An approach to harmonic analysis on non-locally compact groups I: level structures over locally compact groups
AAn approach to harmonic analysis onnon-locally compact groups I: level structuresover locally compact groups
Raven Waller ∗ Abstract
We define a class of spaces on which one may generalise the notionof compactness following motivating examples from higher-dimensionalnumber theory. We establish analogues of several well-known topolog-ical results (such as Tychonoff’s Theorem) for such spaces. We alsodiscuss several possible applications of this framework, including thetheory of harmonic analysis on non-locally compact groups.
The theory of harmonic analysis on locally compact groups is by now entirelyclassical. However, in higher dimensional number theory in particular, manyobjects arise which are no longer locally compact. For example, the field Q p pp t qq of formal Laurent series over Q p is not locally compact in any of the naturaltopologies which take into account both of its residue fields. The loss of localcompactness for such fields is one of the most pervasive problems when onetries to study them.In this paper we consider topological spaces whose topology may be recon-structed in a particular way from a locally compact group. This constructionleads to a very natural generalisation of compactness, and in particular al-lows us to apply certain compactness arguments to groups related to higherdimensional local fields. ∗ This work was completed while the author was supported by an EPSRC Doctoral Train-ing Grant at the University of Nottingham. a r X i v : . [ m a t h . GN ] F e b he main focus of this paper is thus the definition and properties of groupswith a level structure. The main motivation for the definition comes fromthe study of two-dimensional local fields as follows. Consider once again F “ Q p pp t qq , and its rank two ring of integers O F “ Z p ` t Q p rr t ss . One can define thelevel of a subset S Ă F as the least integer j such that S contains a translateof a fractional ideal p i t j O F for some i P Z . The remarkable observation is thenthat, although O F is not compact, if one looks only at its open covers (in aparticular topology) by sets of the same level (level 0), all of them have finitesubcovers. We thus have a weaker substitute which works “on the level”.The definition of a level structure formalises this example, and gives areasonable context in which to study such “level compactness” properties. Itturns out that many properties of compact sets are also shared by those whichare only level-compact. For example, we obtain an analogue of Tychonoff’sTheorem for products of compact spaces.Since this text is dedicated to the development of a new, more generaltheory of compactness, there will be a substantial number of definitions givenin quick succession. We try to give as much motivation as possible to show thateach definition is important and, where possible, we demonstrate the kind ofpathological cases that may arise if one doesn’t take care to make the requiredassumptions.The reader is thus asked to persevere with the abnormal Definition-to-Theorem ratio, if only because of the possible wide-reaching applications. In-deed, the notion of level-compactness is not at all limited to problems relatedto higher-dimensional number theory, and can almost certainly be studied ina variety of other contexts.This paper is organised as follows. In Section 2 we recall the definition ofa higher-dimensional local field, and explain the main motivating example forthe constructions that follow. We then begin Section 3 with the definition ofa level structure on a group G . The remainder of the paper will be devoted tothe study of such groups, and so it is paramount that the reader familiarisesthemselves with this definition as thoroughly as possible, keeping in mind theexample of a higher dimensional local field from Section 2.We continue into 4 with the notion of rigidity, and after several elementaryresults concerning levels we arrive as the definition of level-compactness. Thisis again a definition which the reader should take due time to become familiarwith, as it is not only the main focus of the following sections, but possibly2he most far-reaching idea in the entire text.Following this, we work through several properties of level-compactness in5, including (though not limited to) many elementary topological propertieswhich may be reformulated in this context. For example, one sees that suf-ficiently “large” closed subsets of a level-compact set are level-compact, andthat the product of level-compact spaces is level-compact. If nothing else, thissection should allow the reader to become accustomed to working with all ofthe new definitions.Finally, in Section 7 we discuss ways in which the theory developed in thistext may be applied or further generalised. Indeed, although the author’s mainmotivation for studying level structures comes from compactness problems inhigher dimensional number theory, the notion of a level structure has manypossible applications which are not at all related to compactness. Considerthe following, for instance. Example.
Let G “ Z and let X “ t x u be a one-element group. For γ P Z , wedefine a collection of subsets of G as follows. For γ ě G t x u ,γ “ t u ,and for γ “ ´ n with n a positive integer we set G t x u ,γ “ t , , . . . , n ´ u .In the language of this paper, this defines a level structure for G over X ofelevation 1. This level structure defines a map ´ lv : t subsets of G u Ñ Z , which assigns to a subset S of G the length of the longest chain of an arithmeticprogression of the form t a, a ` , a ` , . . . , a ` r u contained in S . Conjecture.
Let k be an integer, and let P k “ t p prime : p ě k u be the set ofall primes ě k . Then ´ lv p P k q “ k .The above conjecture is a reformulation of the familiar Twin Prime Conjec-ture in the language of level structures. Note that we make no claims of beingable to resolve this conjecture - it merely serves as an example that the frame-work of this paper is not restricted only to the confines of higher dimensionallocal fields. Acknowledgements.
I would like to thank Ivan Fesenko for his manycomments on previous drafts of the current text, as well as various shorterworks that were eventually incorporated here. I am also grateful to the manypeople with whom I have discussed this work during its various stages of com-pletion - in particular Kyu-Hwan Lee, Sergey Oblezin, and Tom Oliver.3
Motivation from higher dimensional num-ber theory
We begin with a few motivating examples from higher dimensional numbertheory which illustrate the usefulness of the general constructions in this paper.First of all, recall the inductive definition of an n -dimensional local field. Definition 2.1. An n -dimensional local field F is defined inductively as fol-lows. If n “ then we take F to be a local field (i.e. either complete discretevaluation field with finite residue field, or an archimedean field F “ R or F “ C ). For n ą we then say that F is an n -dimensional local field if it is acomplete discrete valuation field whose residue field F is an p n ´ q -dimensionallocal field. Finally, we define a -dimensional local field to be a finite field. We will use the following indexing for residue fields. If F is an n -dimensionallocal field, we write F n ´ for the first residue field F “ O F { M F , F n ´ for thesecond residue field O F { M F , and so on. With this convention, the field F i isan i -dimensional local field.Recall that a system of local parameters for F is a collection of elements t , . . . , t n P O F such that the residue of t i generates the maximal ideal of O F i . Definition 2.2.
Let F be an n -dimensional local field. The rank n ring ofintegers of F is the subring O F of the ring of integers O F consisting of theelements x P O F which remain integral under each of the residue maps O F i Ñ F i ´ for ď i ď n . For n ą
1, an n -dimensional field F can be endowed with a natural topol-ogy by lifting the topology of the 1-dimensional residue field F through thesuccessive chain of residue fields. Under this topology (or with any of theother “natural” topologies one may consider), F is not locally compact, andso in particular there is no real-valued Haar measure on F . However, Fesenkonoticed that by relaxing various conditions, it becomes possible to define ameasure on such higher dimensional fields. Theorem 2.3.
Let F be an n -dimensional local field with local parameters t , . . . , t n . There exists a finitely additive, translation-invariant measure onthe ring of subsets of F generated by the sets α ` t i ¨ ¨ ¨ t i n n O F with α P F , i , . . . , i n P Z which takes values in the field R pp X qq ¨ ¨ ¨ pp X n qq . roof. See [F03], [F05].
Example.
Consider the two-dimensional field F “ Q p pp t qq . In this case we havelocal parameters t “ p and t “ t , and we have O F “ Z p ` t Q p rr t ss . Theunique Fesenko measure µ on F subject to the condition µ p O F q “ µ p α ` p i t j O F q “ p ´ i X j . This measure is countably additive except in veryspecific cases - see [F03] for details. Remark.
Let F “ Q p pp t qq as in the above example, and let π : Q p rr t ss Ñ Q p be the residue map. If U is a measurable subset of Q p , the set t j π ´ p U q ismeasurable in F , and satisfies µ p t j π ´ p U qq “ X j µ p p U q , where µ p denotes theHaar measure on Q p normalised so that µ p Z p q “ G is (roughly) equivalentto G being locally compact, Theorem 2.3 thus suggests that a higher dimen-sional local field is “not far” from being locally compact, in a sense which isto be made precise. This is further supported by the following. Proposition 2.4.
Let F be a two-dimensional nonarchimedean local field withparameters t , t . Then every covering of t i t j O F by sets of the form α ` t k t j O F admits a finite subcover.Proof. Assume otherwise, i.e. there is such a cover p V m q m P M which admitsno finite subcover. Since t i t j O F { t i ` t j O F » O F { t O F is finite, there is θ P t i t j O F with θ ` t i ` t j O F not contained in a finite union of the V m (sinceotherwise we would have a finite subcover).Similarly, there are θ , . . . , θ n P t i t j O F such that α n ` t i ` n ` t j O F “ θ ` θ t ` ¨ ¨ ¨ ` θ n t n ` t i ` n ` t j O F is not covered by a finite union of V m . But since F is complete, α “ lim n Ñ8 α n belongs to some V (cid:96) .Now, V (cid:96) “ β ` t r t j O F for some β P F , r P Z . Furthermore, α P V (cid:96) and α P A n “ α n ` t i ` n ` t j O F for all n ě
0. It is known from [F03] that suchsets are closed under finite intersection, and so we have V (cid:96) X A n “ V (cid:96) or A n . If V (cid:96) X A n “ A n for every n , we have β ` t r t j O F Ă Ş A n “ t j ` O F , where O F isthe rank one ring of integers of F , which is clearly impossible. We thus have A n Ă V (cid:96) for n large enough. However, we have previously concluded that no A n can be covered by a finite number of the V m , and so we have reached thedesired contradiction.In other words, F behaves like a locally compact space when we take coverswhich are of a “similar size”. We are thus prompted to look for a class of groups5ore general than those which are locally compact where one can performharmonic analysis using Fesenko-type measures. This motivates the definitionof a level structure. We now come to the most important definition of the entire text.
Definition 3.1.
Let X be a locally compact topological group, and let e ě be an integer. A group G is levelled over X (with elevation e ) if there is acollection L of subsets of G satisfying the following conditions:(1) Each element of L contains the identity element e G of G .(2) L indexed by U p q ˆ Z e , where U p q is a basis of neighbourhoods of theidentity in X and Z e is lexicographically ordered from the right.(3) For any U, V P U p q with V Ă U , if G V,γ , G
U,δ P L with γ ď δ then G V,γ X G U,δ “ G V,δ .(4) For any fixed γ P Z e , G U,γ Y G V,γ “ G U Y V,γ and G U,γ X G V,γ “ G U X V,γ .The collection L is called a level structure.Remark. If U “ V , condition (3) simply says that G U,δ Ă G U,γ for γ ď δ . Forfurther discussion on the generalities of this definition, see Section 7.Before we give examples, let us briefly discuss the importance of the con-ditions (1) to (4) in the definition above. The first two conditions mean thatwe are defining a local lifting of a basis of neighbourhoods at the identity ofthe base X . Moreover, by (2) this lifting is made up of a “continuous” part(coming from U p q ) and a “discrete” part (coming from Z e ). The final twoconditions then describe how the continuous and discrete parts of the structureshould interact; (4) says that on any given discrete “level” the local behaviourof G should mimic the behaviour of X , while (3) says that the discrete com-ponent gives rise to a notion of relative size which respects the idea of “size”encapsulated by the notion of subsets. Examples. (1) Any locally compact group G is levelled over itself with elevation0. In this case L “ U p q .(2) An n -dimensional nonarchimedean local field F is levelled over the one-element group with elevation n , where L consists of all principal fractionalideals of the rank- n ring of integers O F .63) Let F be an n -dimensional local field (which may now be archimedean)and F is its p n ´ q st residue field. Then F is levelled over F with elevation n ´
1. If F and F have the same characteristic, so that F is isomorphic to F pp t qq . . . pp t n qq , L consists of sets of the form t i . . . t i n n B p , r q ` n ÿ j “ t i j ` j t i j ` j ` . . . t i n n F pp t qq . . . pp t j ´ qqrr t j ss , where B p , r q is the open ball of radius r in F . In the mixed characteristic case,we associate to the pair p t i O F , p i , . . . , i n qq P U p qˆ Z n ´ the set t i . . . t i n n O F Ă F . In the nonarchimedean case we may note that the neighbourhoods of theidentity are themselves indexed by the totally ordered group Z ; doing so re-covers the previous example for such fields.(4) Since the form taken by elements of L in the previous example may lookquite complicated, we give a concrete example in dimension 4 to illustrate thegeneral phenomenon. Let F “ Q p pp t qqpp t qqpp t qq , so that F “ Q p . The openballs in F are then simply the fractional ideals p i Z p for i P Z . Elements of L are thus of the form p i t i t i t i Z p ` t i ` t i t i Q p rr t ss ` t i ` t i Q p pp t qqrr t ss ` t i ` Q p pp t qqpp t qqrr t ss . Note that this is exactly the set p i t i t i t i O F . Lemma 3.2.
Let G be levelled over X with elevation e . The set L is closedunder finite intersection.Proof. Let G U,γ , G
V,δ P L , and assume without loss of generality that γ ď δ .In this case, G V,γ X G V,δ “ G V,δ by condition (3), hence G U,γ X G V,δ “ G U,γ X G V,γ X G V,δ “ G U X V,γ X G V,δ by condition (4). But U X V P U p q and is asubset of V , hence by (3) we have G U X V,γ X G V,δ “ G U X V,δ P L .By the above Lemma, the collection L is a basis of neighbourhoods of theidentity for a topology on G . Definition 3.3.
We equip G with the level topology as follows. We take L asa basis of neighbourhoods of the identity, and then extend to other points of G by insisting that multiplication by any fixed element be continuous. If G , as in the first example, is a locally compact group viewed as beinglevelled over itself with e “
0, this is just the original topology on G . On the7ther hand, if G is (the additive group of) a two-dimensional local field as inthe third example, the level topology on G is not the usual two-dimensionaltopology as defined (for example) in [MZ95] - in this topology elements of L are closed but not open, for instance. Definition 3.4.
An element of G L “ t gH : g P G, H P L u is called a distin-guished set. We also allow the empty set to be distinguished.Example. The distinguished subsets of a two-dimensional local field F as de-fined by Fesenko in [F03] are exactly the distinguished sets of the elevation 1level structure of F over its residue field, namely those of the form α ` t i t j O F .The following result shows that the level topology is equivalent to thetopology generated by the distinguished sets. Proposition 3.5. G L Y tHu is closed under finite intersection.Proof.
We want to consider the intersection of gG U,γ and hG V,δ with g, h P G and G U,γ , G
V,δ P L . If the intersection is empty we are done, so assumeotherwise, so that there is an element α ´ P G which is contained in theintersection. Translating by α then implies that e G is contained in both αgG U,γ and αhG
V,δ . By continuity, both of these are basic open neighbour-hoods of e G in the level topology, and hence by the above Lemma so istheir intersection: αgG U,γ X αhG V,δ “ G W,β . Translating back then gives gG U,γ X hG V,δ “ α ´ G W,β .The following definition won’t be of immediate interest to us, but will be auseful tool to have, for example, if one wishes to construct an invariant measureon groups with level structure following [F03] and [W18].
Definition 3.6.
The ring of ddd-sets of G with respect to the level structure L is the minimal ring of sets containing G L . We now arrive at the second most important definition of this text.
Definition 3.7.
For G U,γ P L , we define its level lv p G U,γ q “ max t δ P Z e : G U,γ Ă G U,δ u . We then put lv p gG U,γ q “ lv p G U,γ q for any g P G . For a general subset S Ă G ,the level of S (if it exists) is the minimal level of any subset of S of the form gG U,γ for g P G , U P U p q , γ P Z e . We write lv p S q for the level of S . emark. Note that we in fact havelv p G U,γ q “ max t δ P Z e : G U,γ Ă G U,δ u “ max t δ P Z e : G U,γ “ G U,δ u . One would like to simply define the level of gG U,γ to be γ , but since we havenot ruled out the possibility that, say, G U,γ “ G U,δ for γ ‰ δ , this would not bea consistent definition. The advantage of allowing such ”degeneracy” is thatone may define the induced level structure on a subgroup (which we will do inSection 6) with no additional difficulty.Clearly we have the equalitylv p S q “ min t lv p S q : S Ă S has a level u . This leads to the following first observation concerning levels.
Lemma 3.8. If lv p A q ă lv p B q for two subsets A and B of G then it cannotbe the case that A Ă B . Moreover, there can be no g P G such that A Ă gB .Proof. If lv p A q ă lv p B q then there is γ P Z e such that A contains a distin-guished set gG U,γ but there is no δ ď γ with hG V,δ Ă B for any choice of h and V . In particular, there is at least one element of gG U,γ which is containedin A but not in B . The second statement follows from the fact that level isinvariant under the action of G by translation. Remark.
We may interpret level as being related to the size of a subset, witha higher level indicating a smaller size. Lemma 3.8 is in agreement with thisinterpretation.
Example.
Let F be a two-dimensional nonarchimedean local field with localparameters t , t , and let O F and O F be the rank-one and rank-two rings ofintegers of F . If S Ă F is a finite set then S does not have a level. For i, j P Z the set t i t j O F is a distinguished set of level j , while the set t j O F is anon-distinguished set of level j .One of the main purposes of this paper is to introduce the notion of “level-compactness”, for which the definition of level above will be very important.Using the intuitive interpretation of the level of a subset of G as indicatingits relative size, level-compactness will be equivalent to saying that any opencover by “large enough” sets has a finite subcover (this will be made preciselater, see Definition 4.5). However, our current definition of level is not quitestrong enough for this, as the next example illustrates.9 xample. Let F be a two-dimensional nonarchimedean local field with t , t , O F as in the previous example. For all α, β P F the set A α,β “ p α ` t O F q Yp β ` O F q has level 0. We have O F Ă Ť α A α,β , where the union is over acomplete (and infinite) set of representatives of O F { t O F in O F . Taking any β R O F , we obtain in this way an open cover of O F by sets of level 0 with nofinite subcover.The problem in the above example is that, although the open cover we con-struct is essentially a cover of sets of level 1 (since the p β ` O F q componentscontribute nothing, being disjoint from O F ), the presence of this extra compo-nent formally lowers the level even though it does not contribute. This showsthat in order to make reasonable progress towards any kind of “compactness”,we must consider only those open covers by sets which are “uniformly large”,which motivates the following definition. Definition 3.9.
A subset S Ă G has uniform level γ if lv p S q “ γ and forevery point s P S there is a distinguished set D s of level γ with s P D s and D s Ă S . It is immediate from the definition that any subset with uniform level isopen. The sets A α,β “ p α ` t O F q Y p β ` O F q in the previous example are notuniformly of level 0, since α (for example) is not contained in any distinguishedset of level 0 lying inside A α,β . This additional condition is enough to eliminatesuch pathological examples; Proposition 2.4 is simply the statement that O F is in fact compact with respect to open covers of uniform level 0, formulatedin more familiar terminology. We will see in the following section (Proposition4.6) the same result stated instead in the language of level structures. In the lead up to the third and final “most important definition”, we firstexamine a few more properties of the level of a subset. In particular, wewould like this notion to be well behaved with respect to certain set theoreticoperations. In order to achieve this, it is convenient to include the followingrigidity assumption, which will also be very important in the next section whenwe look at properties of level-compactness.10 efinition 4.1.
A group G levelled over X with elevation e is rigid if it sat-isfies the following condition: for any γ P Z e , if G contains at least one subsetof level γ then lv p G U,γ q “ γ for all U P U p q .Remark. Note that we always have lv p G U,γ q ě γ from the definition of level.The level of a subset was defined in a way as to allow for a consistent definition,for instance, in the case when there are no subsets having level ď δ for some δ by setting the level of all “large enough” distinguished sets to be the maximumpossible. In theory, this definition could also allow pathological cases wherethe level of a distinguished set no longer matches the intuitive notion of its“size”; the notion of rigidity is intended to exclude these possible cases.We now begin our investigation of the interaction between levels and setoperations with the following preliminary Lemma. Lemma 4.2.
Let G be rigid, let S Ă G with lv p S q “ γ , and for any δ ě γ suppose that G contains at least one subset of level δ . Then for every δ ě γ there is a subset S Ă S with lv p S q “ δ . If, moreover, S has uniform level γ ,for each s P S there is a distinguished set D s,δ of level δ with s P D s,δ Ă S .Proof. By rigidity and the definition of level, S contains some distinguishedset gG U,γ of level γ . By property (3) in the definition of level structure wehave gG U,δ Ă gG U,γ Ă S , and by rigidity we have lv p gG U,δ q “ δ . This provesthe first assertion.Now suppose that S has uniform level γ , and take any s P S . Then thereis some g P G such that s P gG U,γ Ă S . On the other hand, s P sG U,δ , andfrom the proof of Proposition 3.5 we see that, if δ ě γ , gG U,γ X sG U,δ is adistinguished set D s,δ which has level δ , and s P D s,δ Ă S by construction. Proposition 4.3. If A and B are both open subsets of G which have a leveland A X B ‰ H has a level, lv p A X B q ě max t lv p A q , lv p B qu . Furthermore, if A and B both have uniform level then so does A X B , and the inequality is infact an equality.Proof. If A and B are both distinguished sets then it follows from the proofsof Lemma 3.2 and Proposition 3.5 that lv p A X B q “ max t lv p A q , lv p B qu . In thegeneral case, let x P A X B . Let D A be a distinguished open neighbourhoodof x in A , and let D B be a distinguished open neighbourhood of x in B . ByProposition 3.5 D “ D A X D B Ă A X B is a distinguished set, and by the firstline it has level max t lv p D A q , lv p D B qu ě max t lv p A q , lv p B qu .11t thus remains to show that A X B cannot contain a distinguished set ofany smaller level. Indeed, if it did contain such a set D of level less thanmax t lv p A q , lv p B qu , then both A and B would also contain D , which is impos-sible since at least one of them has strictly larger level.For the final assertion, note that if A and B have uniform level we maychoose D A and D B to have levels lv p D A q “ lv p A q and lv p D B q “ lv p B q , andthen from what we have already proved it follows thatlv p D q “ max t lv p D A q , lv p D B qu “ max t lv p A q , lv p B qu ď lv p A X B q . But since D Ă A X B we have lv p D q ě lv p A X B q by definition, hence we haveequality. Remark.
Note that this Proposition not prove that lv p A X B q exists in thenon-uniform case, since A X B may still contain distinguished sets of level γ with max t lv p A q , lv p B qu ă γ ă max t lv p D A q , lv p D B qu . In the case that theelevation e “
1, this is indeed enough to prove that the level exists, since theremust be a minimal such γ , but for e ą Z e do not necessarily have extrema. Corollary 4.4.
Let A and B are subsets of G which have a level such that A X B has a level. If int A X int B ‰ H then lv p A X B q ě max t lv p A q , lv p B qu .(Here int S denotes the interior of a subset S Ă G .)Proof. We can apply Proposition 4.3 to int A and int B to find a distinguishedset D inside A X B of level lv p D q ě max t lv p int A q , lv p int B qu ě max t lv p A q , lv p B qu .The same argument in the second paragraph of the proof of Proposition 4.3then shows that A X B cannot contain a distinguished set whose level is lowerthan the latter. Remark.
Unlike with intersections, the level of a subset is not at all well be-haved under unions, and being of uniform level behaves even worse. (Indeed,the notion of uniform level was defined because of the problems that unionsmay cause.) In certain specific cases it is possible to slightly control the be-haviour of unions, but in general it is so wild that hardly anything may besaid at all.We now come to the fundamental notion of level-compactness.12 efinition 4.5.
Let G be a group with level structure, and let γ P Z e . A subset S Ă G is called γ -compact if every open cover (in the level topology) of S bysets of uniform level γ has a finite subcover. We will call S level-compact ifthere is some γ P Z e such that S is γ -compact.Remark. As was hinted previously, it is important that each set in the cover has uniform level γ . Note that although we refer to open covers in the definition(so that the reader may immediately see the connection with compactness), wemay in fact omit the word ”open” since we saw earlier that any set of uniformlevel is necessarily open.Possibly the most important example to keep in mind is the following,which is the main motivating example for the definition of level-compactness(and hence for this entire paper). Compare also with Proposition 2.4. Proposition 4.6.
Let F be a d -dimensional nonarchimedean local field withparameters t , . . . , t d . If F is given the level structure of elevation d ´ overit’s -dimensional residue field then the subset t i ¨ ¨ ¨ t i d d O F is γ -compact with γ “ p i , . . . , i d q .Proof. Assume otherwise, i.e. there is a γ -cover p V m q m P M which admits nofinite subcover. Since t i ¨ ¨ ¨ t i d d O F { t i ` ¨ ¨ ¨ t i d d O F » O F { t O F is finite, there is θ P t i ¨ ¨ ¨ t i d d O F with θ ` t i ` ¨ ¨ ¨ t i d d O F not contained in a finite union of the V m (since otherwise we would have a finite subcover).Similarly, there are θ , . . . , θ n P t i ¨ ¨ ¨ t i d d O F such that α n ` t i ` n ` ¨ ¨ ¨ t i d d O F “ θ ` θ t ` ¨ ¨ ¨ ` θ n t n ` t i ` n ` ¨ ¨ ¨ t i d d O F is not covered by a finite union of V m .But since F is complete, α “ lim n Ñ8 α n belongs to some V (cid:96) .Now, since V (cid:96) has uniform level γ , there is a distinguished set D withlv p D q “ γ and α P D Ă V (cid:96) . On the other hand, we also have α P A n “ α n ` t i ` n ` ¨ ¨ ¨ t i d d O F for n ě
0. (Note that A n has uniform level γ .) ByLemma ?? , the intersection of two distinguished sets is either empty or equalto one of them, and so we have D X A n “ D or A n . If D X A n “ A n for every n , we have D Ă Ş A n “ t i ` . . . t i d d O p d ´ q F , where O p d ´ q F is the rank p d ´ q ring of integers of F . But then we have γ “ lv p D q ě lv p Ş A n q ą γ , which isa contradiction, hence we have A n Ă D Ă V (cid:96) for n large enough. But we havepreviously concluded that no A n can be covered by a finite number of the V m ,and so we have reached the desired contradiction.13 emark. It is essential that the elements of the cover all have the same levelas t i ¨ ¨ ¨ t i d d O F in this Proposition. Indeed, O F “ Ť α ` t O F where α runsthrough an (infinite!) set of representatives for O F { t O F , and since the unionis disjoint there can be no finite subcover. It is equally important that theelements of the cover have uniform level, as we saw in an earlier example. Remark.
The proof of Proposition 4.6 uses the fact that F is complete in anessential way. As some of the consequences of completeness will be cruciallater, it is worthwhile to ask if the completeness property (or perhaps a weakeralternative which still works for the above proof) can be restated purely interms of the level structure. Definition 4.7.
A group G levelled over a locally compact group X is calledlocally level-compact if for every g P G there is some γ P Z e such that g has a γ -compact neighbourhood.Example. An n -dimensional local field F is locally level-compact over its localresidue field. In this section we will study various elementary topological properties of level-compactness. Since we may ask the question “can we replace compactness bylevel-compactness” in almost every definition and theorem concerning com-pactness, we will of course not cover all possibilities here. Instead we focus asmuch as possible on results that are useful from the point of view of poten-tial applications to areas of higher dimensional number theory and arithmeticgeometry.
Proposition 5.1.
Let G be levelled over X with elevation e with the leveltopology, suppose that G is rigid, and suppose that G contains at least one setof level γ . Then the following properties hold.(1) If S Ă G is γ -compact then S is also δ -compact for all δ ď γ .(2) If S Ă G is γ -compact and C is a closed subset of S such that S z C isuniformly of level δ ď γ then C is δ -compact. roof. If G has no subsets of level δ then the result trivially holds. Otherwise,let S “ Ť m U m be a uniform open δ -cover of S . We may assume without loss ofgenerality that each U m is a basic open set of level δ . By rigidity, it follows fromLemma 4.2 that we may write U m “ Ť α P U m V α , where V α is a distinguished setof level γ containing α . Then S “ Ť m,α P U m V α is a uniform open γ -cover of S ,and by γ -compactness it has a finite subcover. It thus follows that for each V α in this finite subcover we may take some U m containing it, and doing so givesa finite δ -subcover of S .The proof of (2) follows the same reasoning as the proof that closed subsetsof compact spaces are compact. Indeed, first note that we know from (1) that S is also δ -compact. If we take any uniform open δ -cover C “ Ť m U m of C ,then G “ p G z C q Y Ť m U m is a uniform open δ -cover of S , hence it has a finitesubcover, and this gives us also a finite subcover of C . Remark.
Property (2) may be thought of as saying that sufficiently small closedsubsets of γ -compact sets are level-compact.It is important that G contains a set of level γ in the above Proposition.Indeed, if there are no sets of level γ then every subset of G is trivially γ -compact, in which case the result may not be true for some δ ă γ where δ -covers exist.We can actually improve property (1) of Proposition 5.1 quite substantially.In order to do this, we first note that we may classify subsets that have nolevel into three distinct categories. Definition 5.2.
A subset A Ă G is of type S is there exists no distinguishedsubset D with D Ă A .Remark. The subsets of type S should be thought of as those which are “toosmall” to have a level. In all examples we have given so far, finite sets havealways been of type S . Definition 5.3.
A subset A Ă G is of type L if for every γ P Z e there is adistinguished set D γ of level δ ď γ with D γ Ă L .Remark. Subsets of type L are the opposite extreme to those of type S ; theyare the subsets which are “too large” to have a level. If the map G L Ñ Z e which sends every distinguished set to its level is surjective, the whole group G is always of type L , although there may be more subsets of this type.15 efinition 5.4. Let A be a subset of G with no level. We say that A is oftype E if it is not of type S or of type L .Remark. The subsets of type E are “exceptional” subsets. If G is of elevation e ď E . For e “ e “ Z which is bounded below has a minimum. When thelevel map is not surjective, many more sets of type E may appear.We noted earlier that level-compactness should be thought of as beingcompactness with respect to open covers by “sufficiently large” sets. Theintuition from the above three definitions suggests that we should also attemptto allow sets of type L in our covers. Of course, due to the same issues whichappeared previously, we are guided towards the following subcollection of setsof type L . Definition 5.5.
Let A Ă G be a subset of type L . For γ P Z e we say that A is γ -uniform if A has an open covering A “ Ť U i with each U i Ă A uniformlyof level γ . The refinement of Proposition 5.1 is the following.
Proposition 5.6.
Let G be levelled over X with elevation e , and suppose that G is rigid. Let A Ă G be γ -compact for some γ P Z e such that G contains aset of level γ , and let A Ă Ť U i be an open cover of A . Suppose that for each i we have either (i) U i is uniformly of level γ i ď γ , or (ii) U i is of type L andis γ -uniform. Then Ť U i has a finite subcover.Proof. If U i is of the form (i), we saw in the proof of Proposition 5.1 that wemay cover U i by distinguished sets t V p i q α u of uniform level γ . If U i is of theform (ii) then from the definition of γ -uniformity we also an open coveringof U i by sets t V p i q α u of uniform level γ . This gives us a uniform open γ -cover A Ă Ť i,α V p i q α , which has a finite subcover by γ -compactness. For each V p i q α in this subcover, choosing one of the U i Ą V p i q α gives the required subcover of Ť U i . Remark.
One may similarly define the notion of γ -uniformity for sets of type E (note that for sets of type S the condition can never be satisfied), andfurther refine Proposition 5.6 to include these sets as well. These exceptionalsets of type E seem quite mysterious, and it may be interesting to study their16roperties. There seems to be some link between the presence of exceptionalsets and how badly G can behave.We end this section with one final result on unions. Proposition 5.7.
Let G be rigid. If K Ă G is γ -compact and K Ă G is δ -compact with γ ě lv p K q , δ ě lv p K q then K “ K Y K is η -compact forsome η ě lv p K q , if this level exists.Proof. Let η “ min t γ, δ u . Then any uniform open η -cover of K is an open coverof each of the η -compact sets K and K , which both have finite subcovers.Taking the union of these subcovers then gives a finite subcover of K , hence K is η -compact.Since K contains both K and K ,lv p K q ď min t lv p K q , lv p K qu ď min t γ, δ u “ η, if the level of K exists. When speaking about compactness one also expects to consider products ofspaces. If G is levelled over X and H is levelled over Y with the same elevation e , then G ˆ H is naturally levelled over the product space X ˆ Y with elevation e via p G ˆ H q p U ˆ V q ,γ “ G U,γ ˆ H V,γ . The level topology on the product coincideswith the product topology.Since we will want to apply the earlier results of this section, the followingeasy Lemma is important.
Lemma 5.8.
Let G be levelled over X and H be levelled over Y , both withelevation e . If G and H are rigid then G ˆ H is rigid over X ˆ Y .Proof. We need to show that lv pp G ˆ H q p U ˆ V q ,γ q “ γ . However, this followsfrom the definitions, since if p G ˆ H q p U ˆ V q ,γ is contained in some other p G ˆ H q p U ˆ V q ,δ with δ ą γ then (for example) G U,γ Ă G U,δ , and so lv p G U,γ q ě δ ą γ ,which contradicts rigidity of G . Remark.
Since the direct product of two spaces is commutative, the proof ofLemma 5.8 actually shows that G ˆ H is rigid as long as at least one of G and H is rigid. 17 roposition 5.9. If G is γ -compact and rigid over X , and if H is γ -compactand rigid over Y , the product G ˆ H is γ -compact over X ˆ Y , where γ “ min t γ , γ u . Proof.
We know from Proposition 5.1 that G and H are both γ -compact. Thusif we take any uniform open γ -cover G ˆ H “ Ť m p U m ˆ V m q , we know that G “ Ť m U m has a finite subcover G “ Ť m P M U m and H “ Ť m V m has a finitesubcover H “ Ť m P M V m . It then follows that Ť m P M Y M U m ˆ V m is a finitesubcover of G ˆ H . Remark.
This, along with the following results, is also true for the appropriatelevel-compact subsets of G and H . However, for the sake of brevity we willformulate the statements only in terms of the full group G , and so on.The definitions also work for infinite products, but for the analogue ofTychonoff’s theorem we will need to do a little more work. Lemma 5.10.
Let G “ ś i P I G i with each G i γ -compact over a space X i . Thenany open cover of G by sets of the form π ´ j p U q with U Ă G j open of uniformlevel γ has a finite subcover. (Here π j : G Ñ G j is the projection map.)Proof. Let U be such a cover, and let U j be the collection of U Ă G j such that π ´ j p U q P U . Suppose that there is no j such that U j covers G j , so that foreach j we find g j P G j with g j not in the union of all elements of U j . But then p g i q i P I is not contained in any element of U , which is not possible since this isa cover of G .We can thus find some j such that U j is a cover of G j , and by γ -compactnesswe can find a finite subcover G j Ă Ť nk “ U k , hence we have a finite subcover G “ Ť nk “ π ´ j p U k q of U .Now we may prove a modification of the Alexander Subbase Theorem. Forthis, the set theory enthusiasts will note that we must assume the axiom ofchoice, since the proof requires the use of Zorn’s Lemma. Lemma 5.11.
Let G be levelled over X , and let V be a subbase for the leveltopology on G . If every collection of sets of uniform level γ from V whichcovers G has a finite subcover then G is γ -compact. roof. Suppose that every such cover has a finite subcover, but G is not γ -compact. Then the collection F of all open γ -covers of G with no finite sub-cover is nonempty and is partially ordered by inclusion. Let t E α u be anytotally ordered subset of F , and put E “ Ť α E α .By definition, E is a uniform open γ -cover of G . For any finite collection U , . . . U n of elements of the cover, we have U j P E α j for some α j , and sincethe E α are totally ordered there is some E α containing all of them. It followsthat E has no finite subcover, and so E is an upper bound for t E α u , thus byZorn’s Lemma there is a maximal element M of F .Let S “ V X M , and suppose there is g P G that is not inside any elementof S . Since M is a cover of G there is some U P M with g P U , and since V is a subbase for the topology there are V , . . . , V n P V with g P Ş ni “ V i Ă U .By assumption none of the V i are in M , so by maximality M Y t V i u containsa finite subcover G “ V i Y U i , where U i is a finite union of sets in M . Then U Y Ť ni “ U i Ą p Ş ni “ V i q Y p Ť ni “ U i q Ą Ş ni “ p V i Y U i q Ą G . But this is a finitecover of G by elements of M , which contradicts M having no finite subcover.It thus follows that S is a cover of G , and since S Ă V it has a finite subcoverby assumption. But S is also contained in M , and so it cannot have a finitesubcover, hence the collection F must he empty, i.e. G is γ -compact.This allows us to deduce the extension of Tychonoff’s Theorem. Theorem 5.12. If G “ ś i P I G i with each G i γ -compact over X i then G is γ -compact over X “ ś i P I X i .Proof. The collection t π ´ j p U j qu is a subbase for the product topology. ByLemma 5.10 any subcollection of this set which covers G has a finite subcover,and then the Alexander Subbase Theorem shows that G is γ -compact. Remark.
If all of the G i are rigid, we don’t need that all of the G i are γ -compact with the same γ . As long as there is a minimum γ min across all G i ,this γ min will be the (maximal) γ that works for the product.For a collection of spaces t A i u with subspaces B i Ă A i , recall that therestricted product ś A i of the A i with respect to B i consists of sequences p a i q i P ś A i such that a i P B i for all but finitely many i . It follows fromTheorem 5.12 that the restricted product of a collection of rigid locally level-compact groups t A i u with respect to a system of γ -compact subgroups t B i u isagain locally level-compact. 19 orollary 5.13. Let k be either a number field or the function field of aproper, smooth, connected curve over a finite field, and let S be an arithmeticsurface which is a proper regular model of a smooth, projective, geometricallyirreducible curve X { k . Then groups A S of geometric adeles and A S of ana-lytic adeles, associated to S and a given set S of all fibres and finitely manyhorizontal curves on S , are locally level-compact.Proof. For a two-dimensional nonarchimedean local field, we saw in Proposi-tion 4.6 that t i t j O F is j -compact, and since t j O F is contained in all of thedistinguished sets t i t j ´ O F and disjoint from any of their F -translates it istrivially p j ´ q -compact. For the archimedean components, the j - compact-ness of the subsets α ` Ct j ` t j ` F rr t ss for C Ă F compact follows immediatelyfrom the compactness of C , and so the archimedean fields F pp t qq are also lo-cally level compact. The adelic spaces are then restricted products of locallylevel-compact spaces with respect to level-compact subgroups (see [F10]). Remark.
The space A S was first considered by Parshin, Beilinson and others,while A S was first defined by Fesenko (see [F10] for the definitions, whichare too lengthy to reproduce here). In particular when S “ E is the minimalregular model of an elliptic curve over a number field, the adelic spaces A E and A E are related to several important open problems in number theory andarithmetic geometry, including the Riemann Hypothesis and the Birch andSwinnerton-Dyer Conjecture. For details see [F10], [F17]. We now consider connections with discreteness. It follows immediately fromthe definitions that a topological space which is both discrete and compactmust be finite. Here we discuss what happens when we replace ”compact”with ” γ -compact”. Remark.
If every element of L is equal to t e G u , discreteness essentially forcesthe elevation e to be equal to 0, since all distinguished sets will have the samelevel. However, if there are non-singleton distinguished sets, this may not bethe case, as the following example illustrates. Example.
Let X “ t x u be the single element group. We may endow G “ Z with an elevation e “ X via G t x u ,γ “ t u for γ ě t x u ,γ “ t , , . . . , n ´ u for γ “ ´ n with n a positive integer. The formersets all have level ´
1, while the latter have level ´ n .The level topology coincides with the discrete topology since t u P L , andin this case ´ lv p U q gives the length of the longest chain of consecutive integersin a subset U of G . In this example, we easily see that G has no subsets oftype S , and that every finite set has a level. Since G has elevation e “ E . We see that G is of type L , but we mayalso construct infinitely many examples of proper subsets of type L . One suchexample is the set t , , , , , , , , , , , . . . u obtained by adding consecutive chains of longer lengths each time ad infinitum . Remark.
Note that not all infinite subsets of G in the above example are oftype L : the set 2 Z of even integers, for example, has (uniform) level ´
1, whilethe set of all prime numbers has (non-uniform) level ´ Remark.
Since, as we have noted now on several occasions, the level of a set isintuitively related to a notion of “size”, introducing various level structures on Z may be of some interest in analytic number theory, as it gives an alternativeway of defining the “volume” of a set of primes. For example, in the intro-duction we stated the twin prime conjecture in terms of the level structure G x u ,γ “ G t x u ,γ , with G t x u ,γ the level structure in the previous example.Suppose G is levelled over X , and that S Ă G is a γ -compact, discrete sub-set. By the results of the previous section, distinguished sets are the smallestopen subsets of G , and so for every s P S there is a distinguished subset D s with D s X S “ t s u .Now, in general the distinguished sets D s may change wildly as s varies, orsimply all be of too high level for γ -compactness to come into play. However,if we have some control over these factors then we can indeed say something. Proposition 5.14.
Let G be levelled over X and let S Ă G be discrete and γ -compact. If there exists a δ ď γ such that for every s P S there is a distin-guished D s Ă G with lv p D s q “ δ and D s X S “ t s u then S is finite.Proof. We have a uniform δ open cover S Ă Ť s P S D s , and since δ ď γ S is δ -compact and so we have a finite subcover. However, since each s P S appearsin exactly one element of the cover (namely D s ), this implies that S must befinite. 21 orollary 5.15. If S is discrete and γ -compact over X for all γ P Z e , andfurthermore S can be completely disconnected by open subsets of some level δ ,then S is finite. Clearly if G is compact then it is γ -compact for all γ . However, it is not clearwhen (if at all) the converse holds. Obtaining results in this direction is almostequivalent to controlling the subsets of type L and type E . Proposition 5.16.
Suppose G is rigid and γ -compact for every γ P Z e , andthat the following conditions are satisfied:(i) G contains no open subsets of type E ;(ii) For all γ P Z e and for every open subset A Ă G of type L , A is γ -uniform.Then any open cover G “ Ť U i such that the set t lv p U i qu i has an upperbound in Z e has a finite subcover. (Here we allow the possibility that U i hasno level.)Proof. Essentially all that we have to do is describe the possible structures ofopen subsets of G . If U Ă G is open and does not have a level, it must be oftype L or of type E , since the only other possibility is that U is of type S andcontains no nonempty open set (which is clearly false). Since by assumptionno subset of type E is open, the open subsets of G either have a level or areof type L .Now, let G “ Ť U i be an open cover such that the levels of the U i arebounded above by γ P Z e . If U i is of type L , by (ii) we can cover U i by opensets t V i,m u which are uniformly of level γ , and if U i is distinguished then wecan do the same by Lemma 4.2. This gives a uniform open γ -cover G “ Ť V i,m ,from which we can take a finite subcover, and then take one U i containing eachelement of this subcover to obtain a finite subcover of t U i u . An alternative characterisation of compactness can be formulated in terms ofthe finite intersection property. Recall that a family t S i u of subsets of a topo-logical space has the finite intersection property if any intersection of finitelymany elements of the family is nonempty. It is well known that a topologicalspace is compact if and only if every family of closed subsets satisfying the22nite intersection property has nonempty intersection. Similarly, we have thefollowing. Lemma 5.17. G is γ -compact if and only if any collection t E i u of closed setswith G z E i uniformly of level γ satisfying the finite intersection property hasnonempty intersection.Proof. First let G be γ -compact, and suppose Ş E i “ H . Then G “ G zH “ G z Ş i E i “ Ť i p G z E i q is a uniform open γ -cover of G , and so there is a finitesubcover G “ Ť ni “ p G z E i q “ G z Ş ni “ E i . But the latter implies that Ş ni “ E i is empty, which is a contradiction.Now suppose that any collection t E i u of closed sets with G z E i uniformlyof level γ satisfying the finite intersection property has nonempty intersection.Let G “ Ť i U i be a uniform open γ -cover, and suppose there is no finitesubcover. Then for any finite subcollection U , . . . , U n we have Ş ni “ p G z U i q “ G zp Ť ni “ U i q ‰ H , and so the collection t G z U i u satisfies the finite intersectionproperty. We thus have H ‰ Ş i p G z U i q “ G zp Ť i U i q “ G z G “ H , which isclearly impossible. Let G be levelled over X , and let H be a subgroup of G . One can define alevel structure on H over X in a canonical way. Definition 6.1. If G has a level structure L over X , and H is a subgroup of G , then the collection L H “ t H U,γ “ H X G Uγ : G U,γ P L u of subsets of H iscalled the induced level structure. Lemma 6.2.
The collection L H is a level structure for H over X .Proof. Property (1) is satisfied since H is a subgroup of G , and property (2)holds by construction. Properties (3) and (4) follow from the associativity anddistributivity properties of union and intersection.It is clear from the definition that the level topology on H given by L H coincides with the topology induced on H as a subspace of G .Up until this point, it would have peen possible to restrict attention onlyto spaces where the level map G L Ñ Z e is surjective. However, if we wantinduces level structures to make sense in general it is important that we do23ot make this assumption. For example, if H is a subgroup of G of level δ , theinduced level structure on H contains no sets of level smaller than δ . Proposition 6.3. If G is rigid and (locally) level-compact and H is a closedsubgroup of G with the induced level structure such that G z H has a level in G then H is (locally) level-compact.Proof. This follows immediately from Proposition 5.1.
Corollary 6.4.
Any algebraic group G over a higher dimensional local field F which is closed in GL n p F q such that GL n p F qz G has a level is locally level-compact. In the case of algebraic subgroups, one may in fact consider an induced(partial) level structure over a more appropriate base than the original one.
Proposition 6.5.
Suppose G “ GL m p F q for an n -dimensional nonarchimedeanfield F , with the partial level structure over GL m p F q given by the distinguishedsubgroups K γ ,...,γ n . Let H be a subgroup of G defined by finitely many polyno-mial equations f , . . . , f k P O F rr X , . . . , X m ss , i.e. H “ tp g r,s q P G : f i pp g r,s qq “ , ď i ď k u , and let H be the subgroup of GL m p F q defined by the reductions ¯ f , . . . , ¯ f k P F rr X , . . . , X m ss . If all of the polynomials ¯ f i are separable (i.e. all the roots are simple), theassociation p U X H, γ q ÞÑ H X G U,γ defines a (partial) level structure for H over H whose distinguished sets coincide with those of the induced level structure.Proof. We must first check that the association is well-defined. In other words,if H X K α “ H X K β for α, β P Z we must make sure that H X K α,γ “ H X K β,γ for all γ P Z n ´ . Without loss of generality, we may assume that β ď α , sothat we have H X K α,γ Ă H X K β,γ .Let g “ p g r,s q P H X K β, . Then the reduction ¯ g “ p ¯ g r,s q P K β is a root ofall of the polynomials ¯ f i , and so we have ¯ g P H X K β “ H X K α . By Hensel’sLemma, there exists a unique g P K β, with ¯ g “ ¯ g and f i p g q “
0, andthere exists a unique g P K α, with ¯ g “ ¯ g and f i p g q “
0. Since f i p g q “ g forces g “ g . Furthermore, since K α, Ă K β, , we have g P K β, , and so uniqueness of g forces also g “ g . We thus have g “ g P X K α, and so we have the required equality for γ “
0. The result for general γ then follows from the fact that the map I m ` t γ . . . t γ n n M ÞÑ I m ` t γ M is anisomorphism K γ ,γ Ñ K γ , .The fact that the distinguished sets coincide with those of the inducedlevel structure is then immediate from the definition, and this implies thatconditions (1) and (2) in the definition of a level structure hold. It remains tocheck (3) and (4).Let U, V P U p q and γ ď δ P Z n ´ . We have G V X H,γ X G U X H,δ “ p H X G V,γ q X p H X G U,δ q “ H X G V,δ “ G V X H,δ , and so (3) is satisfied.Similarly, G U X H,γ Y G V X H,γ “ p H X G U,γ q Y p H X G V,γ q“ H X p G U,γ Y G V,γ q“ H X G U Y V,γ “ G p U Y V qX H,γ , and G U X H,γ X G V X H,γ “ p H X G U,γ q X p H X G V,γ q“ H X p G U,γ X G V,γ q“ H X G U X V,γ “ G p U X V qX H,γ , and so (4) is satisfied. Remark.
It seems to be a very common theme concerning groups with levelstructure that, while the topological and analytic properties of G should beessentially self-contained (since the information is bound to the level structure,which is in principle just a collection of subsets of G ), by choosing the correctbase X for the level structure we may see various analogies between G and X which makes certain properties of G appear more clearly.One may, for example, utilise the induced level structures to study theabove remark. If we consider G with the same level structure over two differentbases X and X , we may take the product to obtain a level structure for G ˆ G over X ˆ X . The induced level structure on the diagonal image of G inside G ˆ G will then be related to both X and X .25o end this section, we return briefly to the problem of degeneracy forinduced level structures. Consider the following example. Example.
Consider G “ Q p pp t qq with its usual level structure of elevation e “ Q p . We consider the induced level structure on H “ Q p . For G i,j “ p i t j Z p ` t j ` Q p rr t ss P L we have H i,j “ j ą H i, “ p i Z p , and H i,j “ Q p for j ă
0. One sees at once that the induced level topology coincides withthe p -adic topology, and that the level structure has essentially fallen awaycompletely. It this example it is indeed true that j -compactness for all j P Z implies compactness, since one only needs to check the j “ Proposition 6.6.
Let G be levelled over X with elevation e , and let L be thelevel structure. Then for any w P Z the collection L of subsets of G indexedby U p q ˆ Z e ` » U p q ˆ Z e ˆ Z given by G U,γ,j “ G for j ă w , G U,γ,w “ G U,γ ,and G U,γ,j “ t e G u for j ą w defines a level structure for G over X of elevation e ` .Proof. It follows immediately from the definitions and the fact that we orderlexicographically from the right.By iterating the above procedure we may inflate the elevation from e to e ` n for any positive integer n . While inflation may not produce anythingparticularly interesting, it does allow us to be precise about saying that thelevel structure on Q p induced from Q p pp t qq is essentially of elevation 0. It alsoallows us to take products of groups with different elevations, since we mayinflate everywhere to the maximal elevation among the components. We end with a few ideas that aren’t necessarily important for the overall themeof the text, but may nonetheless be interesting to think about.First, the reader will surely have noted that, we do not use anywhere thefact that the base space X of a group with level structure is locally compact.One may thus relax the requirements on the base X to the following much lessstringent conditions: X must be a pointed set, and the set must come equipped26ith a collection U p q of subsets which contain the distinguished point of X and satisfies the defining condition to be a “basis of neighbourhoods” at thispoint. (The set X itself need not be a topological space, it just needs to looklike one around the distinguished point.)One possibly important example is to take X to be a group levelled over asecond space X . Lemma 7.1.
Let X have level structure L X over X with elevation e , andlet G be levelled over X with elevation e and level structure L G indexed by L X ˆ Z e . Then G is levelled over X with elevation e ` e and level structure L G .Proof. To an open neighbourhood U of 1 in X and to p γ, δ q P Z e ` e with γ P Z e and δ P Z e we associate the distinguished set G U, p γ,δ q “ G X U,γ ,δ . Thefact that this defines a level structure immediately follows from the fact that L X and L G are both level structures. Remark.
In an upcoming paper, we will construct an invariant measure on acertain class of groups with level structure, and in this case we do use the factthat X is locally compact. This is one of the main reasons that the requirementwas written into the original definition.While considering generalities regarding the base space, it would also beinteresting to investigate also the special case where the elevation e “
1. Inthis case, the indexing group is simply Z (or, if we only take a partial levelstructure, a subset of this). In particular, we have the following elementaryLemma which may have some interesting consequences. Lemma 7.2.
Any sequence in Z which is bounded below has a minimum ele-ment. We have already noted this previously in passing, when we remarked thatno subsets of type E can exist in the case e “ Z : the sequence p´ n, q n ě is bounded below by p , ´ q since we order fromthe right, but it is strictly decreasing. This may have important consequencesregarding the difference between arithmetic geometry in dimension 2 ( e “ ě e ě G we canextract from the base. We noted earlier that, while various properties are27ntrinsic to G , a convenient choice of base can make these properties easier tostudy.One possibility in this direction (for which the author thanks Tom Oliverfor the suggestion!) is the following. Consider the two-dimensional local field F “ Q p pp t qq . We obtain the natural level structure of F over the residue field F “ Q p via the association p p i Z p , j q ÞÑ p i t j O F . Alternatively, we obtain thesame level structure on F over the field F p pp t qq via the association p t j F p rr t ss , i q ÞÑ p i t j O F . Depending on how much information may be “shared” between agroup and its base, it may thus be possible to obtain new analogies betweenthe two different local fields Q p and F p pp t qq . Alternatively, by going up onemore level to a three-dimensional field, it may be interesting to investigatepossible analogies between two-dimensional fields that arise in a similar way.The information contained within the base also has some ramifications forthe potential study of harmonic analysis. For example, for F an n -dimensionallocal field, it is possible to construct a finitely additive, translation-invariantmeasure on F (see [F03], [F05], [M10]) and on GL p F q (see [M08], [W18]). Ineach case, it was noted that the measure was (in an appropriate sense) a liftof the measure on the residue field.The initial expectation was that this should always be the case. Moreprecisely, one can conjecture that for a rigid, locally level-compact G levelledover a locally compact group X we should have such a measure µ G on G whichsatisfies (for example) µ G p G U,γ q “ µ X p U q Y γ , where µ X is a Haar measure on X . Unfortunately, we know that in general this cannot be the case. Indeed,we saw previously that for an arithmetic surface S both the geometric andanalytic adelic spaces A S and A S may be given a level structure of elevation1 over a locally compact group. However, Fesenko showed in [F10] that therecan be no measure on A S which appropriately lifts the one-dimensional adelicmeasure.On the other hand, the space A S was constructed exactly so the analyticadelic measure does lift the one-dimensional measure. It would thus be in-teresting to consider possible conditions on a general group G levelled over alocally compact X which allow such lifting properties. One possible possiblestarting point is to examine the difference in the level-compactness proper-ties of the two different adelic spaces above, which will have some relation tothe fact that the rank one ring of integers O F of a two-dimensional field is28evel-compact but not lv p O F q -compact.Finally, we note that the current text was written mainly in order to for-malise certain behaviours of higher dimensional local fields. To this end, our“justification” for many of the definitions and additional assumptions here isthat they hold for such fields and groups closely related to them. In particu-lar, we have not been particularly diligent in trying to find examples of spaceswhich do not satisfy the properties we require.It would thus be interesting to see whether some of our definitions may infact be turned into theorems, and if this is not the case one should see someinteresting counterexamples. For example, in almost all of the current paperwe assume that our groups are rigid in order to manipulate levels in the correctway, but it may be possible that our version of rigidity can be deduced from aweaker assumption.In any case, the author would be interested to see constructions of inter-esting examples of spaces with level structure which are not related to higherdimensional local fields. References [F03] I. Fesenko. Analysis on arithmetic schemes. I.
Docum. Math. ExtraKato Volume , pages 261–284, 2003.[F05] I. Fesenko. Measure, integration and elements of harmonic analysis ongeneralized loop spaces.
Proc. of St. Petersburg Math. Soc. , 12:179–199, 2005.[F10] I. Fesenko. Analysis on arithmetic schemes. II.
J. K-theory , pages437–557, 2010.[F17] Ivan Fesenko. Analysis on arithmetic schemes. III. 2017.[M08] Matthew Morrow. Integration on product spaces and GL n of a valu-ation field over a local field. Communications in Number Theory andPhysics , vol. 2(no. 3):563–592, 2008.[M10] Matthew Morrow. Integration on valuation fields over local fields.
Tokyo J. Math , 33(1):235–281, 2010.29MZ95] A. I. Madunts and I. B. Zhukov. Multidimensional complete fields:topology and other basic constructions.