the etale theta function and its frobenioid-theoretic manifestations

8/13/2019 The Etale Theta Function and Its Frobenioid-Theoretic Manifestations

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THE ETALE THETA FUNCTION AND

ITS FROBENIOID-THEORETIC MANIFESTATIONS

Shinichi Mochizuki

December 2008

We develop the theory of the tempered anabelianand Frobenioid-theore-

ticaspects of the etale theta function, i.e., the Kummer class of the classical formal

algebraic theta function associated to a Tate curve over a nonarchimedean mixed-

characteristic local field. In particular, we consider a certain natural environment

for the study of the etale theta function, which we refer to as a mono-theta environ-

ment essentially a Kummer-theoreticversion of the classical theta trivialization

and show that this mono-theta environment satisfies certain remarkable rigidity

properties involving cyclotomes, discreteness, and constant multiples, all in a fash-

ion that is compatiblewith the topologyof the tempered fundamental group and the

extension structureof the associated tempered Frobenioid.

Contents:

0. Notations and Conventions1. The Tempered Anabelian Rigidity of the Etale Theta Function2. The Theory of Theta Environments3. Tempered Frobenioids4. General Bi-Kummer Theory5. The Etale Theta Function via Tempered Frobenioids

Introduction

The fundamental goalof the present paper is to study the tempered anabelian[cf. [Andre], [Mzk14]] and Frobenioid-theoretic [cf. [Mzk17], [Mzk18]] aspects ofthe theta function of a Tate curveover a nonarchimedean mixed-characteristic lo-cal field. The motivation for this approach to the theta function arises from thelong-term goal of overcoming various obstacles that occur when one attempts toapply the Hodge-Arakelov theory of elliptic curves [cf. [Mzk4], [Mzk5]; [Mzk6],[Mzk7], [Mzk8], [Mzk9], [Mzk10]] to the diophantine geometry of elliptic curvesover number fields. That is to say, the theory of the present paper is motivated

2000 Mathematical Subject Classification: Primary 14H42; Secondary 14H30.

Typeset by AMS-TEX

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2 SHINICHI MOCHIZUKI

by the expectation that these obstacles may be overcome by translating the [es-sentially] scheme-theoreticformulation of Hodge-Arakelov theory into the languageof the geometry of categories[e.g., the temperoids of [Mzk14], and the Frobe-nioids of [Mzk17], [Mzk18]]. In certain respects, this situation is reminiscent ofthe well-known classical solution to the problem of relating the dimension of the

first cohomology group of the structure sheaf of a smooth proper variety in positivecharacteristic to the dimension of its Picard variety a problem whose solutionremained elusive until the foundations of the algebraic geometry ofvarietieswerereformulated in the language ofschemes[i.e., one allows for the possibility of nilpo-tent sections of the structure sheaf].

Since Hodge-Arakelov theory centers around the theory of thetheta functionofan elliptic curve with bad multiplicative reduction [i.e., aTate curve], it is naturalto attempt to begin such a translation by concentrating on such theta functionson Tate curves, as is done in the present paper. Indeed, Hodge-Arakelov theorymay be thought of as a sort ofcanonical analytic continuation of the theory of

theta functions on Tate curves to elliptic curves over number fields. Here, we recallthat although classically, the arithmetic theory of theta functions on Tate curves isdeveloped in the language of formal schemes in, for instance, [Mumf] [cf. [Mumf],pp. 306-307], this theory only addresses the slope zero portion of the theory i.e., the portion of the theory that involves the quotient of the fundamental groupof the generic fiber of the Tate curve that extendsto an etale covering in positivecharacteristic. From this point of view, the relation of the theory of1,2 of thepresent paper to the theory of [Mumf] may be regarded as roughly analogous tothe relation of the theory ofp-adic uniformizations of hyperbolic curves developedin [Mzk1] to Mumfords theory of Schottky uniformations of hyperbolic curves [cf.,

e.g., [Mzk1], Introduction,0.1; cf. also Remark 5.10.2 of the present paper].Frequently in classical scheme-theoretic constructions, such as those that ap-

pear in the scheme-theoreticformulation of Hodge-Arakelov theory, there is a ten-dency to make various arbitrary choicesin situations where, a priori, some sort ofindeterminacyexists, without providing any sort ofintrinsic justificationfor thesechoices. Typical examples of such choices involve the choiceof a particular rationalfunction or section of a line bundle among various possibilities related by a constantmultiple, or thechoiceof a natural identification between variouscyclotomes[i.e.,isomorphic copies of the module ofN-th roots of unity, for N 1 an integer] ap-pearing in a situation [cf., e.g., [Mzk13], Theorem 4.3, for an example of a crucial

rigidityresult in anabelian geometry concerning this sort of choice].One important theme of the present paper is the study of phenomena that

involve some sort ofextraordinary rigidity[cf. Grothendiecks famous use of thisexpression in describing his anabelian philosophy]. This sort of extraordinary rigid-ity may be thought of as a manifestation of the very strong canonicalitypresentin the theory of theta functions on Tate curves that allows one, in Hodge-Arakelovtheory, to effect a canonical analytic continuation of this theory on Tate curvesto elliptic curves over global number fields. The various examples of extraordinaryrigidity that appear in the theory of the etale theta function may be thought ofas examples ofintrinsic, category-theoretic justificationsfor thearbitrary choices

that appear in classical scheme-theoretic discussions. Such category-theoretic jus-


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THE ETALE THETA FUNCTION 3

tifications depend heavily on the proper category-theoretic formulation of vari-ous scheme-theoretic venues. In the theory of the present paper, onecentralsuchcategory-theoretic formulation is a mathematical structure that we shall refer to asa mono-theta environment [cf. Definition 2.13, (ii)]. Roughly speaking:

The mono-theta environment is essentially a Kummer-theoretic versioni.e., a Galois-theoretic version obtained by extracting various N-th roots[forN 1 an integer] of the theta trivializationthat appears in classicalformal scheme-theoretic discussions of the theta function on a Tate curve.

A mono-theta environment may be thought of as a sort of common core for, orbridgebetween, the [tempered] etale- and Frobenioid-theoretic approaches to theetale theta function [cf. Remarks 2.18.2, 5.10.1, 5.10.2, 5.10.3].

We are now ready to discuss the main resultsof the present paper. First, we

remark that

the reader who is only interested in thedefinition and basic tempered an-abelian propertiesof theetale theta functionmay restrict his/her attentionto the theory of1, the main result of which is a tempered anabelian con-struction of the etale theta function [cf. Theorems 1.6, 1.10].

On the other hand, the main results of the bulk of the paper, which concernmoresubtle rigidity propertiesof the etale theta function involving associatedmono-thetaenvironmentsand tempered Frobenioids, may be summarized as follows:

A mono-theta environment is a category/group-theoretic invariantof thetempered etale fundamental group[cf. Corollary 2.18] associated to [certaincoverings of] a punctured elliptic curve [satisfying certain properties] overa nonarchimedean mixed-characteristic local field, on the one hand, and ofa certain tempered Frobenioid [cf. Theorem 5.10, (iii)] associated to sucha curve, on the other. Moreover, a mono-theta environment satisfies thefollowingrigidityproperties:

(a) cyclotomic rigidity [cf. Corollary 2.19, (i); Remark 2.19.4];

(b) discrete rigidity [cf. Corollary 2.19, (ii); Remarks 2.16.1, 2.19.4];

(c) constant multiple rigidity [cf. Corollary 2.19, (iii); Remarks5.12.3, 5.12.5]

all in a fashion that is compatiblewith the topology of the temperedfundamental group as well as with the extension structure of the tem-pered Frobenioid [cf. Corollary 5.12 and the discussion of the followingremarks].

In particular, the phenomenon ofcyclotomic rigiditygives a category-theoretic

explanation for the special role played by the first power [i.e., as opposed to the


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M-th power, forM >1 an integer] of [the l-th root, when one works withl-torsionpoints, for l 1 an odd integer, of] the theta function [cf. Remarks 2.19.2, 2.19.3,5.10.3, 5.12.5] a phenomenon which may also be seen in scheme-theoreticHodge-Arakelov theory [a theory that also effectively involves [sections of] the firstpowerof some ample line bundle, and which does not generalize in any evident way

to arbitrary powers of this ample line bundle].

One further interesting aspect ofcyclotomic rigidityin the theory of the presentpaper is that it is obtained essentially as the result of a certain computation in-volving commutators[cf. Remark 2.19.2]. Put another way, one may think of thiscyclotomic rigidity as a sort of consequence of thenonabelian structureof [what es-sentially amounts, from a more classical point of view, to] the theta-group. Thisapplication of the structure of the theta-group differs substantially from the wayin which the structure of the theta-group is used in more classical treatments ofthe theory of theta functions namely, to show that certain representations, suchas those arising from theta functions, are irreducible. One way to understand thisdifference is as follows. Whereas classical treatments that center around such ir-reducibility results only treat certainl-dimensional subspacesof thel2-dimensionalspace of set-theoretic functions on the l-torsion points of an elliptic curve, Hodge-Arakelov theoryis concerned with understanding [modulo the l-dimensional sub-space which has already been well understood in the classical theory!] theentirel2-dimensional function spacethat appears [cf. [Mzk4], 1.1]. Put another way, thel-dimensional subspace which forms the principal topic of the classical theory maybe thought of as corresponding to the space of holomorphic functions [cf. [Mzk4],1.4.2]; by contrast, Hodge-Arakelov theory cf., e.g., the arithmetic Kodaira-Spencer morphism of [Mzk4],

1.4 is concerned with understanding deforma-

tions of the holomorphic structure, from an arithmetic point of view. Moreover,from the point of view of considering such deformations of holomorphic structure,it is convenient, and, indeed, more efficient, to work modulo variations containedwithin the subspace corresponding to the holomorphic functions which, at anyrate, may be treated, as a consequence of the classical theory of irreducible repre-sentations of the theta-group, as a single irreducible unit! This is precisely thepoint of view of theLagrangian approachof [Mzk5], 3, an approach which allowsone to work modulo the l-dimensional subspace of the classical theory, by apply-ing various isogeniesthat allow one to replacethe entire l2-dimensional functionspace associated to the original elliptic curve by an l-dimensional function space

that is suited to studying arithmetic deformations of holomorphic structure inthe style of Hodge-Arakelov theory. In the present paper, these isogenies of the La-grangian approach to Hodge-Arakelov theory correspond to the various coveringsthat appear in the discussion at the beginning of2; the resulting l-dimensionalfunction space then corresponds, in the theory of the present paper, to the spaceof functions on the labelsthat appear in Corollary 2.9.

Here, it is important to note that although the above three rigidity proper-ties may be stated and understood to a certain extent without reference to theFrobenioid-theoretic portion of the theory [cf. 2], certain aspects of the interde-pendenceof these rigidity properties, as well as the meaning of establishing these

rigidity properties under the condition of compatibility with the topology of the


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tempered fundamental groupas well as with theextension structure of the temperedFrobenioid, may only be understood in the context of the theory of Frobenioids[cf.Corollary 5.12 and the discussion of the following remarks]. Indeed, one importanttheme of the theory of the present paper which may, roughly, be summarized asthe idea that sometimes

less(respectively,more) data yieldsmore(respectively,less) information

[cf. Remark 5.12.9] is precisely the study of the ratherintricateway in whichthese various rigidity/compatibility properties are related to one another. Typicallyspeaking, the reason for this [at first glance somewhat paradoxical] phenomenon isthat more data means more complicatedsystemsmade up of variouscomponents,hence obligates one to keep track of the various indeterminacies that arise in re-lating [i.e., without resorting to the use ofarbitrary choices] these componentsto one another. If these indeterminacies are sufficiently severe, then they may have

the effect of obliterating certain structures that one is interested in. By contrast,if certain portions of such a system are redundant, i.e., in fact uniquely and rigidly i.e., canonically determinedby more fundamental portions of the system,then one need not contend with the indeterminacies that arise from relating theredundant components; this yields a greater chance that the structures of interestare notobliterated by the intrinsic indeterminacies of the system.

One way to appreciate the tension that exists between the various rigidityproperties satisfied by the mono-theta environment is by comparing the theory ofthe mono-theta environment to that of the bi-theta environment [cf. Definition2.13, (iii)]. The bi-theta environment is essentially a Kummer-theoretic versionof

thepair of sections corresponding to the numeratorand denominatorof thetheta function of a certain ample line bundle on a Tate curve. One of these twosections is the theta trivializationthat appears in the mono-theta environment; theother of these two sections is the algebraic sectionthat arises tautologically fromthe original definition of the ample line bundle.

The bi-theta environment satisfies cyclotomic rigidity and constant multiplerigidity properties for somewhat more evident reasons than the mono-theta envi-ronment. In the case of constant multiple rigidity, this arises partly from the factthat the bi-theta environment involves working, in essence, with a ratio [i.e., inthe form of a pair] of sections, hence is immuneto the operation of multiplying

both sections by a constant [cf. Remarks 5.10.4; 5.12.7, (ii)]. On the other hand,the bi-theta environment fails to satisfy the discrete rigidityproperty satisfied bythe mono-theta environment [cf. Corollary 2.16; Remark 2.16.1]. By contrast, themono-theta environment satisfies all three rigidity properties, despite the fact thatcyclotomic rigidity[cf. the proof of Corollary 2.19, (i); Remark 2.19.2] and constantmultiple rigidity [cf. Remarks 5.12.3, 5.12.5] are somewhat moresubtle.

Here, it is interesting to note that the issue of discrete rigidityfor the mono-

theta and bi-theta environments revolves, in essence, around the fact thatZ/Z = 0,i.e., the gap betweenZ and its profinite completionZ cf. Remark 2.16.1. Onthe other hand, the subtlety ofconstant multiple rigidity for mono-theta environ-

ments revolves, in essence, around the nontriviality of the extension structure of


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a Frobenoid [i.e., as an extension by line bundles of the base category cf.Remarks 5.10.2, 5.12.3, 5.12.5, 5.12.7]. Put another way:

The mono-theta environment may be thought of as a sort oftranslationapparatus that serves to translate the global arithmetic gap between

Z andZ [cf. Remark 2.16.2 for more on the relation of this portionof the theory to global arithmetic bases] into the nontriviality of thelocal geometric extension constituted by the extension structure ofthe tempered Frobenioidunder consideration.

This sort of relationship between the global arithmetic gap between Z andZ andthe theory of theta functions is reminiscent of the point of view that theta functionsare related to splittings of the natural surjection Z Z/NZ, a point of view thatarises in Hodge-Arakelov theory [cf. [Mzk4],1.3.3].

The contents of the present paper are organized as follows. In 1, we discuss thepurely tempered etale-theoretic anabelianaspects of the theta function and show,in particular, that theetale theta function i.e., theKummer classof the usualformal algebraic theta function is preserved by isomorphisms of the temperedfundamental group [cf. Theorems 1.6; 1.10]. In 2, after studying various coveringsand quotient coverings of a punctured elliptic curve, we introduce the notions ofa mono-theta environment and a bi-theta environment [cf. Definition 2.13] andstudy the group-theoretic constructibilityand rigidityproperties of these notions[cf. Corollaries 2.18, 2.19]. In3, we define the tempered Frobenioids in whichwe shall develop the Frobenioid-theoretic approach to the etale theta function; inparticular, we show that these tempered Frobenioids satisfy various nice properties

which allow one to apply the extensive theory of [Mzk17], [Mzk18] [cf. Theorem 3.7;Corollary 3.8]. In 4, we developbi-Kummer theory i.e., a sort of generalizationof the Kummer class associated to a rational function to the bi-Kummer dataassociated to a pair of sections [corresponding to thenumeratorand denominatorofa rational function] of a line bundle in acategory-theoretic fashion [cf. Theorem4.4] for fairly general tempered Frobenioids. Finally, in 5, we specialize the theoryof3, 4 to the case of theetale theta function, as discussed in 1. In particular, weobserve that a mono-theta environmentmay also be regarded as a mathematicalstructure naturally associated to a certain tempered Frobenioid [cf. Theorem 5.10,(iii)]. Also, we discuss certain aspects of the constant multiple rigidity[as well as, to

a lesser extent, of thecyclotomicand discrete rigidity] of a mono-theta environmentthat may only be understood in the context of the Frobenioid-theoreticapproachto the etale theta function [cf. Corollary 5.12 and the discussion of the followingremarks].

Acknowledgements:

I would like to thank Akio Tamagawa, Makoto Matsumoto, Minhyong Kim,Seidai Yasuda,Kazuhiro Fujiwara, andFumiharu Kato for many helpful commentsconcerning the material presented in this paper.


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Section 0: Notations and Conventions

In addition to theNotations and Conventionsof [Mzk17], 0, we shall employthe following Notations and Conventions in the present paper:

Monoids:

We shall denote by N1 themultiplicative monoidof [rational] integers 1 [cf.[Mzk17],0].

Let Q be a commutative monoid [with unity]; P Q a submonoid. If Qis integral [so Q embeds into its groupification Qgp; we have a natural inclusionPgp Qgp], then we shall refer to the submonoid

P

gp

Q ( Qgp

)

ofQ as the group-saturationofP in Q; ifP is equal to its group-saturation in Q,then we shall say that P is group-saturatedin Q. IfQ is torsion-free[so Qembedsinto itsperfectionQpf; we have a natural inclusion Ppf Qpf], then we shall referto the submonoid

Ppf

Q ( Qpf)ofQ as theperf-saturationofP in Q; ifPis equal to its perf-saturation in Q, thenwe shall say thatP isperf-saturatedin Q.

Topological Groups:

Let be a topological group. Then let us write

Btemp()

for thecategorywhoseobjectsarecountable[i.e., of cardinality the cardinality ofthe set of natural numbers], discretesets equipped with a continuous -action andwhosemorphismsare morphisms of -sets [cf. [Mzk14], 3]. If may be written asan inverse limit of an inverse system of surjections of countable discrete topological

groups, then we shall say that is tempered [cf. [Mzk14], Definition 3.1, (i)].

We shall refer to a normal open subgroup H such that the quotient group/H is free as co-free. We shall refer to a co-free subgroup H as minimal ifevery co-free subgroup of contains H. Thus, a minimal co-free subgroup of isnecessarily uniqueand characteristic.

Categories:

We shall refer to an isomorphic copy of some object as an isomorph of the

object.


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LetC be a category; A Ob(C). Then we shall write

CA

for the category whose objectsare morphisms B

A of

C and whose morphisms

[from an object B1 Ato an object B2 A] are A-morphismsB1 B2 inC [cf.[Mzk17],0] and

C[A] Cfor thefull subcategoryofC determined by the objects ofC that admit a morphismto A. Given two arrows fi : Ai Bi (where i = 1, 2) inC , we shall refer to acommutative diagram

A1 A2f1 f2

B1

B

2 where the horizontal arrows are isomorphisms in C as anabstract equivalencefrom f1 to f2. If there exists an abstract equivalence from f1 to f2, then we shallsay that f1, f2 are abstractly equivalent.

Let :C D be a faithful functorbetween categoriesC,D. Then we shallsay that is isomorphism-full if every isomorphism (A)

(B) ofD, whereA, B Ob(C), arises by applying to an isomorphism A B ofC. Suppose that is isomorphism-full. Then observe that theobjectsofD that are isomorphic toobjects in the image of , together with the morphismsofD that are abstractlyequivalent to morphisms in the image of , form a subcategory

C

D such that

induces an equivalence of categoriesC C . We shall refer to this subcategoryC D as the essential imageof . [Thus, this terminology is consistent with theusual terminology of essential image in the case where is fully faithful.]

Curves:

We refer to [Mzk14],0, for generalities concerning [families of ] hyperboliccurves,smooth log curves,stable log curves,divisors of cusps, anddivisors of markedpoints.

IfClog

Slog

is astable log curve, and, moreover,Sis the spectrum of afield,then we shall say thatClog issplitif each of the irreducible components and nodesofC is geometrically irreducibleover S.

A morphism of log stacksClog Slog

for which there exists an etale surjection S1 S, whereS1 is a scheme, such thatClog1

def= ClogSS1may be obtained as the result of forming the quotient [in the sense

of log stacks!] of a stable (respectively, smooth) log curveClog2 Slog1 def= Slog SS1by theaction of a finite groupof automorphisms ofClog2 overS

log1 which actsfreely

on a dense open subset of every fiber ofC2 S1 will be referred to as a stable log


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orbicurve(respectively, smooth log orbicurve) over Slog. Thus, the divisor of cusps

ofClog2 determines a divisor of cuspsofClog1 , C

log. Here, ifClog2 Slog1 is of type(1, 1), and the finite group of automorphisms is given by the action of 1 [i.e.,relative to the group structure of the underlying elliptic curve ofClog2 Slog1 ], thenthe resulting stable log orbicurve will be referred to as being oftype(1, 1).

IfSlog is the spectrum of a field, equipped with the trivial log structure, then ahyperbolic orbicurveX Sis defined to be the algebraic [log] stack [with trivial logstructure] obtained by removing the divisor of cusps from some smooth log orbicurveClog Slog over Slog. IfX(respectively, Y) is a hyperbolic orbicurve over a fieldK(respectively, L), then we shall say that X is isogenous to Y if there exists ahyperbolic curve Z over a field M together with finite etale morphismsZ X,Z Y. Note that in this situation, the morphisms Z X, Z Y induce finiteseparable inclusions of fieldsK M, L M. [Indeed, this follows immediatelyfrom the easily verified fact that every subgroup G(Z, OZ) such that G

{0}determines a fieldis necessarily contained in M

.]


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Section 1: The Tempered Anabelian Rigidity of the Etale Theta Function

In this, we construct a certain cohomology class in the [continuous] groupcohomology of the tempered fundamental group of a once-punctured elliptic curve

which may be regarded as a sort oftempered analytic representation of the thetafunction. We then discuss various properties of this etale theta function. Inparticular, we apply the theory of [Mzk14], 6, to show that it is, up to certain rela-tively mildindeterminacies, preserved by arbitraryautomorphisms of the temperedfundamental group [cf. Theorems 1.6, 1.10].

LetKbe a finite extension ofQp, with ring of integers OK;Kan algebraic clo-sure ofK;Sthe formal scheme determined by the p-adic completionof Spec(OK);Slog the formal log scheme obtained by equipping S with the log structure deter-mined by the unique closed point of Spec(OK); Xlog astable log curveover Slog oftype (1, 1). Also, we assume that the special fiberofXissingularand split[cf.

0],

and that the generic fiberof the algebrization ofXlog is a smooth log curve. Write

Xlog def= Xlog OK K for the ringed space with log structure obtained by tensoring

the structure sheaf ofX over OKwithK. In the following discussion, we shall often[by abuse of notation] use the notation Xlog also to denote the generic fiber of thealgebrization ofXlog. [Here, the reader should note that these notational conven-tionsdiffersomewhat from notational conventions typically employed in discussionsofrigid-analytic geometry.]

Let us write

tpX

for the tempered fundamental group associated to Xlog [cf. [Andre],4; the grouptemp1 (X

logK ) of [Mzk14], Examples 3.10, 5.6], with respect to somebasepoint. The

issue of how our constructions are affected when the basepointvarieswill be studiedin the present paper by considering to what extent these constructions arepreservedby inneror, more generally, arbitrary isomorphismsbetween fundamental groups[cf. Remark 1.6.2 below]. Here, despite the fact that the [tempered] fundamentalgroup in question is best thought of not as the fundamental group ofX but ratheras the fundamental group ofXlog, we use the notation tpX rather than

tpXlog

in order to minimize the number of subscripts and superscriptsthat appear in the

notation [cf. the discussion to follow in the remainder of the present paper!]; thus,the reader should think of theunderlinednotation tp() as an abbreviation for the

logarithmictempered fundamental group of the scheme (), equipped with the logstructure currently under consideration, i.e., an abbreviation for temp1 (()log).

Denote by tpX tpX the geometric tempered fundamental group. Thus, wehave a natural exact sequence

1 tpX tpX GK 1

whereGKdef

= Gal(K/K).


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whereZZ denotes the module of functions Z Z; the additive structure on thismodule is induced by the additive structure on the codomain Z. Write

LN

for the line bundle on YNdetermined by the constant functionZ Zwhose valueis 1. Also, we observe that it follows immediately from the above explicit descriptionof the special fiber ofYN that (YN, OYN) = OKN.

Next, write

JNdef= KN(a

1/N)aKN K where we note that [since KN is topologically finitely generated] JN is a finiteGalois extensionofKN. Observe, moreover, that we have an exact sequence

1

Z/NZ (= Z/NZ(1))

(tpYN)

/N

(tpY)

GKN

1

[cf. the construction ofYN]. Sinceany two splittingsof this exact sequence differ bya cohomology class H1(GKN,Z/NZ(1)), it follows [by the definition ofJN] thatall splittingsof this exact sequence determine the same splitting over GJN. Thus,the imageof the resulting open immersion

GJN (tpYN)/N (tpY)

is stabilizedby the conjugation action of tpX , hence determines a Galois covering

ZN YNsuch that the resulting surjection tpYN Gal(ZN/YN), whose kernel we denote

by tpZN, induces a natural exact sequence 1 Z/NZ Gal(ZN/YN)Gal(JN/KN) 1. Also, we shall write

tpZN (tpZN

) (tpZN)ell; tpZN (

tpZN

) (tpZN)ell

for the quotients of tpZN, tpZN

induced by the quotients of tpYN, tpYN

with similarsuperscripts and

Zlog

N

for the object obtained by equipping ZNwith the log structure determined by the[manifestly] JN-valued points ofZN lying over the cusps ofY. Set

ZN YN

equal to the normalization ofY in ZN. Since Y is generically of characteristiczero [i.e., Y is of characteristic zero], it follows that ZN is finiteover Y.

Next, let us observe that there exists a section

s1 (Y= Y1,L1)


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well-defined up to anOK-multiple whose zero locus on Y is precisely thedivisor of cuspsofY. Also, let us fix an isomorphismofLNN with L1|YN, whichwe use toidentifythese two bundles. Note that there is anatural actionof Gal(Y /X)on L1 which isuniquelydetermined by the condition that it preserve s1. Thus, weobtain a natural actionof Gal(YN/X) on L1

|YN.

Proposition 1.1. (Theta Action of the Tempered Fundamental Group)

(i) The section

s1|YN (YN,L1|YN =LNN )admits an N-th root sN (ZN,LN|ZN) over ZN. In particular, if we denoteassociated geometric line bundles by the notation V(), then we obtain a com-mutative diagram

ZN YN = YN Y1 V(LN|ZN) V(LN) V(LNN ) V(L1)

ZN YN = YN Y1

where the horizontal morphisms in the first and last lines are the natural mor-phisms; in the second line of horizontal morphisms, the first and third horizontalmorphisms are the pull-back morphisms, while the second morphism is given byraising to theN-th power; in the first row of vertical morphisms, the morphism onthe left (respectively, in the middle; on the right) is that determined bysN (respec-tively, s1|YN; s1); the vertical morphisms in the second row of vertical morphismsare the natural morphisms; the vertical composites are the identity morphisms.

(ii) There is a unique action of tpX on LNOKN OJN [a line bundle onYNOKN OJN] that iscompatible with the morphismZN V(LNOKN OJN)determined by sN [hence induces the identity on s

NN = s1|ZN]. Moreover, this

action of tpX factors through tpX/

tpZN

= Gal(ZN/X), and, in fact, induces a

faithful action oftpX/tpZN

onLNOKN OJN.

Proof. First, observe that by the discussion above [concerning the structure of

the special fiber ofYN], it follows that the action of tpX (Gal(YN/X)) on YN

preserves the isomorphism class of the line bundle LN, hence also the isomorphismclass of the line bundle LNN [i.e., theidentification ofL

NN with L1|YN, up to

multiplication by an element of (YN, OYN) = OKN]. In particular, if we denoteby GNthegroup of automorphismsof the pull-back ofLNto YNKNJNthatlie overthe JN-linear automorphisms ofYNKNJN induced by elements of tpX/tpYN Gal(YN/X) and whose N-th tensor power fixes the pull-back of s1|YN, then one


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verifies immediately [by recalling the definitionofJN] thatGNfits into a naturalexact sequence

1 N(JN) GN tpX/tpYN 1 where N(JN)

JNdenotes the group ofN-th roots of unity in JN.

Now Iclaimthat the kernelHN GNof the composite surjection

GN tpX/tpYN tpX/

tpY

= Z

where we note that Ker(tpX/tpYN

tpX/tpY) =

tpY/

tpYN

= Z/NZ(1) is anabelian group annihilated by multiplication byN. Indeed, one verifies immediately,by considering various relevant line bundles on Gm, that [if we write U for thestandard multiplicative coordinate on Gm and for a primitiveN-th root of unity,then] this follows from the identity of functionson Gm

N1j=0

f(j U) = 1

where f(U) def= (U 1)/(U ) represents an element ofHN that maps to a

generator of tpY/tpYN

.

Now consider thetautologicalZ/NZ(1)-torsorRN YNobtained by extract-ing anN-th root ofs1. More explicitly,RN YNmay be thought of as the finiteYN-scheme associated to theOYN-algebra

N1j=0

LjN

where the algebra structure is defined by the morphism LNN OYN given bymultiplying by s1|YN. In particular, it follows immediately from thedefinitionofGN thatGN acts naturally on (RN)JN def= RNOKN JN. Since s1|YN has zeroesof order1 at each of the cusps ofYN, it thus follows immediately that (RN)JN is

connectedand Galoisover XJNdef= X

KJN, and that one has an isomorphism

GN Gal((RN)JN/XJN)

arising from the natural action ofGNon (RN)JN. Since the abelian group tpX/tpYNactstriviallyon N(JN), and HN isannihilated byN, it thus follows formally fromthe definition of (X)

[i.e., as the quotient by a certain double commutatorsubgroup] that at least geometrically, there exists a map fromZN to RN. Moreprecisely, there is a morphism ZNOJN K RN over YN. That this morphismin fact factors through ZN inducing an isomorphism

ZN

RNOKN OJN


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follows from the definitionof the open immersion GJN (tpYN)/N (tpY)

whose image was used to define ZN YN [together with the fact that s1|YN isdefined over KN]. This completes the proof of assertion (i).

Next, we consider assertion (ii). Since thenatural actionof tpX ( Gal(YN/X))

on L1|YN =LNN preservess1|YN, and the action of

tpX on YNpreserves the iso-

morphism classof the line bundle LN, the existence and uniqueness of the desiredaction of tpX on LNOKN OJN follow immediately from the definitions [cf. espe-cially the definition ofJN]. Moreover, since sNis defined over ZN, it is immediate

that this action factors through tpX/tpZN

. Finally, the asserted faithfulnessfollowsfrom the fact that s1 has zeroes of order1 at the cusps ofYN [together with thetautological fact that tpX/

tpYN

acts faithfully on YN].

Next, let us set

KN def= K2N; JN def= KN(a1/N)aKN KYN

def= Y2NOKN OJN; YN

def= Y2NKN JN; LN

def= LN|YN =L22NOKN OJN

and write ZNfor the composite of the coverings YN,ZN ofYN; ZN for thenormal-

izationofZN in ZN; Y def

= Y1=Y2; Ydef= Y1 =Y2; K

def= K1= J1 =K2. Thus, we

have a cartesian commutative diagram

V(LN) V(LN)

YN YN

where, tpX acts compatibly onYN, YN, and [by Proposition 1.1, (ii)] on

LNOKN OJN. Thus, since this diagram is cartesian [and JN JN], we ob-tain anatural action oftpX on

LN whichfactorsthrough tpX/

tp

ZN. Moreover, we

have a natural exact sequence

1 tpZN/tp

ZN tpY/tpZN Gal(ZN/Y) 1

where tpZN/tpZN

Gal(YN/YN) which is compatiblewith the conjugationactions by tpX on each of the terms in the exact sequence.

Next, let us choose an orientationon the dual graph of the special fiber ofY.Such an orientation determines a specific isomorphism Z

Z, hence alabel Z foreach irreducible component of the special fiber ofY. Note that this choice of labelsalso determines a label Z for each irreducible component of the special fiber ofYN, YN. Now we define DNto be the effective divisoron YNwhich is supportedon the special fiber and corresponds to the function

Z j j2 log(qX)/2N


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i.e., at the irreducible component labeledj , the divisor DN is equal to the divisor

given by the schematic zero locus ofqj2/2NX . Note that since the completion of

YNat each node of its special fiber is isomorphic to the ring

OJN[[u, v]]/(uv q1/2N

X )

where u, v are indeterminates it follows that this divisor DN is Cartier.Moreover, a simple calculation of degrees reveals that we have an isomorphism ofline bundles on YN

OYN(DN) =LN i.e., there exists a section, well-defined up to anO

JN-multiple, (YN,LN)

whose zero locus is precisely the divisorDN. That is to say, we have a commutativediagram

YN N V(LN) V(LN) V(LNN ) V(L1)id YN = YN YN = YN Y1

in which the second square is the cartesian commutative diagram discussed above;the third and fourth squares are the lower second and third squares of the diagramof Proposition 1.1, (i); N which we shall refer to as the theta trivializationof

LN is a section whose zero locus is equal to DN. Moreover, since the action oftpY on

YN clearlyfixes the divisorDN, we conclude that the action of tpY on

YN,

V(LN) alwayspreservesN, up to anOJN-multiple.Next, let M 1 be an integer that dividesN. ThenYM Y (respectively,

ZM Y; YM Y) may be regarded as a subcoveringofYN Y (respectively,ZN Y; YN Y). Moreover, we have natural isomorphismsLM|YN =LN/MN ;LM|YN =L

N/MN . Thus, we obtain adiagram

YNN V(LN) YN

()N/MYM M V(LM) YM

in which the second square consists of the natural morphisms, hence commutes;the first square commutes up to anO

JN-,O

JM-multiple, i.e., commutes up to

composition, at the upper right-hand corner of the square, with an automorphismof V(LN) arising from multiplication by an element ofOJN, and, at the lowerright-hand corner of the square, with an automorphism of V(LM) arising frommultiplication by an element ofO

JM. [Indeed, this last commutativity follows from

the definitionofJN, and the easily verified fact that there exist Ns which are

defined overY2N.]


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Thus, since by the classical theory of the theta function[cf., e.g., [Mumf], pp.

306-307; the relation (U) =(U) given in Proposition 1.4, (ii), below], itfollows that one may choose 1 so that the natural action of

tpY on V(L1) [arising

from the fact that L1 is the pull-back of the line bundle L1 on Y; cf. the actionof Proposition 1.1, (ii)] preserves

1, we conclude, in light of the definition ofJ

N,

the following:

Lemma 1.2. (Compatibility of Theta Trivializations) By modifying the

variousNby suitableOJN-multiples, one may assume thatN1/N2N1

=N2 , for all

positive integersN1,N2 such thatN2|N1. In particular, there exists acompatiblesystem[asNvaries over the positive integers] ofactionsoftpY (respectively,

tp

Y)

onYN, V(LN) which preserve N. Finally, each action of this system differsfrom the action determined by the action of Proposition 1.1, (ii), by multiplicationby a(n) 2N-th root of unity (respectively, N-th root of unity).

Thus, by taking the N to be as in Lemma 1.2 and applying the natural iso-

morphism=Z(1) to thedifferencebetween the actions of tpY, tpY arising from

Proposition 1.1, (ii), and Lemma 1.2, we obtain the following:

Proposition 1.3. (TheEtale Theta Class)The difference between the naturalactions oftpY arising from Proposition 1.1, (ii); Lemma 1.2, on constant multiplesofNdetermines acohomology class

N H1(tpY, ( 12Z/NZ)(1)) =H1(tpY, ( 12Z/NZ))

which arises from a cohomology class H1(tpY/tpZN, (12Z/NZ)) whose re-

striction to

H1(tpYN

/tpZN

, (1

2Z/NZ)) = Hom(tp

YN/tp

ZN, (

1

2Z/NZ))

is the composite of the natural isomorphism tpYN

/tpZN

Z/NZ with thenatural inclusion (Z/NZ) ( 12Z/NZ). Moreover, if we set

OK/K

def={a O

K| a2 K}

and regardOK/K

as acting onH1(tpY, (12Z/NZ)(1)) via the composite

OK/K

H1(GK, ( 12Z/NZ)(1)) H1(tpY, (

1

2Z/NZ)(1))

where the first map is the evident generalization of the Kummer map, whichis compatible with the Kummer mapOK H1(GK, (Z/NZ)(1)) relative to thenatural inclusionO

K O

K/K and the morphism induced on cohomology by the


19/112


natural inclusion(Z/NZ) ( 12Z/NZ); the second map is the natural map thentheset of cohomology classes

OK/K

N H1(tpY, (1

2Z/NZ))

is independent of the choices of s1, sN, N. In particular, by allowing N tovary among all positive integers, we obtain aset of cohomology classes

OK/K

H1(tpY ,1

2)

each of which is a cohomology class H1((tpY), 12 ) whose restriction to

H1((tpY),

1

2) = Hom((

tpY)

,1

2)

is the composite of the natural isomorphism(tpY)

with the natural inclusion 12 . Moreover, the restricted classes

OK/K

|Y H1(tpY,1

2)

arise naturally from classes

OK H1(tp

Y, )

whereOK

acts via the composite of the Kummer mapOK

H1(GK, ) withthe natural map H

1

(GK, ) H1

(tp

Y, ) without denominators. Byabuse of the definite article, we shall refer to any element of the setsO

K/K N,

OK/K

,OK as theetale theta class.

Remark 1.3.1. Note that the denominators 12 in Proposition 1.3 are by nomeans superfluous: Indeed, this follows immediately from the fact that the divisorD1 on Y clearly does not descend to Y.

Let us denote by

U Ythe open formal subschemeobtained by removing the nodes from the irreduciblecomponent of the special fiber labeled 0 Z. If we take the unique cusp lying inUas the origin, then as is well-known from the theory of the Tate curve[cf., e.g.,[Mumf], pp. 306-307] the group structure on the underlying elliptic curve ofXlog

determines a group structureon U, together with a unique [in light of our choiceof an orientation on the dual graph of the special fiber ofY] isomorphism ofUwith thep-adic formal completion ofGm overOK. In particular, this isomorphismdetermines a multiplicative coordinate

U (U, OU )


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which, as one verifies immediately from the definitions, admits a square root

U (U, OU

)

on Udef= U

Y

Y.

Proposition 1.4. (Relation to the Classical Theta Function) Set

= (U)def= q

18

X nZ

(1)n q12 (n+ 12 )2

X U2n+1 (U, OU)

so extends uniquely to ameromorphic function onY [cf., e.g., [Mumf], pp.306-307]. Then:

(i) The zeroes of onY are precisely the cusps ofY; each zero has multi-

plicity1. The divisor ofpoles of onY is precisely the divisorD1.

(ii) We have

(U) = (U1); (U) = (U);(q

a/2X U) = (1)a qa

2/2X U2a (U)

fora Z.(iii) The classes

O

K

H1

(

tp

Y, )

of Proposition 1.3 are precisely theKummer classesassociated to OK

-multiples

of, regarded as a regular function onY. In particular, ifL is a finite extensionofK, andy Y(L) is anon-cuspidal point, then therestricted classes

OK |y H1(GL, ) =H1(GL,Z(1)) =(L)

where the denotes the profinite completion lie in L (L) and areequal to thevaluesO

K (y)ofand itsO

K-multiples aty. A similar statement

holds if y Y(L) is a cusp, if one restricts first to the associated decompositiongroup Dy and then to a sectionGL Dy compatible with thecanonical integral

structure [cf. [Mzk13], Definition 4.1, (iii)] onDy. In light of this relationship

between the cohomology classes of Proposition 1.3 and the values of, we shallsometimes refer to these classes asthe etale theta function.

Proof. Assertion (ii) is a routine calculation involving the series used to define

. A similar calculation shows that (1) = 0. The formula given for (qa/2X U)in assertion (ii) shows that the portion of the divisor of poles supported in the

special fiber ofY is equal to D1. This formula also shows that to complete the

proof of assertion (i), it suffices to show that the given description of the zeroes


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and poles of is accurate over the irreducible component of the special fiber ofY labeled 0. But this follows immediately from the fact that the restriction ofto this irreducible component is the rational function U U1. Finally, in lightof assertion (i) [and the fact, observed above, that (YN, OYN) = OKN], assertion(iii) is a formal consequence of the construction of the classes

O

K.

Proposition 1.5. (Theta Cohomology)

(i) The Leray-Serre spectral sequences associated to the filtration of closed sub-groups

(tpY) (tpY)

determine a natural filtration 0 F2 F1 F0 = H1((tpY), ) on thecohomology moduleH1((tpY)

, ) with subquotients

F0/F1 = Hom(, ) =Z log()F1/F2 = Hom((tpY)

ell/, ) =Z log(U)F2 =H1(GK, )

H1(GK,Z(1)) (K) where we use the symbollog() to denote the identity morphism andthe symbollog(U)to denote the standard isomorphism(tpY)

ell/Z(1) .

(ii) Similarly, the Leray-Serre spectral sequences associated to the filtration ofclosed subgroups

(tp

Y)

(tp

Y)

determine a natural filtration 0 F2 F1 F0 = H1((tpY

), ) on the

cohomology moduleH1((tpY

), ) with subquotients

F0/F1 = Hom(, ) =Z log()F1/F2 = Hom((tp

Y)ell/, ) =

Z log(U)F2 =H1(GK, )

H1(GK,Z(1)) (K)

where we write log(U) def= 12 log(U).(iii) Any class H1(tp

Y, ) arises from a unique class [which, by abuse

of notation, we shall denote by] H1((tpY

), ) that maps to log() in the

quotientF0/F1 and on whicha Z = Z =tpX/tpY acts as follows:

2a log(U) a2

2 log(qX) + log(OK)

where we use the notation log to express the fact that we wish to write the

group structure of(K) additively. Similarly, any inversion automorphism


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oftpY i.e., an automorphism lying over the action of 1 on the underlyingelliptic curve ofXlog which fixes the irreducible component of the special fiber ofYlabeled0 fixes +log(O

K), but mapslog(U)+log(O

K)tolog(U)+log(O

K).

Proof. Assertions (i), (ii) follow immediately from the definitions. Here, in (i)(respectively, (ii)), we note that the fact that F0 (respectively, F0) surjects ontoHom(, ) follows, for instance, by considering theKummer classof the mero-

morphic function U1 on Y(respectively, on Y cf. Proposition 1.4, (iii)).Assertion (iii) follows from Propositions 1.3; 1.4, (ii), (iii).

Theorem 1.6. (Tempered Anabelian Rigidity of the Etale Theta Func-

tion) Let Xlog (respectively, Xlog ) be a smooth log curve of type (1, 1) over a

finite extension K (respectively, K) ofQp; we assume that Xlog (respectively,

X

log

) hasstable reduction overOK (respectively,OK ), and that the resultingspecial fibers are singular and split. We shall use similar notation for objectsassociated to Xlog , X

log [but with a subscript or] to the notation that was used

for objects associated to Xlog. Let

: tpX tpX

be anisomorphism of topological groups. Then:

(i) We have: (tpY

) = tpY

.

(ii) induces an isomorphism

() ()

that iscompatible with the surjections

H1(GK , ()) H1(GK ,Z(1)) (K) Z

H1(GK , ()) H1(GK ,Z(1)) (K) Z

determined by the valuations onK

, K

. That is to say, induces an isomor-

phismH1(GK , ()) H1(GK , ())that preserves both the kernel of these

surjections and the elements 1 Z in the resulting quotients.(iii) The isomorphism of cohomology groups induced by maps the classes

OK

H1(tpY , ())

of Proposition 1.3 forX to sometpX

/tpY= Z-conjugate of the corresponding

classes

O

K

H1

(tp

Y , ())


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of Proposition 1.3 forX.

Proof. Assertion (i) is immediate from the definitions; thediscretenessof the topo-logical group Z; and the fact that maps tpX onto

tpX

[cf. [Mzk2], Lemma

1.3.8] and preservesdecomposition groups of cusps[cf. [Mzk14], Theorem 6.5, (iii)].As for assertion (ii), the fact that induces an isomorphism ()

() isimmediate [in light of the argument used to verify assertion (i)] from the defini-tions. The assertedcompatibilitythen follows from [Mzk14], Theorem 6.12; [Mzk2],Proposition 1.2.1, (iv), (vi), (vii).

Next, we consider assertion (iii). By composing with an appropriate in-ner automorphismof tpX , it follows from [Mzk14], Theorem 6.8, (ii), that we may

assume that the isomorphism tpY

tpY

iscompatible with suitable inversion au-

tomorphisms , [cf. Proposition 1.5, (iii)] on both sides. Next, let us observe

that it is a tautologythatis compatible with thesymbolslog() of Proposition1.5, (i), (ii). On the other hand, by Proposition 1.5, (ii), (iii), the property of map-

ping to log() in the quotient F0/F1 and being fixed, up to a unit multiple, under

an inversion automorphism completely determinesthe classes up to a (K)-multiple. Thus, to complete the proof, it suffices to reducethis indeterminacy upto a(K)-multiple to an indeterminacy up to aO

K-multiple.

This reduction of indeterminacy from (K) toOK

may be achieved [in

light of the compatibility shown in assertion (ii)] by evaluating the classes at

a cusp that maps to the irreducible component of the special fiber ofY labeled

0 [e.g., a cusp that is preserved by the inversion automorphism] via the canonicalintegral structure, as in Proposition 1.4, (iii), and applying the fact that, by [Mzk14],Theorem 6.5, (iii); [Mzk14], Corollary 6.11, preserves both the decompositiongroupsand thecanonical integral structures[on the decomposition groups] of cusps.

Remark 1.6.1. In the proof of Theorem 1.6, (iii), we eliminated the indeter-minacy in question by restricting to cusps, via the canonical integral structure.Another way to eliminate this indeterminacy is to restrict to non-cuspidal torsionpoints, which are temp-absolute by [Mzk14], Theorem 6.8, (iii). This latter ap-

proach amounts to invoking the theory of [Mzk13], 2, which is, in some sense, lesselementary [for instance, in the sense that it makes use, in a much more essentialway, of the main result of [Mzk11]] than the theory of [Mzk13], 4 [which one is, ineffect, applying if one uses cusps].

Remark 1.6.2. One way of thinking about isomorphisms of the tempered funda-mental group is that they arise from variation of the basepoint, or underlying settheory, relative to which one considers the associated temperoids [cf. [Mzk14],3]. Indeed, this is the point of view taken in [Mzk12], in the case ofanabelioids.From this point of view, the content of Theorem 1.6 may be interpreted as stating

that:


24/112


Theetale theta functionis preservedby arbitrarychanges of the underly-ing set theoryrelative to which one considers the tempered fundamentalgroup in question.

When viewed in this way, Theorem 1.6 may be thought of especially if one

takes the non-cuspidal approach of Remark 1.6.1 as a sort ofnonarchimedeananalogueof the so-called functional equationof the classicalcomplex theta function,which also states that the theta function is preserved, in effect, by changesof the underlying set theory relative to which one considers the integral singularcohomologyof the elliptic curve in question, i.e., more concretely, by the action of themodular groupSL2(Z). Note that in the complex case, it iscrucial, in order to provethe functional equation, to have not only the Schottky uniformization C C/qZ by C which naturally gives rise to the analytic series representation ofthe theta function, but isnot, however, preserved by the action of the modular group but also the full uniformization of an elliptic curve byC [whichispreserved by

the action of the modular group]. This preservation of the full uniformization byC in the complex case may be regarded as being analogous to the preservation ofthe non-cuspidal torsion points in the approach to proving Theorem 1.6 discussedin Remark 1.6.1.

Remark 1.6.3. The interpretation of Theorem 1.6 given in Remark 1.6.2 is rem-iniscent of the discussion given in the Introduction of [Mzk7], in which the authorexpresses his hope, in effect, that some sort of p-adic analogue of the functionalequation of the theta functioncould be developed.

Remark 1.6.4. One verifies immediately that there are [easier]profinite versionsof the constructions given in the present1: That is to say, if we denote by

(Ylog) (Ylog) Xlog

the profinite etale coveringsdetermined by the tempered coverings Ylog Ylog Xlog [so X/Y

=Z], then the set of classesOK

H1(tpY

, ) determines,

by profinite completion, a set of classes

O

K()

H1(

Y ,

)

on which any X/Y=Z aacts via

() () 2a log(U) a2

2 log(qX) + log(OK)

[cf. Proposition 1.5, (iii)]. Moreover, given X, X as in Theorem 1.6, any isomor-phism

X X

preservesthese profinite etale theta functions.


25/112


In fact, it is possible to eliminate theOK

-indeterminacyof Theorem 1.6, (iii),

to a substantial extent [cf. [Mzk13], Corollary 4.12]. For simplicity, let us assumein the following discussion that the following two conditions hold:

(I) K= K.

(II) The hyperbolic curve determined byXlog is not arithmetic overK [cf.,e.g., [Mzk3], Remark 2.1.1].

As is well-known, condition (II) amounts, relative to the j-invariant of the ellipticcurve underlying X, to the assertion that we exclude four exceptionalj-invariants[cf. [Mzk3], Proposition 2.7].

Now let us write Xlog Xlog for the Galois [by condition (I)] covering of degree4 determined by the multiplication by2 map on the elliptic curve underlying X;writeXlog

Clog for the stack-theoretic quotientofXlog by the natural action of

1 on [the underlying elliptic curve of] X. Thus, [by condition (II)] the hyperbolicorbicurve determined by Clog is a K-core [cf. [Mzk3], Remark 2.1.1]. Observe

that the covering Xlog Clog isGalois, with Galois group isomorphic to (Z/2Z)3.Moreover, we have two natural automorphisms

Gal(X/X) Gal(X/C); Gal(X/C)

i.e., respectively, the unique nontrivial element of Gal(X/X) that acts triviallyon the set of irreducible components of the special fiber; the unique nontrivialelement of Gal(X/C) that acts trivially on the set of cusps ofX.

Now suppose that we are given a nontrivial element

Z Gal(X/X)

which is=. ThenZ determines a commutative diagram

Ylog Xlog Xlog Xlog

C

log

Clog

where Ylog Xlog, Xlog Clog are the natural morphisms; Xlog Xlog isthe quotient by the action ofZ; X

log Xlog is the quotient by the action of;Xlog Clog is the [stack-theoretic] quotient by the action of ; the square iscartesian.

Definition 1.7. We shall refer to a smooth log orbicurve over K that arises, upto isomorphism, as the smooth log orbicurve Xlog (respectively, Clog) constructedabove for some choice of Z as being of type (1,2) (respectively, (1,2)). We

shall also apply this terminology to the associated hyperbolic orbicurves.


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Proposition 1.8. (Characteristic Nature of Coverings)For= , , let

Xlog

be a smooth log curve of type (1,2) over a finite extension K ofQp;

writeYlog

, Xlog

, Xlog

, Clog

, Clog

for the related smooth log orbicurves [as in theabove discussion]. Then any isomorphism of topological groups

: tpX

tpX

(respectively,: tpC

tpC

)

induces an isomorphism between the commutative diagrams of outer homomor-phisms of topological groups

tpY

tpX

tpX

tpX tpC

tpC where =, . A similar statement holds when tp is replaced by .

Proof. First, we consider the tempered case. By [Mzk2], Lemma 1.3.8, it followsthatis compatible with the respective projections to GK . By [Mzk14], Theorem6.8, (ii) [cf. also [Mzk3], Theorem 2.4], it follows from condition (II) that induces

an isomorphism tpC tpC that is compatible with. Since [as is easily verified]

tpX tpC

may be characterized as the unique open subgroup of index 2 that

corresponds to a double covering which is a scheme [i.e., open subgroup of index2 whose profinite completion contains no torsion elements cf., e.g., [Mzk16],

Lemma 2.1, (v)], it follows that determines an isomorphism tpX tpX , hence

also an isomorphism (tpX)ell (tpX )ell. Moreover, by considering the discrete-

ness of Gal(Y/X)= Z, or, alternatively, the trivialityof the action of GKon Gal(Y/X), it follows that this last isomorphism determines an isomorphism

tpX/tpY

tpX/tpY

= Z , hence [by considering the kernel of the action oftpC on

tpX

/tpY ] an isomorphism tpX

tpX . Since Xlog

Clog

may be char-

acterized as the quotient by the unique automorphism of Xlog

over Clog

that acts

nontrivially on the cusps of Xlog

[where we recall that preserves decomposition

group of cusps cf. [Mzk14], Theorem 6.5, (iii)] but does not lie over X

log

, wethus conclude thatinduces isomorphisms between the respective tpX

, tpC

, tpX ,

tpC that are compatible with the natural inclusions among these subgroups [for a

fixed ]. Moreover, since preserves the decomposition groups of cusps of tpX[cf. [Mzk14], Theorem 6.5, (iii)], we conclude immediately that is also compatible

with the subgroups tpY

tpX

tpX , as desired. The profinite case is provensimilarly [or may be derived from the tempered case via [Mzk14], Theorem 6.6].

Next, let us suppose that

1 K


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where we note that 1 determines a 4-torsion point of [the underlying

elliptic curve of] X whose restriction to the special fiber lies in the interior of[i.e., avoids the nodes of] the unique irreducible componentof the special fiber; the4-torsion point 1 determined by 1 admits a similar description. Let

H1(tpY

, )

be a class as in Proposition 1.3; write ,Z for the tpX

/tpY

= Z-orbitof .

Definition 1.9. Suppose that1 K.

(i) We shall refer to either of the following two sets of values [cf. Proposition1.4, (iii)] of ,Z

,Z

|,

,Z

|1

K

as a standard set of values of,Z.

(ii) If ,Z satisfies the property that the unique value OK [cf. the valueat

1 of the series representation of given in Proposition 1.4; Proposition 1.4,(ii)] ofmaximal order [i.e., relative to the valuation on K] of some standard set ofvalues of ,Z is equal to1, then we shall say that ,Z is of standard type.

Remark 1.9.1. Observe that it is immediate from the definitions that anyinner automorphism of tp

C arising from tp

X acts trivially on ,Z, and that the

automorphisms , map ,Z ,Z [cf. Proposition 1.4, (ii)]. In particular,anyinner automorphismof tp

C maps ,Z ,Z.

The point of view of Remark 1.6.1 motivates the following result:

Theorem 1.10. (Constant Multiple Rigidity of the Etale Theta Func-

tion) For =, , letClog

be asmooth log curve of type(1,2) over a finiteextensionK ofQp that contains a square root of1. Let

: tpC

tpC

be an isomorphism of topological groups. Suppose that the isomorphismtpX tpX

induced by [cf. Proposition 1.8] maps,Z ,Z [cf. Theorem 1.6, (iii)]. Then:

(i) The isomorphismpreserves the property that,Z

beof standard type,a property that determines this collection of classes up to multiplication by1.

(ii) The isomorphism

K

K


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where we regard K

(K

) as a subset of (K

)= H1(GK , ())H1(tp

C, ()) induced by [an arbitrary] preserves the standard sets of

values of,Z

.

(iii) Suppose that ,Z

is of standard type, and that the residue charac-teristic of K is odd. Then

,Z

determines a{1}-structure [cf. [Mzk13],Corollary 4.12; Remark 1.10.1, (ii), below] on the(K

)-torsor at the unique cusp

ofClog

that is compatible with thecanonical integral structure and, moreover,preservedby [arbitrary].

Proof. First, we observe that assertions (i), (iii) follow formally from assertion(ii) [cf. also the series representation of Proposition 1.4]. Now we verify assertion(ii), as follows. By applying [Mzk14], Theorem 6.8, (iii), together with the factthatinduces isomorphisms between thedual graphs[cf. Theorem 1.6, (i); [Mzk2],Lemma 2.3] of the special fibers of the various corresponding coverings ofC, Cthat appear in the proof of [Mzk14], Theorem 6.8, (iii), it follows that maps [the

decomposition group of points ofY lying over] to [the decomposition group of

points ofY lying over] 1. Now assertion (ii) follows immediately.

Remark 1.10.1.

(i) The -indeterminacy of Theorem 1.10, (i), (iii), is reminiscent of, butstronger than, the indeterminacy up to multiplication by a 12-th root of unity of[Mzk13], Corollary 4.12. Also, we note that from the point of view of the techniqueof the proof of loc. cit., applied in the present context ofTate curves, it is thefact that there is a special2-torsion point, i.e., the 2-torsion point whose imagein the special fiber lies in the same irreducible component as the origin, that allowsone to reduce the 12 = 2 (3!) of loc. cit. to 2.

(ii) We take this opportunity to remark that in [Mzk13], Corollary 4.12, theauthor omitted the hypothesis that Kcontain a primitive 12-th root of unity.The author apologizes for this omission.

Remark 1.10.2. One observation that one might make is that since Theorem

1.10 depends on the theory of [Mzk13],2 [i.e., in the tempered version of thistheory given in [Mzk14],6], one natural approach to further strengthening Theo-rem 1.10 is to consider applying the absolute p-adic version of the GrothendieckConjectureof [Mzk16], Corollary 2.3, which may be regarded as a strengtheningof the theory of [Mzk13], 2. One problem here, however, is that unlike the portionof the theory of [Mzk13],2, that concerns [non-cuspidal] torsion points of once-punctured elliptic curves [i.e., [Mzk13], Corollary 2.6], the absolute p-adic versionof the Grothendieck Conjecture of [Mzk16], Corollary 2.3, only holds for ellipticcurves which are defined over number fields. Moreover, even if, in the future, thishypothesis should be eliminated, the [somewhat weaker] theory of [Mzk13],

2, fol-

lows formally [cf. [Mzk13], Remark 2.8.1] from certain general nonsense-type


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group ofSell\tors may be thought of as corresponding to the Heisenberg group, ortheta-group i.e., the theta quotientdiscussed in (i) which plays a fundamentalrole in the theory of theta functions, hence in the scheme-theoretic Hodge-Arakelovtheory reviewed in [Mzk4] [cf. especially the discussion of [Mzk4],1.3.5]. On theother hand, Sdisc\tors is homotopically equivalent to a bouquet of circles, where the

circles correspond naturally to the l-torsion pointsofE. Thus, from the point ofview of algebraic topology, the Sdisc\tors portion ofSell\tors may be thought of asthe suspensionof the discrete set ofl-torsion points [together with some additionalbasepoint]. On the other hand, from the point of view of arithmetic geometry,the circle may be thought of as a sort of Tate motive and the arithmetic suspen-sion constituted by the Sdisc\tors portion ofSell\tors as a sort ofanabelian Tatetwist of the l-torsion points of E. Moreover, from the point of view of Hodge-Arakelov theory, the universal covering of the bouquet constituted by Sdisc\tors maybe thought of as the result of filling in the discrete set of l-torsion points byjoining these points via continuous line segments cf. the original point of view

of Hodge-Arakelov theory [discussed in [Mzk4],1.3.4] to the effect that the set oftorsion points is to be regarded as a high resolution approximationof the under-lying real analytic manifold ofE. Put another way, this sort of continuous versionof thel-torsion points may be thought of as a sort ofcyclotomic blurringof thel-torsion points i.e., a sort of space of infinitesimal deformations of the l-torsionpoints that allows one to consider derivatives [e.g., of the theta function] at thel-torsion points. Thus, in summary, relative to these geometric considerations, thetopological surface Sell\tors = Sell\disc

Sdisc\tors, hence also its group-theoretic

counterpart [i.e., the fullfundamental group under consideration] may be thoughtof as being precisely a geometric realizationof the content

(theta functions and their derivatives)|l-torsion points[i.e., where theta functions correspond, via theta-groups, toSell\disc, and deriva-tives at l-torsion points correspond to Sdisc\tors] of Hodge-Arakelov theory.

Remark 1.10.4.

(i) As observed in Proposition 1.4, (iii), by restricting the etale theta functionto various [e.g., torsion!] points, one obtains a Kummer-theoretic approach toconsidering the theta function as a functionon points. In the context of the theoryof the present

1 [and indeed of the present paper], in which one does notassume [cf.

Remark 1.10.2] that themultiplicativegroups associated to the base fields involvedthat appear in the absolute Galois groups of these fields e.g., the (L) thatappears in Proposition 1.4, (iii) are equipped withadditivestructures [i.e., arisingfrom the addition operation on the field], the functions that may be obtained inthis way are very special. Indeed, if one may avail oneself of both the additive andmultiplicative structures i.e., the ring structures of the fields involved, thenit is not difficult to give various group-theoretic algorithms for constructing allsorts of such functions. On the other hand:

If one may only avail oneself of the multiplicative structure, then it is

difficult to construct such functions, except via considering directly the


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functions obtained by restricting Kummer classes of meromorphic func-tions.

[Of course, the multiplicative structure allows one to construct N-th powers, forN

1 an integer, of the values obtained by restricting Kummer classes, but such

N-th powers are simply the values obtained by restricting the Kummer classesobtained by multiplying the original Kummer classes by N.] Finally, we recall thatthe theta function satisfies the unusual property, among meromorphic functionson tempered coverings of pointed stable curves, of having a divisor of poles thatis contained in the special fiber and a divisor zeroes that does not contain anyirreducible components of the special fiber [cf. Proposition 1.4, (i)].

(ii) Even if one allows oneself to consider Kummer classes

H1(,Z(1))

for anarbitrary topological groupthat surjects onto the absolute Galois groupGKof a finite extension K ofQp, it is not difficult to see that it is a highly nontrivialoperation to construct functions on the set of [conjugacy classes of] sections of GK. Indeed, if is to give rise to a nontrivial function, then it is natural

to assume that it must induce an isomorphismof some isomorph [cf. 0] ofZ(1)inside ontoZ(1) [cf. the role of in Proposition 1.3 in the case of the etaletheta function]. On the other hand, if, for instance, =Z(1) GK, then [cf.the discussion of the theta quotient in Remark 1.10.3, (i)] it is easy to see thatthe resulting fails to satisfy the analogue of the rigidity property of Theorem

1.10, (i). Thus, just as in the discussion of Remark 1.10.3, one is ultimately lednaturally to consider the case where is some sort of [e.g., tempered or profinite]arithmetic group of a hyperbolic orbicurve, in which case various strong anabelianrigidity properties are known. Moreover, it is difficult to see how to develop thetheory of2 below which makes essential use, in so many ways, of the theoryand structure oftheta-groups [i.e., the theta quotients of Remark 1.10.3, (i)] for hyperbolic orbicurves that are not isogenous to once-punctured elliptic curves.

This state of affairs again serves to highlight the fact that the function[on, say, torsion points] determined by the etale theta functionshould be

regarded as a very specialand unusualobject.


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Section 2: The Theory of Theta Environments

In this, we begin by discussing various general nonsense complements [cf.Corollaries 2.8, 2.9] to the theory of etale theta functions of1 involving coverings[cf. the discussion of the Lagrangian approach to Hodge-Arakelov theory inthe Introduction]. This discussion leads naturally to the theory of thecyclotomicenvelope[cf. Definition 2.10] and the associatedmono- and bi-theta environments[cf. Definition 2.13], whose tempered anabelian rigidity properties [cf. Corollaries2.18, 2.19] we shall use in5, below, to relate the theory of the present2 to thetheory oftempered Frobenioidsto be discussed in3,4, below.

Let Xlog be a smooth log curve of type (1, 1) over a field K ofcharacteristiczero. For simplicity, we assume that the hyperbolic curve determined byXlog isnotK-arithmetic[i.e., admits a K-core cf. [Mzk3], Remark 2.1.1]. As in1, weshall denote the (profinite) etale fundamental groupofXlog by X . Thus, we have

a natural exact sequence:

1 X X GK 1

whereGKdef= Gal(K/K); X is defined so as to make the sequence exact. Since

X is a profinite free group on2 generators, the quotient

Xdef= X/[X , [X , X ]]

fits into a natural exact sequence

1 X ellX 1

where ellXdef= abX = X/[X , X ]; we write for the imageof2 ellX in

X . Also, we shall write X X for the quotient whose kernel is the kernel of

the quotient X X .

Now letl 1 be an integer. One verifies easily by considering the well-knownstructure of X that the subgroup of

X generated by l-th powers of elements of

X is normal. We shall write X X for the quotient of

X by this normal

subgroup. Thus, the above exact sequence for X determines a quotient exactsequence

1 X ellX 1

where =(Z/lZ)(1); ellX is a free (Z/lZ)-module of rank 2. Also, we shallwrite X X for the quotient whose kernel is the kernel of the quotient X

X and ell

Xdef= X/.

Let us write x for the unique cusp ofXlog. Then there is a natural injective[outer] homomorphism

Dx X


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whereDx X is the decomposition group associated to x which maps theinertia group Ix Dx isomorphically onto . Thus, we have exact sequences

1 X X GK 1; 1 Dx GK 1

where we writeDx X for the image ofDx in X .Next, let

ell

X Q be a quotientonto a free (Z/lZ)-module Q of rank 1 such

that the restricted map ell

X Qis stillsurjective, but the restricted map Dx Qis trivial. Denote the corresponding covering by Xlog Xlog; write X X ,X X , ellX

ell

X for the corresponding open subgroups. Observe that ourassumption that the restricted map Dx Q is trivial implies that every cusp ofXlog is K-rational. Let us write (respectively, ) for the automorphismofXlog

(respectively,Xlog) determined by multiplication by 1 on the underlying elliptic

curve relative to choosing the unique cusp ofX

log

(respectively, relative to somechoice of a cusp of Xlog) as the origin. Thus, if we denote the stack-theoretic

quotient of Xlog (respectively, Xlog) by the action of (respectively, ) by Clog

(respectively,Clog), then we have a cartesian commutative diagram:

Xlog Xlog Clog Clog

We shall write C, C for the respective (profinite) etale fundamental groups ofClog, Clog. Thus, we obtain subgroups C C, C C [i.e., the kernels ofthe natural surjections to GK]; moreover, by forming the quotient by the kernelsof the quotients X X , X X , we obtain quotients C C, C

C, C C, C C. Similarly, the quotient X ell

X determines a

quotient Cell

C . Let C be an element that lifts the nontrivial elementof Gal(X/C) = Z/2Z.

Definition 2.1. We shall refer to a smooth log orbicurve over K that arises,

up to isomorphism, as the smooth log orbicurve Xlog (respectively, Clog) con-

structed above for some choice of ell

X Qas being of type(1, l-tors) (respectively,(1, l-tors)). We shall also apply this terminology to the associated hyperbolicorbicurves.

Remark 2.1.1. Note that although Xlog Xlog is [by construction] Galois,with Gal(X/X) =Q, the covering Clog Clog fails to be Galois in general. Moreprecisely, no nontrivial automorphism Gal(X/X) of, say, oddorder descends toan automorphism ofClog over Clog. Indeed, this follows from the fact that acts

on Qbymultiplication by1.


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Proposition 2.2. (The Inversion Automorphism) Suppose that l isodd.Then:

(i) The conjugation action of on the rank two (Z/lZ)-module X deter-mines a direct product decomposition

X=ellX into eigenspaces, with eigenvalues1 and 1, respectively, that is compatible withthe conjugation action ofX . Denote by

s: ell

X X

the resulting splitting of the natural surjectionX ell

X.

(ii) In the notation of (i), the normal subgroup Im(s)X induces an iso-morphismDx

X/Im(s)over GK. In particular, any section of the H

1(GK, )= K/(K)l-torsor ofsplittings ofDx GK determines a covering

Xlog Xlog

whose corresponding open subgroup we denote byX X . Here, the geometricportionX ofX mapsisomorphicallyonto

ell

X [hence is a cyclic group of order

l], i.e., we haveX = Im(s), X = X . Finally, the image of inC/Xmay be characterized as theuniquecoset ofC/X that lifts the nontrivial element

ofGal(X/C) = C/X andnormalizes the subgroup X C.

(iii) There exists aunique coset C/X such that hasorder 2 if andonly if it belongs to this coset. If we chooseto have order2, then the open subgroup

generated by X and in C [or, alternatively, the open subgroup generated by

GK=X/X and in C/X] determines a double covering Xlog Clogwhich fits into a cartesian commutative diagram

Xlog Xlog Clog Clog

whereXlog is as in (ii).

Proof. Assertions (i) and (ii) are immediate from the definitions. To verify asser-tion (iii), we observe that Dx=X/X is of index 2 in C/X . Thus, normal-izes Dx=X/X . Since, moreover, l is odd [so H

1

(GK, ), /H0

(GK, )


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have no elements of order 2], and conjugation by induces the identity on and

GK, it follows that centralizesDx=X/X , hence [a fortiori] GK=X/X .Now assertion (iii) follows immediately.

Remark 2.2.1. We shall not discuss the case ofevenl in detail here. Nevertheless,we pause briefly to observe that ifl = 2, then [since lies in thecenterof X ] the

automorphism Gal(X/C)=ellC of1 acts naturallyon the exact sequence1 X ellX 1. Since this action is clearly trivial on ,

ell

X , one

verifies immediately that this action determines a homomorphism ell

X , i.e.,in effect, a 2-torsion point[so long as the homomorphism is nontrivial] of the ellipticcurve underlyingXlog. Thus, by considering the case whereKis thefield of moduliof this elliptic curve [so that GK permutes the 2-torsion points transitively], we

conclude that this homomorphism must be trivial, i.e., that every element of ell

X

admits an -invariant liftingto X .

Definition 2.3. We shall refer to a smooth log orbicurve over K that arises, upto isomorphism, as the smooth log orbicurve Xlog (respectively,Clog) constructed

in Proposition 2.2 above as being of type (1, l-tors) (respectively, (1, l-tors)).We shall also apply this terminology to the associated hyperbolic orbicurves.

Remark 2.3.1. Thus, one may think of the single underline in the notationXlog, Clog as denoting the result of extracting a single copy of Z/lZ, and the

double underline in the notation Xlog, Clog as denoting the result of extracting

two copies ofZ/lZ.

Proposition 2.4. (Characteristic Nature of Coverings) For = , ,

let Xlog

be a smooth log curve of type (1, l-tors) over a finite extension K

ofQp, where l is odd; write Clog

, Xlog

, Clog

, Xlog

, Clog

for the related smooth

log orbicurves [as in the above discussion]. Assume further thatXlog

has stablereductionoverOK , withsingularandsplitspecial fiber. Then any isomorphismof topological groups

: tp

X

tp

X (respectively, : tp

X

tp

X ;

: tpC

tpC

; : tpC tpC )

induces isomorphisms compatible with the various natural maps between the respec-

tive tps of Xlog

, Xlog

, Clog

, Clog

, Clog

, Ylog

(respectively, Xlog

, Clog

, Ylog

;

Clog

, Clog

, Xlog

, Xlog

, Xlog

, Ylog

; Clog

, Xlog

, Xlog

, Ylog

) where =, .

Proof. As in the proof of Proposition 1.8, it follows from our assumption thatthe hyperbolic curve determined by Xlog

admits a K-corethat induces an iso-

morphism tpC

tpC [cf. [Mzk3], Theorem 2.4] which [cf. [Mzk2], Lemma 1.3.8]


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induces an isomorphism tpC tpC ; moreover, this last isomorphism induces

[by considering open subgroups of index 2 whose profinite completions contain no

torsion elements] an isomorphism tpX tpX , hence also [by considering the con-

jugation action of tpC on an appropriate abelian quotient of tpX

as in the proof

of Proposition 1.8] an isomorphism tpX

tpX , which preserves the decomposi-

tion groups of cusps [cf. [Mzk14], Theorem 6.5, (iii)]. Also, by the definition of

X , the isomorphism tpX

tpX determines an isomorphism X X . In

light of these observations, the various assertions of Proposition 2.4 follow immedi-ately from the definitions [cf. also Proposition 2.2; Theorem 1.6, (i); the proof ofProposition 1.8].

Now, we return to the discussion of1. In particular, we assume thatK is afinite extension ofQp.

Definition 2.5. Suppose thatl and the residue characteristicofKare odd, andthat K= K [cf. Definition 1.7 and the preceding discussion].

(i) Suppose, in the situation of Definitions 2.1, 2.3, that the quotient ell

X Q

factorsthrough the natural quotient X Zdetermined by the quotient tpX Z

discussed at the beginning of1, and that thechoice of a splittingofDx GK [cf.Proposition 2.2, (ii)] that determined the covering Xlog Xlog is compatiblewiththe {1}-structure of Theorem 1.10, (iii). Then we shall say that the orbicurveof type (1, l-tors) (respectively, (1, l-tors); (1, l-tors); (1, l-tors

)) under consid-eration isof type(1,Z/lZ) (respectively, (1, (Z/lZ)); (1,Z/lZ); (1, (Z/lZ)

)).

(ii) In the notation of the above discussion and the discussion at the end of1,we shall refer to a smooth log orbicurve isomorphic to the smooth log orbicurve

Xlog

(respectively, Xlog

; Clog

; Clog

)

obtained by taking the composite of the covering

Xlog (respectively,Xlog; Clog; Clog)

ofClog with the covering Clog Clog, as being of type(1,2 Z/lZ) (respectively,(1,2 (Z/lZ)); (1,2 Z/lZ); (1,2 (Z/lZ))).

Remark 2.5.1. Thus, theirreducible componentsof the special fiber ofC, Cmaybe naturally identifiedwith the elements of (Z/lZ)/{1} cf. Corollary 2.9 belowfor more details.

Proposition 2.6. (Characteristic Nature of Coverings)For= , , letus assume that we havesmooth log orbicurves as in the above discussion, overa finite extensionK ofQp. Then any isomorphism of topological groups

: tpX

tpX

(respectively, : tpX

tpX

;

: tpC

tpC

; : tpC tpC )


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induces isomorphisms compatible with the various natural maps between the respec-tive tps ofXlog

(respectively, Xlog

; Clog

; Clog

) andClog

, where=, . A

similar statement holds when tp is replaced by .

Proof. The proof is entirely similar to the proofs of Propositions 1.8, 2.4.

Remark 2.6.1. Suppose, for simplicity, that K contains a primitive l-th rootof unity. Then we observe in passing that by applying the Propositions 2.4, 2.6to isomorphisms of fundamental groups arising from isomorphisms of the orbi-curves in question [cf. also Remark 2.1.1], one computes easily that the groupsofK-linear automorphismsAutK() of the various smooth log orbicurves underconsideration are given as follows:

AutK(Xlog) = l

{1

}; AutK(X

log) = Z/lZ

{1

}AutK(Clog) = l; AutK(Clog) = {1}

where l denotes the group of l-th roots of unity in K, and the semi-directproduct is with respect to the natural multiplicative action of1 onZ/lZ; theAutK()s of the various once-dotted versions of these orbicurves [cf. Defini-tion 2.5, (ii)] are given by taken the direct product of the AutK()s listed abovewith Gal(Clog/Clog) = {1}.

Next, we consider etale theta functions. First, let us observe that the coveringY

log

Clog

factors naturally through X

log

. Thus, the class [which is only well-defined up to aOK-multiple]

H1(tpY

, )

of1 as well as the corresponding tpX

/tpY

= Z-orbit,Z may be thought ofas objects associated to the tp ofX

log, C

log,Xlog,Clog. On the other hand, the

compositesof the coverings Ylog Clog, Ylog Clog with Clog Clog determinenew coverings

Ylog

Ylog; Ylog

Ylog

of degree l. Moreover, the choice of a splitting of Dx GK [cf. Proposition2.2, (ii)] that determined the covering Xlog Xlog determines [by consideringthe natural map Dx tpY (

tp

Y) cf. Proposition 1.5, (ii)] a specific class

H1(tpY

, Z/lZ), which may be thought of as achoice of up to an(OK)l-multiple[i.e., as opposed to only up to aOK-multiple]. Now it is a tautologythat,upon restriction to the covering Y

log Ylog [which was determined, in effect, bythe choice of a splittingofDx GK], the class determines a class

H1

(

tp

Y, l )


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as well as a corresponding tpX

/tpY

=tpX

/tpY

=l Z-orbit,lZ which maybe thought of as objects associated to the tp of X

log, C

log, Xlog, Clog, and

which satisfy the following property:

H1

(tp

Y , l )

|Y H1

(tp

Y, )

[relative to the natural inclusion l ]. That is to say, at a more intuitivelevel, may be thought of as an l-th root of the etale theta function. In the

following, we shall also consider thel Z-orbit,lZ of , as well as the tpX/tpY =tpX/

tp

Y= {(l Z) 2}-orbits

,lZ2 , ,lZ2

of , , and the tpX/tpY =(Z 2)-orbit,Z2 of .

Definition 2.7. If ,Z is of standard type, then we shall also refer to ,lZ,,lZ, ,lZ2 , ,lZ2 , ,Z2 as being of standard type.

Corollary 2.8. (Constant Multiple Rigidity of Roots of theEtale ThetaFunction) For =, , let us assume that we havesmooth log orbicurves asin the above discussion, over a finite extensionK ofQp. Let

: tpX

tpX

(respectively, : tpX

tpX ; : tpC

tpC

; : tpC tpC )

be an isomorphism of topological groups. Then:

(i) The isomorphism preserves the property [cf. Theorem 1.6, (iii)] that

,lZ2

(respectively, ,Z2

; ,lZ2

; ,lZ2

) beof standard type a

property that determines this collection of classes up to multiplication by a rootof unity of order l (respectively, 1; l; 1).

(ii) Suppose further that the cusps of X are rational over K, that theresidue characteristic ofK isprime tol, and thatK contains aprimitivel-throot of unity. Then the{1}-[i.e., 2-] structure of Theorem 1.10, (iii), deter-mines a2l (respectively, 2; 2l; 2)-structure [cf. [Mzk13], Corollary 4.12] on

the(K

)-torsor at the cusps ofXlog

(respectively, Xlog

; Xlog

; Xlog

). Moreover,

this2l (respectively, 2; 2l; 2)-structure is compatible with thecanonical in-tegral structure[cf. [Mzk13], Definition 4.1, (iii)] determined by the stable model

ofXlog

andpreserved by.

(iii) If the data for = , are equal, and arises [cf. Proposition 2.6]

from an inner automorphism of

tp

X (respectively,

tp

X ;

tp

C ;

tp

C), then


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prese

the etale theta function and its frobenioid-theoretic manifestations

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