of [IUTchI]. Since this data isonly preserved up to isomorphism, it follows that the topological group Gv must

be regarded as beingonly known up to isomorphism, while the monoid

O

Fv

must be

regarded as beingonly known up to [the automorphisms of this monoid determined

by the natural action of]Z. That is to say, one must regardthe data Gv OFv as subject to Aut(Gv)-,Z-indetermnacies.


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These indeterminacies will play an important role in the theory of the present seriesof papers cf. the indeterminacies (Ind1), (Ind2) of [IUTchIII], Theorem 3.11,(i).

Now let us return to our discussion of the various mono-theta-theoretic rigidityproperties. Thekey observation concerning these rigidity properties, as reviewed

above, in the context of the Aut(Gv)-,Z-indeterminacies just discussed, is thefollowing:

thecanonical splittings, viaevaluation at the zero section, of thethetamonoids, together with the construction of the mono-theta-theoreticcyclotomic rigidity isomorphism, are compatible with, in the sense

that they areleft unchangedby, the Aut(Gv)-,

Z-indeterminaciesdis-

cussed above

cf. Corollaries 1.10, 1.12; Proposition 3.4, (i). Indeed, this observation consti-tutes the substantive content of the multiradiality of mono-theta-theoretic con-stant multiple/cyclotomic rigidity [cf. Fig. I.6] and will play an important rolein the statements and proofs of the main results of the present series of papers[cf. [IUTchIII], Theorem 2.2; [IUTchIII], Corollary 2.3; [IUTchIII], Theorem 3.11,(iii), (c); Step (ii) of the proof of [IUTchIII], Corollary 3.12]. At a technical level,this key observation simply amounts to the observation that the only portion ofthe monoidO

Fvthat isrelevantto the construction of the canonical splittingsand

cyclotomic rigidity isomorphismunder consideration is the torsion subgroup

O

Fv

,

which [by definition!] maps to the identity elementofOFv

, hence is immuneto

the variousindeterminaciesunder consideration. That is to say, the multiradialityof mono-theta-theoretic constant multiple/cyclotomic rigidity may be regarded asan essentially formal consequence of the triviality of the natural homomorphism

OFv

OFv

[cf. Remark 1.10.2].

After discussing, in

1, the multiradiality theory surrounding the various rigid-ity properties of the mono-theta environment, we take up the task, in2 and3, ofestablishing the theory ofHodge-Arakelov-theoretic evaluation, i.e., of passing[for v Vbad]

OFv

v O

Fv {qj

2

v}j=1,... ,l

fromtheta monoids as discussed above [i.e., the monoids on the left-hand side ofthe above display] toGaussian monoids[i.e., the monoids on the right-hand sideof the above display] by means of the operation ofevaluation atl-torsion points.Just as in the case of theta monoids, Gaussian monoids admit both etale-likever-sions, which constitute the main topic of

2, and Frobenius-like [i.e., Frobenioid-

theoretic] versions, which constitute the main topic of3. Moreover, as discussed atthe beginning of the present Introduction, it is of crucial importance in the theoryof the present series of papers to be able to relatetheseetale-likeand Frobenius-likeversions to one another via Kummer theory. One important observation in this


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context which we shall refer to as the principle of Galois evaluation isthe following: it is essentially a tautologythat

this requirement ofcompatibilitywithKummer theoryforcesany sortofevaluation operationto arise fromrestrictiontoGalois sectionsofthe [arithmetic] tempered fundamental groups involved

[i.e., Galois sections of the sort that arise from rational points such as l-torsionpoints!] cf. the discussion of Remarks 1.12.4, 3.6.2. This tautology is interestingboth in light of the history surrounding the Section Conjecturein anabelian geom-etry [cf. [IUTchI],I5] and in light of the fact that the theory of [SemiAnbd] that isapplied to prove [IUTchI], Theorem B a result which plays an important role inthe theory of

2 of the present paper! [cf. the discussion below] may be thought

of as a sort of Combinatorial Section Conjecture.

At this point, we remark that, unlike the theory of theta monoids discussedabove, the theory of Gaussian monoids developed in the present paper does not,by itself, admit a multiradial formulation[cf. Remarks 2.9.1, (iii); 3.4.1, (ii); 3.7.1].In order to obtain a multiradial formulation of the theory of Gaussian monoids which is, in some sense, the ultimate goal of the present series of papers! itwill be necessary to combine the theory of the present paper with the theory ofthe log-link developed in [IUTchIII]. This will allow us to obtain a multiradialformulationof the theory of Gaussian monoids in [IUTchIII], Theorem 3.11.

One important aspect of the theory of Hodge-Arakelov-theoretic evaluation isthe notion ofconjugate synchronization. Conjugate synchronization refers to acollection of symmetrizing isomorphisms between the various copies of the localabsolute Galois group Gv associated to the labels Fl at which one evaluates thetheta function [cf. Corollaries 3.5, (i); 3.6, (i); 4.5, (iii); 4.6, (iii)]. We shall alsouse the term conjugate synchronization to refer to similar collections of sym-metrizing isomorphisms for copies of various objects [such as the monoidO

Fv]

closely related to the absolute Galois group Gv. With regard to the collections ofisomorphisms between copies ofGv, it is ofcrucial importancethat these isomor-phisms be completely well-defined, i.e., without any conjugacy indeterminacies!

Indeed, if one allows conjugacy indeterminacies [i.e., put another way, if one allowsoneself to work with outer isomorphisms, as opposed to isomorphisms], then onemust sacrifice either

the distinct nature of distinct labels |Fl| which is necessary inorder to keep track of the distinct theta valuesq

j2

for distinct j or

the crucialcompatibilityof etale-like and Frobenius-like versions of thesymmetrizing isomorphisms with Kummer theory

cf. the discussion of Remark 3.8.3, (ii); [IUTchIII], Remark 1.5.1; Step (vii)of the proof of [IUTchIII], Corollary 3.12. In this context, it is also of interest toobserve that it follows from certain elementary combinatorial considerations thatone must require that


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these symmetrizing isomorphisms arise from a group action, i.e., suchas the Fl -symmetry

cf. the discussion of Remark 3.5.2. Moreover, since it will be of crucial impor-tance to apply these symmetrizing isomorphisms, in [IUTchIII],1 [cf., especially,[IUTchIII], Remark 1.3.2], in the context of the log-link whose definition de-

pends on the localring structuresat v Vbad [cf. the discussion of [AbsTopIII],I3] it will be necessary to invoke the fact that

the symmetrizing isomorphisms at v Vbad arise from conjugation op-erations within a certain [arithmetic] tempered fundamental group namely, the tempered fundamental group of Xv [cf. the notation of

[IUTchI],I1] that contains v as an open subgroup of finite index

cf. the discussion of Remark 3.8.3, (ii). Here, we note that these conjugationoperations related to the Fl -symmetry may be applied to establish conjugatesynchronizationprecisely because they arise from conjugation by elements of thegeometrictempered fundamental group [cf. Remark 3.5.2, (iii)].

The significanceof establishing conjugate synchronization i.e., subjectto the various requirements discussed above! lies in the fact that the resultingsymmetrizing isomorphisms allow one to

construct the crucialcoricF-prime-strips

i.e., theF

-prime-strips that are preserved, up to isomorphism, by the modi-fied versions of the-link of [IUTchI] [cf. the discussion of the -, gau-linksbelow] that are introduced in4 of the present paper [cf. Corollary 4.10, (i), (iv);[IUTchIII], Theorem 1.5, (iii); the discussion of [IUTchIII], Remark 1.5.1, (i)].

In4, the theory of conjugate synchronization established in3 [cf. Corollaries3.5, (i); 3.6, (i)] is extended so as to apply to arbitraryv V, i.e., not justv Vbad[cf. Corollaries 4.5, (iii); 4.6, (iii)]. In particular, in order to work with the thetavalue labels Fl in the context of the Fl -symmetry, i.e., which involves theaction

Fl Fl

on the labels Fl, one must avail oneself of the global portionof theell-Hodgetheaters that appear. Indeed, this global portion allows one to synchronizethe apriori independent indeterminacies with respect to the action of{1} on thevariousX

v [forv Vbad], Xv [forv V

good] cf. the discussion of Remark 4.5.3,

(iii). On the other hand, the copy of the arithmetic fundamental group ofXK thatconstitutes this global portion of the ell-Hodge theater is profinite, i.e., it doesnot admit a globally tempered version whose localization at v Vbad is naturallyisomorphic to the corresponding tempered fundamental group at v. One importantconsequence of this state of affairs is that

in order to apply the global-synchronizationafforded by the ell-Hodge theater in the context of the theory of Hodge-Arakelov-theoreticevaluationat v Vbad relative to labels Fl that correspond to conju-gacy classes of cuspidal inertia groups oftemperedfundamental groups at


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v Vbad, it is necessary to compute the profinite conjugates of suchtempered cuspidal inertia groups

cf. the discussion of [IUTchI], Remark 4.5.1, as well as Remarks 2.5.2 and 4.5.3,(iii), of the present paper, for more details. This is precisely what is achieved bythe application of [IUTchI], Theorem B [i.e., in the form of [IUTchI], Corollary 2.5]in2 of the present paper.

As discussed above, the theory of Hodge-Arakelov-theoretic evaluation devel-oped in1,2,3 is strictly local [at v Vbad] in nature. Thus, in4, we discussthe essentially routine extensions of this theory, e.g., of the theory of Gaussianmonoids, to the remaining portion of the ell-Hodge theater, i.e., to v Vgood,as well as to the case ofglobal realified Frobenioids[cf. Corollaries 4.5, (iv), (v); 4.6,(iv), (v)]. We also discuss the corresponding complements, involving the theory of

[IUTchI],5, for NF-Hodge theaters[cf. Corollaries 4.7, 4.8]. This leads naturallyto the construction of modified versions of the -link of [IUTchI] [cf. Corollary4.10, (iii)]. These modified versions may be described as follows:

The -linkis essentially the same as the -link of [IUTchI], TheoremA, except thatF-prime-strips are replaced byF-prime-strips[cf.[IUTchI], Fig. I1.2] i.e., roughly speaking, the various local O arereplaced by O =O/O. The gau-link is essentially the same as the -link, except that thetheta monoids that give rise to the -link are replaced, via composition

with a certain isomorphism that arises fromHodge-Arakelov-theoretic eval-uation, byGaussian monoids[cf. the above discussion!] i.e., roughly

speaking, the various v

at v Vbad are replaced by {qj2

v}j=1,... ,l.

The basic properties of the -, gau-links, including the correspondingFrobenius-and etale-pictures, are summarized in Theorems A, B below [cf. Corollaries 4.10,4.11 for more details]. Relative to the analogy between the theory of the presentseries of papers and p-adic Teichmuller theory, the passage from the -link tothegau-linkvia Hodge-Arakelov-theoretic evaluation may be thought of ascorresponding to the passage

MF-objects Galois representationsin the case of thecanonical indigenous bundlesthat occur inp-adic Teichmullertheory cf. the discussion of Remark 4.11.4, (ii), (iii). In particular, the corre-sponding passage from the Frobenius-picture associated to the -link to theFrobenius-picture associated to the gau-link or, more properly, relative to thepoint of view of [IUTchIII] [cf. also the discussion of [IUTchI],I4], from thelog-theta-latticearising from the -link to the log-theta-lattice arising from thegau-link corresponds [i.e.., relative to the analogy withp-adic Teichmuller the-ory] to the passage

from thinking ofcanonical liftings as being determined by canonicalMF-objects to thinking of canonical liftings as being determined bycanonical Galois representations [cf. Fig. I.7 below].


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In this context, it is of interest to note that this point of view is precisely the point ofview taken in the absolute anabelian reconstruction theory developed in [CanLift],3 [cf. Remark 4.11.4, (iii), (a)]. Finally, we observe that from this point of view,the important theory of conjugate synchronization via F

l -symmetrymay be

thought of as corresponding to the theory of thedeformation of the canonical Galoisrepresentation from modpn to modpn+1 [cf. Fig. I.7 below; the discussion ofRemark 4.11.4, (iii), (d)].

Property related to Corresponding phenomenonHodge-Arakelov-theoretic in

evaluation in inter-universal p-adic Teichmuller theory

Teichmuller theory

passage from passage from -link canonicality viaMF-objects

to to canonicality via gau-link crystalline Galois representations

Fl -, Fl - ordinary, supersingular monodromy

symmetries of canonical Galois representation

conjugate deformation ofsynchronization canonical Galois representationviaFl -symmetry from mod p

n to modpn+1

Fig. I.7: Properties related to Hodge-Arakelov-theoretic evaluation ininter-universal Teichmuller theory and corresponding phenomena in

p-adic Teichmuller theory

Certain aspects of the various constructions discussed above are summarizedin the following two results, i.e., Theorems A, B, which are abbreviated versionsof Corollaries 4.10, 4.11, respectively. On the other hand, many important aspects such as multiradiality! of these constructions do not appear explicitly inTheorems A, B. The main reason for this is that it is difficult to formulate finalresults concerning such aspects as multiradiality in the absence of the frameworkthat is to be developed in [IUTchIII].

Theorem A. (Frobenioid-pictures ofellNF-Hodge Theaters)Fix a col-lection of initial -data (F/F, XF, l , C K, V, ) as in [IUTchI], Definition

3.1. Let HTellNF; HTellNF beellNF-Hodge theaters [relative to thegiven initial -data] cf. [IUTchI], Definition 6.13, (i). Write HTD-ellNF,


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HTD-ellNF for the associatedD-ellNF-Hodge theaters cf. [IUTchI],Definition 6.13, (ii). Then:

(i)(Constant Prime-Strips)By applying thesymmetrizing isomorphisms,with respect to the Fl -symmetry, of Corollary 4.6, (iii), to the data of the un-

derlyingell-Hodge theater of HTellNF that is labeled bytLabCusp(D),one may construct, in a natural fashion, anF-prime-strip

F = (C, Prime(C) V, F,{,v}vV)

that is equipped with a natural identification isomorphism ofF-prime-stripsF

Fmod between F and theF-prime-strip Fmod of [IUTchI], TheoremA, (ii); this isomorphism induces anatural identification isomorphism ofD-prime-strips D

D> between the

D-prime-strip D associated to

F and

theD-prime-strip D> of [IUTchI], Theorem A, (iii).(ii) (Theta and Gaussian Prime-Strips) By applying the constructions

of Corollary 4.6, (iv), (v), to the underlying-bridge andell-Hodge theater ofHTellNF, one may construct, in a natural fashion,F-prime-strips

Fenv = (Cenv, Prime(Cenv) V, Fenv,{env,v}vV)

Fgau = (Cgau, Prime(Cgau) V, Fgau,{gau,v}vV)

that are equipped with anatural identification isomorphism ofF-prime-strips

F

env

F

tht between

F

env and theF

-prime-strip

F

tht of [IUTchI], TheoremA, (ii), as well as anevaluation isomorphism

Fenv Fgau

ofF-prime-strips.(iii)(- and gau-Links) Write

F (respectively,

Fenv ; Fgau )

for theF-prime-strip associated to theF-prime-strip F (respectively,Fenv;

Fgau). We shall refer to the full poly-isomorphism Fenv

F asthe-link

HTellNF

HTellNF

[cf. the -link of [IUTchI], Theorem A, (ii)] from HTellNF to HTellNF,and to the full poly-isomorphism Fgau

F which may be regarded asbeing obtained from the full poly-isomorphism Fenv

F by compositionwith the inverse of theevaluation isomorphism of (ii) as thegau-link

HTellNF gau HTellNF

from HTellNF to HTellNF.

(iv) (CoricF

-Prime-Strips) The definition of theunit portion of thetheta andGaussian monoids that appear in the construction of theF-prime-strips Fenv,

Fgau of (ii) gives rise to natural isomorphisms

F

Fenv Fgau


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of theF-prime-strips associated to theF-prime-strips F, Fenv, Fgau.Moreover, by composing these natural isomorphisms with the poly-isomorphismsinduced on the respectiveF-prime-strips by the - and gau-links of (iii),one obtains a poly-isomorphism

F

Fwhich coincides with the full poly-isomorphism between these twoF-prime-strips that is to say, ()F is aninvariantof both the

- andgau-links.Finally, this full poly-isomorphism induces thefull poly-isomorphism

D D

between the associatedD-prime-strips; we shall refer to this poly-isomorphism as

theD-ellNF-link from HTD-ellNF to HTD-ellNF.

(v)(Frobenius-pictures) Let{nHTellNF}nZ be acollection of distinctellNF-Hodge theaters indexed by the integers. Then by applying the -andgau-links of (iii), we obtain infinite chains

. . . (n1)HTellNF

nHTellNF

(n+1)HTellNF

. . .

. . .gau (n1)HTellNF

gau nHTellNF

gau (n+1)HTellNF

gau . . .

of -/gau-linked -Hodge theaters cf. Fig. I.8 below, in the case ofthegau-link. Either of these infinite chains may be represented symbolically as an

oriented graph

. . . . . .

i.e., where the arrows correspond to either the s or the

gau s, and

the s correspond to the nHTellNF. This oriented graph admits a naturalaction byZ i.e., atranslation symmetry but it doesnot admit arbitrarypermutation symmetries. For instance,does not admit an automorphism that

switches two adjacent vertices, but leaves the remaining vertices fixed.

. . .

- -

nHTellNF

nqv nq

12...(l)2

v

- -

(n+1)HTellNF

(n+1)qv (n+1)q

12...(l)2

v

- -

. . .

Fig. I.8: Frobenius-picture associated to the gau-link


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Theorem B. (Etale-pictures of Base-ellNF-Hodge Theaters) Supposethat we are in the situation of Theorem A, (v).

(i) Write

. . . D nHTD-ellNF D (n+1)HTD-ellNF D . . .

where n Z for the infinite chain ofD-ellNF-linkedD-ellNF-Hodge theaters [cf. Theorem A, (iv), (v)] induced by either of the infinitechains of Theorem A, (v). Then this infinite chain induces achain of full poly-isomorphisms

. . . nD (n+1)D . . .

[cf. Theorem A, (iv)]. That is to say, ()D forms aconstant invariant

i.e., amono-analytic core [cf. the discussion of [IUTchI],I1] of the aboveinfinite chain.(ii) If we regard each of theD-ellNF-Hodge theaters of the chain of (i) as a

spoke emanating from the mono-analytic core ()D discussed in (i), then we

obtain adiagram i.e., anetale-picture ofD-ellNF-Hodge theaters asin Fig. I.9 below [cf. the situation discussed in [IUTchI], Theorem A, (iii)]. Thus,each spoke may be thought of as a distinct arithmetic holomorphic struc-ture on the mono-analytic core. Finally, [cf. the situation discussed in [IUTchI],Theorem A, (iii)] this diagram satisfies the important property of admittingarbi-trary permutation symmetries among the spokes [i.e., the labelsn

Z of the

D-ellNF-Hodge theaters].(iii) The constructions of (i) and (ii) arecompatible, in the evident sense,

with the constructions of [IUTchI], Theorem A, (iii), relative to thenatural iden-

tification isomorphisms ()D ()D> [cf. Theorem A, (i)].

nHTD-ellNF

. . .|

. . .

nHTD-ellNF

. . .

()D

|

nHTD-ellNF

. . .

nHTD-ellNF

Fig. I.9: Etale-picture ofD-ellNF-Hodge theaters


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Acknowledgements:

I would like to thank Fumiharu Kato and Akio Tamagawa for many helpful

discussions concerning the material presented in this paper.

Notations and Conventions:

We shall continue to use the Notations and Conventions of [IUTchI],0.


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Section 1: Multiradial Mono-theta Environments

In the present

1, we review the theory ofmono-theta environmentsdevel-oped in [EtTh] and give a multiradial interpretation of this theory which willbe of substantial importance in the present series of papers. Roughly speaking, inthe language of [AbsTopIII],I3, this interpretation consists of the computation ofwhich portion of the various objects constructed from the arithmetic holomorphicstructures of various -Hodge theaters may be glued together, in a consistentfashion, via a mono-analytic [i.e., arithmetic real analytic] core. Put an-other way, this computation may be thought of as the computation of

what one arithmetic holomorphic structure looks like from the point ofview of adistinctarithmetic holomorphic structurethat is only related to

the orginal arithmetic holomorphic structure via the mono-analytic core.

In fact, this sort of computation forms one of the central themes of the presentseries of papers.

LetN N1 be a positive integer;l a prime number; k an MLF ofodd residuecharacteristicpthat contains a primitive4l-th root of unity; k an algebraic closureofk;

Xk

a hyperbolic curve of type (1, l-tors) over k that admits a stable modelover thering of integersOk ofk ; XkCk the k-coredetermined by Xk [cf. the discussionpreceding [EtTh], Definition 1.7]. Write tpX

k

for thetempered fundamental groupof

Xk

; Gkdef= Gal(k/k); tpX

k

def= Ker(tpX

k

Gk)tpXk

for the geometric tempered

fundamental groupofXk

. We shall use similar notation for objects associated toCk.

Definition 1.1. LetM

be a mod N mono-theta environment [cf. [EtTh], Definition 2.13, (ii)] which isisomorphic to the mod Nmodelmono-theta environment determined by X

k; write

M for the underlying topological group of M [cf. [EtTh], Definition 2.13, (ii),

(a)]. Then:

(i) There exist functorial algorithms

M Y(M); M X(M); M G(M); M M ;M Y(M); M X(M); M (l )(M); M (M)

for constructing from M a quotient M Y(M) [cf. [EtTh], Corollary

2.18, (iii)]; a topological group X(M) which is isomorphic to tpX

k

and con-

tains Y(M) as a normal open subgroup [cf. [EtTh], Corollary 2.18, (iii)]; a


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quotient X(M) G(M) corresponding to Gk [cf., [EtTh], Corollary 2.18, (i)],

which may also be thought of as a quotient M Y(M) G(M); a closed

normal subgroup M

def

= Ker(M

G(M

)) M

; a closed normal sub-group Y(M

)def= Ker(Y(M

) G(M))Y(M); a closed normal subgroupX(M

) def

= Ker(X(M) G(M)) X(M) corresponding to tpX

k

[cf.,

[EtTh], Corollary 2.18, (i)]; a subquotient (l )(M) of Y(M) which admits anatural X(M

)-action [hence also a Y(M)-action, as well as, by composition, a

M-action] relative to which it is abstractly isomorphic toZ(1) [cf., [EtTh], Corol-lary 2.18, (i)]; a closed normal subgroup (M

) M [cf., [EtTh], Corollary2.19, (i)] which admits a natural X(M

)-action [hence also a Y(M)-action, as

well as, by composition, a M-action] relative to which it is abstractly isomorphic

to (Z/NZ)(1). Also, we recall that the structure ofM determines a liftingof thenatural outer action of

(l Z)(M) def= X(M)/Y(M)=X(M)/Y(M)

on Y(M) to an outer action of (l Z)(M) on M [cf. [EtTh], Definition 2.13,

(i), (ii), and the preceding discussion; [EtTh], Proposition 2.14, (i)].

(ii) We shall refer to (l)(M) (respectively, (M)) as the interior(respectively,exterior) cyclotome associated to M. By [EtTh], Corollary 2.19,

(i), there is afunctorial algorithmfor constructing from M acyclotomic rigidityisomorphism

(l )(M) (Z/NZ) (M)between the reductions modulo Nof the interior and exterior cyclotomes associatedto M.

One key property of mono-theta environments is that they may be constructedeither group-theoreticallyfrom tpX

k

orcategory-theoreticallyfrom certain tempered

Frobenioids related to Xk

.

Proposition 1.2. (Group- and Frobenioid-theoretic Constructions ofMono-theta Environments)

(i) Let be a topological group isomorphic to tpXk

. Then there exists a

functorial group-theoretic algorithm

M()

for constructing from the topological group a mod N mono-theta environ-ment up to isomorphism [cf. [EtTh], Corollary 2.18, (ii)] such that thecomposite of this algorithm with the algorithmM()X(M()) discussed inDefinition 1.1, (i), admits a functorial isomorphism

X(M()). Here,


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the isomorphism indeterminacy of M() is with respect to a group of N-conjugacy classes of automorphisms which is of order1 (respectively, 2) if N isodd (respectively, even) [cf. [EtTh], Corollary 2.18, (iv)].

(ii) LetC be acategory equivalent to thetempered Frobenioid determinedbyX

k[i.e., the Frobenioid denoted C in the discussion at the beginning of [EtTh],

5; the Frobenioid denoted Fv

in the discussion of [IUTchI], Example 3.2, (i)].

Thus,C admits a natural Frobenioid structure over abase categoryD equivalenttoBtemp(tpX

k

)0 [cf. [FrdI], Corollary 4.11, (ii), (iv); [EtTh], Proposition 5.1].

Then there exists a functorial algorithm

C M(C)

for constructing from the categoryC

a mod N mono-theta environment [cf.[EtTh], Theorem 5.10, (iii)] such that the composite of this algorithm with the algo-rithm M(C)X(M(C)) discussed in Definition 1.1, (i), admits afunctorialisomorphismD Btemp(X(M(C)))0.

Proof. The assertions of Proposition 1.2 follow immediately from the results of[EtTh] that are quoted in the statements of these assertions.

The cyclotomic rigidity isomorphismof Definition 1.1, (ii), that arises in thecase of the mono-theta environment M(

C) constructed from the tempered Frobe-

nioidC [cf. Proposition 1.2, (ii)] is compatible with a certain cyclotomic rigidityisomorphism that arises in the theory of [AbsTopIII] [cf. also [FrdII], Theorem 2.4,(ii)] in the following sense.

Proposition 1.3. (Compatibility of Cyclotomic Rigidity Isomorphisms)In the situation of Proposition 1.2, (ii):

(i)(Mono-theta Environments Associated to Tempered Frobenioids)For a suitable objectS Ob(C) [cf. [EtTh], Lemma 5.9, (v)], whose image inDwe denote by Sbs

Ob(

D), theinterior cyclotome (l

)(M

(

C))

(Z/NZ)

corresponds to a certainsubquotient ofAut(Sbs), which we denote by(l )S(Z/NZ), while theexterior cyclotome (M

(C)) (Z/NZ) corresponds to thesubgroup N(S) O(S) Aut(S). In particular, the cyclotomic rigidityisomorphism of Definition 1.1, (ii), takes the form of an isomorphism

(l )S (Z/NZ) N(S) (mono-)

[cf. [EtTh], Proposition 5.5; [EtTh], Lemma 5.9, (v)].

(ii) (MLF-Galois Pairs) Relative to the formal correspondence between p-adic Frobenioids [such as the base-field-theoretic hull

Cbs-fld associated to

C cf. [EtTh], Definition 3.6, (iv)] and MLF-Galois TM-pairs in the theoryof [AbsTopIII] [cf. [AbsTopIII], Remark 3.1.1], N(S) [cf. (i)] corresponds toZ(MTM) (Z/NZ) in the theory of [AbsTopIII],3, while (l )S (Z/NZ)[cf. (i)] corresponds to Z(X) (Z/NZ) in the theory of [AbsTopIII],1 [cf.


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[IUTchI], Remark 3.1.2, (iii)]. In particular, by composing the inverse of the iso-

morphism

Z

(Gk)

Z

(X) of [AbsTopIII], Corollary 1.10, (c), with the in-

verse of the isomorphism Z(MTM)

Z(G) of [AbsTopIII], Remark 3.2.1, weobtainanother cyclotomic rigidity isomorphism

(l )S (Z/NZ) N(S) (bs-Gal)

[cf. the various identifications/correspondences of notation discussed above].

(iii)(Compatibility)The cyclotomic rigidity isomorphisms(mono-),(bs-Gal)of [EtTh], [AbsTopIII] [cf. (i), (ii)]coincide.

Proof. Assertions (i), (ii) follow immediately from the results and definitions of[EtTh], [AbsTopIII] that are quoted in the statements of these assertions. Assertion(iii) follows immediately from the fact that in the situation where the FrobenioidC involved is not just some abstract category, but rather arises from familiar ob-jects of scheme theory [cf. the theory of [EtTh],1!], both isomorphisms (mono-),(bs-Gal) coincide with the conventional identification between the cyclotomes in-volved that arises from conventional scheme theory.

Proposition 1.4. (Etale Theta Functions of Standard Type) Let be asin Proposition 1.2, (i). Then there arefunctorial group-theoretic algorithms

[cf. [EtTh], Corollary 2.18, (i)]

Y(); (l )()

for constructing from the open subgroup Y() corresponding to the tem-pered covering Y [cf. [EtTh],2] and a certain subquotient (l)() of [cf. the subquotient (l )(M) of Definition 1.1, (i)], as well as a functorialgroup-theoretic algorithm

()

H1(Y

(), (l

)())

cf. theconstant multiple rigidity property of [EtTh], Corollary 2.19, (iii) for constructing from the set() ofl-multiples [i.e., wherel denotes the

group of l-th roots of unity] of the reciprocal of the (l Z 2)-orbit,lZ2 ofanl-th root of theetale theta function of standard type of [EtTh], Definition2.7. In this context, we shall write

() limJ

H1(Y()|J, (l )())

where () denotes the subset of elements of the direct limit of cohomologymodules in the display for which some [positive integer]multiple [i.e., some [posi-tive integer]power, if one writes these modules multiplicatively] belongs to ();Jranges over the finite index open subgroups of.


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Proof. The assertions of Proposition 1.4 follow immediately from the results anddefinitions of [EtTh] that are quoted in the statements of these assertions.

Remark 1.4.1. Before proceeding, let us recall from [EtTh],1,2, the theorysurrounding theetale theta functions of standard typethat appeared in Proposi-tion 1.4.

(i) WriteX

kXkCk

for the hyperbolic orbicurves of type (1, l-tors), (1, l-tors) determined by Xk [cf.

[EtTh], Proposition 2.4]. Thus, Xk has a unique zero cusp [i.e., unique cusp thatlies over precisely one cusp ofCk]. Write

Xk(k)

for the unique torsion point of order 2 whose closure in any stable model of XkoverOk intersects the same irreducible component of the special fiber of the stablemodel as the zero cusp [cf. the discussion of [IUTchI], Example 4.4, (i)].

(ii) The unique order two automorphismX ofXk over k [cf. [EtTh], Remark

2.6.1] corresponds [cf., e.g., [SemiAnbd], Theorem 6.4] to the unique order two

tpXk

-outer automorphism of tpXk

over Gk, which, by abuse of notation, we shall

also denote by X . Write

YkY

kX

k

for the tempered coverings ofXk that correspond, respectively, to the open sub-

groups tpYk

def= Y(

tpXk

) tpXk

[cf. Proposition 1.4], tpYk

def= Y(

tpXk

) def

=

Y(M(tpX

k

)) tpXk

[cf. Definition 1.1, (i); Proposition 1.2, (i)]. Since k con-

tains aprimitive4l-th root of unity, it follows from the definition of an etale thetafunction of standard type [cf. [EtTh], Definition 1.9, (ii); [EtTh], Definition 2.7]that there exists a rational point

()Y Yk(k)

that lifts. Moreover, there exists an order two automorphismY of Yk liftingX

which isuniquely determined up to l Z-conjugacy and composition with an elementGal(Y

k/Y

k) by the condition that it fixthe Gal(Y

k/Y

k)-orbit of ()Y. Here,

we think ofl Z, Gal(Yk

/Yk

) (= Z/2Z) as the subquotients appearing in thenaturalexact sequence

1Gal(Yk

/Yk

)Gal(Yk

/Xk

)l Z 1determined by the coverings Y

k Y

k X

k. Again, by abuse of notation, we

shall also denote by Y the corresponding tp

Yk

(= tpXk

tpYk

)-outer automor-

phism of tpYk

. We shall refer to the various automorphisms X , Y as inversion

automorphisms [cf. [EtTh], Proposition 1.5, (iii)]. Write

DYk


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for the decomposition group of ()Y [which is well-defined up to tp

Yk

-conjugacy]

soD is determined byYup to tpYk

(= tpXk

tpYk

)-conjugacy. We shall refer

to either of the pairs(YAut(Yk), ()Y); (YAut(

tp

Yk

)/Inn(tpYk

), D)

as apointed inversion automorphism. Again, we recall from [EtTh], Definition1.9, (ii); [EtTh], Definition 2.7, that

an etale theta function of standard type is defined precisely by the con-dition that its restriction to D be a2l-th root of unity.

Proposition 1.5. (Projective Systems of Mono-theta Environments)Inthe notation of the above discussion, let

M = {. . . MM MM . . . }be aprojective system of mono-theta environments whereMM is a modMmono-theta environment [which is isomorphic to the modM model mono-thetadetermined byX

k], and the indexM of the projective system variesmultiplica-

tivelyamong the elements ofN1 [cf. [EtTh], Corollary 2.19, (ii), (iii)]. Then:

(i) Such a projective system isuniquely determined, up to isomorphism,byX

k[cf. Remark 1.5.1 below; thediscrete rigidityproperty of [EtTh], Corollary

2.19, (ii)].

(ii) The transition morphisms of the resulting projective system of topologicalgroups{. . . X(MM) X(MM) . . . } [cf. the notation of Definition1.1, (i)] are all isomorphisms. Moreover, any isomorphism of topological groups

X(MM)

X(MM), whereM dividesM, lifts to a morphism of mono-thetaenvironmentsMM MM [cf. [EtTh], Corollary 2.18, (iv)]. Thus, to simplify thenotation, we shallidentify these topological groups via these transition morphismsand denote the resulting topological group by the notationX(M

). In particular,

we have anopen subgroupY(M)X(

M), asubquotient(l )(

M) of

X(M), and aquotient X(M

) G(M

) [cf. Definition 1.1, (i); Proposition

1.4].

(iii) The projective system of exterior cyclotomes{. . . (MM) (M

M) . . . } [cf. the notation of Definition 1.1, (i)] determines a projective

limitexterior cyclotome (M) which is equipped with a uniquely determined

cyclotomic rigidity isomorphism

(l )(M) (M)

[i.e., obtained by applying the cyclotomic rigidity isomorphisms of Definition 1.1,(ii), to the various members of the projective system M]. In particular, [cf. Propo-sition 1.4] we obtain a functorial algorithm

M env(M) H1(Y(M), (M))


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where one may think of the env as an abbreviation of the term [mono-theta]environment for constructing from M an exterior cyclotome versionenv

(M) of() [i.e., by transporting() via the above cyclotomic rigidity iso-

morphism] cf. [EtTh], Corollary 2.19, (iii). In this context, we shall write

env(M) lim

J

H1(Y(M)|J, (M))

whereenv(M) denotes the subset of elements of the direct limit of cohomol-

ogy modules in the display for which some [positive integer]multiple [i.e., some[positive integer]power, if one writes these modules multiplicatively] belongs to

env(M); Jranges over the finite index open subgroups ofX(M

).

(iv) Suppose thatM arises from atempered FrobenioidC [cf. Propositions1.2, (ii); 1.3]. Then this construction ofenv(M

) [cf. (iii)] iscompatible with

the Kummer-theoretic construction of the etale theta function i.e., by con-sidering Galois actions on roots of theFrobenioid-theoretic theta function[cf.the theory of [EtTh],5]. In particular, it is compatible with theKummer theoryof the base-field-theoretic hullCbs-fld [cf. [FrdII], Theorem 2.4; [AbsTopIII],Proposition 3.2, (ii); [AbsTopIII], Remark 3.1.1].

Proof. The assertions of Proposition 1.5 follow immediately from the results anddefinitions of [EtTh] [as well as [FrdII], [AbsTopIII]] that are quoted in the state-ments of these assertions.

Remark 1.5.1. We recall in passing that one important consequence of thediscrete rigidity property established in [EtTh], Corollary 2.19, (ii) which, ineffect, allows one to restrict ones attention to l Z-translates [i.e., as opposedto lZ-translates] of the usual theta function is the resulting compatibility ofprojective systems of mono-theta environments [as in Proposition 1.5] with thediscrete structureinherent in the various isomorphs of the monoid N that appearin the structure of the tempered Frobenioidsthat arise in the theory [cf., [EtTh],Remark 2.19.4; [EtTh], Remark 5.10.4, (i), (ii)].

Remark 1.5.2. Note that, in the notation of Proposition 1.5, (iii), by consider-ing tautological Kummer classes of elements of (M

), one obtains a natural

Galois-equivariant injection

(M) Q/Z lim

J

H1(Y(M)|J, (M))

whose image may is equal to the torsion subgroup of the codomain of the injection.Indeed, it follows immediately from the fact that (M

) is torsion-free that the

torsion subgroup of the codomain of the displayed injection may be identified withthe torsion subgroup of

limJ

H1(JG, (M))

whereJranges over the finite index open subgroups of X(M); we writeJGfor

the image ofJ in G(M). The desired conclusion thus follows immediately from


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the well-known Kummer theory of MLFs, i.e., the fact that the Kummer map((M

)) Q/Z)J H1(JG, (M)) [where the superscript J denotes the

submodule ofJ-invariants] is injective with image equal to the torsion subgroup of

the codomain.

Before proceeding, we review a certain portion of the theory of [AbsTopII] thatis relevant to the content of the present1.

Proposition 1.6. (Cores and Cuspidalizations) Let be as in Proposition1.2, (i). Write for the [group-theoretic! cf., e.g., [AbsAnab], Lemma1.3.8] subgroup corresponding to tpX

k

. Then:

(i)(Cores) There exists a functorial group-theoretic algorithm [cf. [Ab-sTopII], Corollary 3.3, (i); [AbsTopII], Remark 3.3.3]

() C() /

for constructing from a topological group C() equipped with an augmentation[i.e., a surjection] C() / whose kernel we denote by C() thatcontains as an open subgroup in a fashion that is compatible with the respec-tive surjections to / and which satisfies the property that when = tpX

k

, the

inclusionC() may be naturally identified with the inclusiontpXk

tpCk .

(ii) (Elliptic Cuspidalizations) Let N be a positive integer. Then thereexists afunctorial group-theoretic algorithm [cf. [AbsTopII], Corollary 3.3,(iii); [AbsTopII], Remark 3.3.3]

UN()

for constructing from a topological group UN() equipped with a surjectionUN() [so the augmentation /determines, by composition, an aug-mentationUN() /] such that when =

tpXk

, the surjectionUN()

may be naturally identified with a certain surjection i.e., elliptic cuspidaliza-tion that arises from a certain open immersion determined by the N-torsionpoints of a once-punctured elliptic curve that forms a double covering of Ck [cf.[AbsTopII], Corollary 3.3, (iii)].

Proof. The assertions of Proposition 1.6 follow immediately from the results of[AbsTopII] that are quoted in the statements of these assertions [cf. also Remark1.6.1 below].

Remark 1.6.1. We recall in passing that the construction of Proposition 1.6,(i), amounts, in effect, to the computation of various centralizers of the image ofvarious open subgroups of G in the outer automorphism groups of various opensubgroups of. In a similar vein, the construction of Proposition 1.6, (ii), amountsto the computation of various outer isomorphisms between various subquotients of


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that are compatible with the outer actions of various open subgroups ofG. Moregenerally, although in Proposition 1.6, we restricted our attention to the construc-tion ofcoresand el liptic cuspidalizations, an analogous result may be obtained for

more general functorial group-theoretic algorithms involvingchains of elementaryoperations, as discussed in [AbsTopI],3 e.g., for Belyi cuspidalizations, asdiscussed in [AbsTopII], Corollary 3.7.

Next, we proceed to discuss the multiradial interpretation of the theory of[EtTh] that is of interest in the context of the present series of papers. We beginby examining various examples of the sort of situation that gives rise to such aninterpretation.

Ri. . . . . .

Ri C Ri

. . . . . .

Ri

Fig. 1.1: Radial functors valued in a single coric category

Example 1.7. Radial and Coric Data I: Generalities.

(i) In the following discussion, we would like to consider a certain type ofmathematical data, which we shall refer to as radial data. This notion of a typeof mathematical data may be formalized cf. [IUTchIV],3, for more details.From the point of view of the present discussion, one may think of a type ofmathematical data as the input or output data of a functorial algorithm [cf. thediscussion of [IUTchI], Remark 3.2.1]. At a more concrete level, we shall assumethat this type of mathematical data gives rise to a category

R

i.e., each of whoseobjectsis a specific collection of radial data, and each of whosemorphisms is an isomorphism. In the following discussion, we shall also consideranother type of mathematical data, which we shall refer to as coric data. Write

C

for the category obtained by considering specific collections of coric data and iso-morphisms of collections of coric data. In addition, we shall assume that we aregiven afunctorial algorithm which we shall refer to asradial whoseinput dataconsists of a collection of radial data, and whose output dataconsists of a collectionof coric data. Thus, this functorial algorithm gives rise to a functor :R C. In


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the following discussion, we shall assume that this functor is essentially surjective.We shall refer to the categoryR and the functor as radialand to the categoryC as coric. Finally, if I is some nonempty index set, then we shall often considercollections {i :Ri C}iIof copies of i andRi, such that the various copies of i have the same codomainC cf. Fig. 1.1. Thus, one may think of eachRi as the category of radial dataequipped with a labeliI, and isomorphisms of such data.

(ii) We shall refer to a triple (R, C, :R C) [or to the triple consisting ofthe corresponding types of mathematical objects and functorial algorithm] ofthe sort discussed in (i) as a radial environment. If is full, then we shall referto the radial environment under consideration as multiradial. We shall refer to a

radial environment which is not multiradial as uniradial. Suppose that the radialenvironment (R, C, :R C) under consideration is uniradial. Then an object ofR may, in general, losea certain portion of its rigidity i.e., may be subject to acertain additional indeterminacy when it is mapped toC. Put another way,in general, an object ofC is impartedwith a certain additional rigidity i.e.,losesa certain portion of itsindeterminacy when one fixes aliftingof the objecttoR. Thus, in summary, the condition that (R, C, :R C) be multiradial maybe thought of as a condition to the effect that the application of the radial algorithmdoes not result in any loss of rigidity.

(iii) In passing, we pause to observe that one way to think of the significanceof the multiradialityof a radial environment (R, C, :R C) is as follows: Write

R CR

for the category whose objectsare triples (R1, R2, ) consisting of a pair of objects

R1, R2 ofR and an isomorphism : (R1) (R2) between the images ofR1,R2 via , and whose morphismsare the morphisms [in the evident sense] betweensuch triples [cf. the discussion of the categorical fiber productgiven in [FrdI],0].Write sw :RCR RCRfor the functor (R1, R2, )(R2, R1, 1) obtainedbyswitchingthe two factors of

R. Then

one formal consequence of the multiradiality of a radial environment(R, C, :R C) is the property that the switching functor sw :RCR RCRpreserves theisomorphism classof objects ofRCR.

Indeed, one verifies immediately that this multiradiality is, in fact, equivalent tothe condition that every object (R1, R2, ) ofR CRbe isomorphic to the object(R1, R1, id : (R1)

(R1)) [which is manifestly left unchanged by the switchingfunctor].

(iv) Next, suppose that we are given another radial environment (R, C, :R C). We shall refer to the type of mathematical object/functorial algo-rithm that gives rise toR (respectively,C; ) as daggered radial data (respec-tively, daggered coric data; the daggered radial functorial algorithm). Also, let us


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suppose that we are given a 1-commutative diagram

R R R

C C C

where R and C arise from functorial algorithms. If (R, C, :R C)is multiradial (respectively, uniradial), then we shall refer to R asmultiradiallydefined(respectively,uniradially defined), or [when there is no fear of confusionbetween and R] as multiradial (respectively, uniradial). If R admits a 1-factorization R for some R:C R that arises from a functorial algorithm,then we shall say that R is corically defined, or [when there is no fear of

confusion] coric. Thus, by considering the case whereR =C, = idR, one maythink of the notion of a corically definedR as a sort ofspecial caseof the notionof a multiradialR.

(v) Suppose that we are in the situation of (iv), and that Rismultiradiallydefined. Then one way to think of the significance of the multiradiality of R isas follows:

The multiradiality of Rrenders it possible to consider the simultaneousexecution of the functorial algorithm corresponding to R relative tovarious collections of radial input data indexed by the set I [cf. Fig.1.1] in a fashion that is compatible with the identification of the coricportions [i.e., corresponding to ] of these collections of radial input data

cf. Remark 1.9.1 below for more on this point of view. That is to say, at a moretechnical level, if one implements this identification of the various coric portions bymeans of various gluing isomorphisms inC, then the multiradiality of R impliesthat one may liftthese gluing isomorphisms inC to gluing isomorphisms inR; onemay then apply R to these gluing isomorphisms inR to obtain gluing isomor-phisms of the output data of R. Put another way, if one assumes instead thatR is uniradial, then the output data of R depends, a priori, on the additional

rigidity[cf. (ii)] of objects ofRrelative to these images inC; thus, if one attemptsto identify these images inCvia arbitrarygluing isomorphisms inC, then one doesnot have any way to compute the effect of such gluing isomorphisms on the outputdata of R.

Remark 1.7.1. One way to understand the significance of the fullness conditionin the definition of amultiradial environmentis as a condition that allows one to ex-ecute a sort ofparallel transportoperation betweenfibersof the radial functor :R C [cf. the notation of Example 1.7, (iv)] i.e., by lifting isomorphisms in

Cto isomorphisms in

R [cf. the discussion of Example 1.7, (v)]. Here, it is perhaps

of interest to make the tautological observation that, up to an indeterminacy arisingfrom the extent that fails to be faithful, such liftings are unique. That is to say,whereas a uniradial environment may be thought of as a sort of abstraction ofthe geometric notion of a fibration that is not equipped with a connection,


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a multiradial environment may be thought of as a sort of abstractionof the geometric notion of a fibration equipped with a connection i.e., that allows one to execute parallel transport operations between the

fibers.

Relative to this point of view, one may think of the coric data as the portion ofthe radial data of a multiradial environment that is horizontalwith respect to theconnection structure. We refer to Remarks 1.9.1, 1.9.2 below for more on thesignificance of multiradiality.

i

. . .

. . .

i G i

. . . . . .

i

Fig. 1.2: Different arithmetic holomorphic structures on a single coric G

Example 1.8. Radial and Coric Data II: Concrete Examples.

(i) From the point of view of the theory to be developed in the remainder of thepresent1, perhaps the most basic example of a radial environment is the following.We define a collection ofradial data

(, G , )

to consist of a topological group isomorphic to tpXk

, a topological group G iso-

morphic to Gk, and the full poly-isomorphism [cf. [IUTchI],0] of topologicalgroups : /

G, where we write for the [group-theoretic! cf.,e.g., [AbsAnab], Lemma 1.3.8] subgroup corresponding to tpX

k

. An isomorphism

of collections of radial data (, G , ) (, G, ) is defined to be a pair of

isomorphisms of topological groups , G G [which are necessarily com-

patible with , !]. A collection ofcoric datais defined to be a topological groupisomorphic to Gk; an isomorphism of collections of coric data is defined to be anisomorphism of topological groups. Theradial algorithm is the algorithm givenby the assignment

(, G , )G whose associated radial functor isfullandessentially surjective, hence determinesa multiradial environment. Note that this example may be thought of as a sort offormalization in the present context of the situation depicted in [IUTchI], Fig. 3.2,


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at vVbad cf. Fig. 1.2. Here, we recall that the topological group G [whichis isomorphic to Gk] that appears in the center of Fig. 1.2 is regarded as beingknownonly up to isomorphism, and that the various isomorphs of X

k

that appear

in thespokesof Fig. 1.2 may be regarded as various arithmetic holomorphicstructures on G [cf. [IUTchI], Remark 3.8.1, (iii)].

(ii) Recall the functorial group-theoretic algorithm

( MTM()) (TM)

of [AbsTopIII],3 [cf., especially, the functors An, An of [AbsTopIII], Definition3.1, (vi); [AbsTopIII], Corollary 3.6, (ii); [IUTchI], Remark 3.1.2] that assigns to

a topological group isomorphic to tpXk

an MLF-Galois TM-pair, which we shall

denote MTM(), and which is isomorphic tothe model MLF-Galois TM-pair

determined by the natural action of tpXk

on the topological monoidOk

. In fact,

[the union with{0}of] the underlying topological monoidMTM() is also equippedwith a natural ring structure [cf. [AbsTopIII], Proposition 3.2, (iii)]. On the otherhand, if one is willing to sacrifice this ring structure, then there exists a functorialgroup-theoretic algorithm

G (G O(G)) ()

[cf. [AbsTopIII], Proposition 5.8, (i)] that assigns to a topological group G isomor-phic toGk an MLF-Galois TM-pair, which we shall denoteG

O(G), and which

is isomorphic to the MLF-Galois TM-pair determined by the natural action ofGkon the topological monoidO

k. Moreover, by [AbsTopIII], Proposition 3.2, (iv)

[cf. also Remark 1.11.1, (i), (a), below], there is a [uniquely determined] functorialtautological isomorphism of MLF-GaloisTM-pairs

( MTM()) (/ O(/))| (TM)

where is as in (i), and the notation | denotes the restriction of theaction of / to an action of . Then another important example of a radialenvironment is the following. We define a collection ofradial data

( MTM(), G O(G), )

to consist of the output data of the algorithm (TM) associated to a topologicalgroup isomorphic to tpX

k

, the output data of the algorithm() associated to atopological group G isomorphic to Gk, and the poly-isomorphism [cf. [IUTchI],0]of MLF-Galois TM-pairs

: ( MTM()) (G O(G))|

determined [in light of [AbsTopIII], Proposition 3.2, (iv)] by the composite of the

natural surjection / with the full poly-isomorphism of topological groups/

G [where is as in (i)]. An isomorphism of collections of radial data( MTM(), G O(G), ) ( MTM(), G O(G), ) is de-fined to be a pair of isomorphisms of MLF-Galois TM-pairs ( MTM())

(


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MTM()), (G O(G)) (G O(G)) [which are necessarily compatible

with , !]. A collection ofcoric data is defined to be the output data of the

algorithm () for some topological group isomorphic to Gk; an isomorphism ofcollections of coric data is defined to be the isomorphism between collections ofoutput data of () associated to an isomorphism of topological groups. Thera-dial algorithm is the algorithm given by the assignment

( MTM(), G O(G), )(G O(G))

whose associated radial functor isfullandessentially surjective, hence determinesa multiradial environment

(iii) Let

Z

be aclosed subgroup[cf. Remark 1.11.1, (i), (ii), below, for more on the significanceof ]. Then by considering the subgroups of invertible elements of the varioustopological monoids that appeared in (ii), one obtains functorial group-theoreticalgorithms

( MTM()); G (G O(G)) ()

defined, respectively, on topological groups isomorphic to tpXk

andG isomorphic

to Gk. Here, we note that we may think of as acting on the output data of the

second algorithm of () by means of the trivial action on G and the natural actionofZ onO(G). Then one obtains another example of a radial environment asfollows. We define a collection of radial data

( MTM(), G O(G), )

to consist of the output data of the first algorithm of() associated to a topolog-ical group isomorphic to tpX

k

, the output data of the second algorithm of ()associated to atopological groupG isomorphic toGk, and thepoly-isomorphism[cf.[IUTchI],0] of topological modules equipped with topological group actions

: ( MTM

()) (G O(G))|determined by the -orbit of the poly-isomorphism | induced by the poly-isomorphismof (ii). Anisomorphism of collections of radial data( M

TM

(),

G O(G), ) ( MTM(), G O(G), ) is defined to consist ofthe isomorphism of topological modules equipped with topological group actions( MTM())

( MTM()) induced by an isomorphism of topologicalgroups

, together with a -multipleof the isomorphism of topological mod-ules equipped with topological group actions (G O(G)) (G O(G))induced by an isomorphism of topological groups G

G [so one verifies immedi-

ately that these isomorphisms are compatible with , in the evident sense]. Acollection ofcoric data is defined to be the output data of the second algorithm of() for some topological group isomorphic to Gk; an isomorphism of collectionsof coric data is defined to be a -multipleof the isomorphism between collections


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of output data of () associated to an isomorphism of topological groups. Theradial algorithm is the algorithm given by the assignment

( MTM(), G O

(G), )(G O

(G))

whose associated radial functor isfullandessentially surjective, hence determinesa multiradial environment.

(iv) By considering thesubgroups of torsion elementsof the various topologicalmonoids that appeared in (ii), one obtains functorial group-theoretic algorithms

( MTM()); G (G O(G)) ()


andG isomorphic

to Gk i.e., a cyclotomic versionof the algorithms of () [cf. (iii)]. Moreover,by forming the quotients MTM()/MTM(),O()/O(), one obtainsfunctorialgroup-theoretic algorithms

( MTM()); G (G O(G)) ()


andG isomorphic

to Gk i.e., a co-cyclotomic version of the algorithms of () [cf. (iii)]. Nowone verifies easily that

by replacing the symbol in (iii) by the symbol or, alternatively,by the symbol ,

one obtains, respectively,cyclotomicand co-cyclotomicversions of the exampletreated in (iii). In the case of , let us write

Ism(G)

for thecompact topological group ofG-isometriesofO(G), i.e.,G-equivariantautomorphisms of the topological module

O(G) that, for each open subgroup

H G, preserve the lattice inO(G)H determined by the imageofO(G)H[i.e., where the superscript H denotes the submodule ofH-invariants]. Let

Ism()

be a closed subgroup, i.e., a collection of closed subgroups of each Ism(G) that is

preserved by arbitrary isomorphisms of topological groups G1 G2. Then one

verifies easily that, in the co-cyclotomic version discussed above of the exampletreated in (iii),

one may replace the in (iii) by such a

.

Finally, we observe that one example of such a which we shall denote bymeans of the notation

Ism


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is the case where one takes to be the entire group Ism(); anotherexample of such a is the image Im(

Z) of the natural homomorphism

Z

Zp Ism.(v) Another example of a radial environment may be obtained as follows. We

define a collection ofradial data

( MTM(), G O(G), ,)

to consist of theoutput data of the first algorithm of() associated to atopologicalgroup isomorphic to tpX

k

, the output data of the second algorithm of ()associated to a topological group G isomorphic to Gk, and the poly-morphism [cf.[IUTchI],0] of topological modules equipped with topological group actions

,: ( M

TM()) (G O(G))|

determined by the full poly-isomorphism / G [cf. (i)] and the trivial ho-

momorphism MTM() O(G) i.e., the composite of the natural homomor-phisms MTM() MTM() O(G) O(G) [where the arises fromthe poly-isomorphism of (iii)]. An isomorphism of collections of radial data

( MTM(), G O(G), ,) ( MTM(), G O(G), ,)is defined to consist of the isomorphism of topological modules equipped with topo-logical group actions ( MTM())

( MTM()) induced by an isomor-

phism of topological groups

, together with a

-multiple of the iso-morphism of topological modules equipped with topological groups actions (G

O(G)) (G O(G)) induced by an isomorphism of topological groupsG G [so one verifies immediately that these isomorphisms are compatible with

,, , in the evident sense]. A collection ofcoric data is defined to be the

output data of the second algorithm of() for some topological group isomorphicto Gk; an isomorphism of collections of coric datais defined to be a

-multipleof the isomorphism between collections of output data of () associated to anisomorphism of topological groups. [That is to say, the definition of the coric datais the same as in the co-cyclotomic version discussed in (iv).] Theradial algo-rithm is the algorithm given by the assignment

( MTM(), G O(G), ,)(G O(G))

whose associated radial functor isfullandessentially surjective, hence determinesa multiradial environment.

(vi) By replacing the notation MTM() in the discussion of (v) by the no-tation (M

()) Q/Z [cf. Propositions 1.2, (i); 1.5, (i), (iii)], one verifies

immediately that one obtains an exterior-cyclotomic version of the multiradialenvironmentconstructed in (v).

(vii) In the discussion to follow, we shall also consider the functorial group-theoretic algorithms

( MgpTM()); G (G Ogp(G)) (gp)


36/

inter-universal teichmuller theory ii

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