aa r X i v : . [ m a t h . L O ] J u l MODEL THEORY OF ADELES AND NUMBER THEORY
JAMSHID DERAKHSHAN
Abstract.
This paper is a survey on model theory of adeles and applications tomodel theory, algebra, and number theory. Sections 1-12 concern model theoryof adeles and the results are joint works with Angus Macintyre. The topicscovered include quantifier elimination in enriched Boolean algebras, quantifierelimination in restricted products and in adeles and adele spaces of algebraicvarieties in natural languages, definable subsets of adeles and their measures,solution to a problem of Ax from 1968 on decidability of the rings Z /m Z forall m > , definable sets of minimal idempotents (or "primes of the numberfield" ) in the adeles, stability-theoretic notions of stable embedding and treeproperty of the second kind, elementary equivalence and isomorphism for adelerings, axioms for rings elementarily equivalent to restricted products and for theadeles, converse to Feferman-Vaught theorems, a language for adeles relevantfor Hilbert symbols in number theory, imaginaries in adeles, and the space adeleclasses.Sections 13-18 are concerned with connections to number theory around zetaintegrals and L -functions. Inspired by our model theory of adeles, I proposea model-theoretic approach to automorphic forms on GL (Tate’s thesis) and GL (work of Jacquet-Langlands), and formulate several notions, problems andquestions. The main idea is to formulate notions of constructible adelic integralsand observe that the integrals of Tate and Jacquet-Langlands are constructible.These constructible integrals are related to the p -adic and motivic integrals inmodel theory.This is a first step in a model-theoretic approach to the Langlands conjec-tures. I also formulate questions on Artin’s reciprocity for ideles, p -adic fieldsand adeles with additive and multiplicative characters in continuous logic, adelicPoisson summation and transfer principles, identities of adelic integrals, andquestions related to model-theoretic aspects of Archimedean integrals (whichrelate to O-minimality). Contents Introduction
History of adeles and role in number theory
Model theory
Acknowledgments
Adeles, idempotents, and Boolean values Mathematics Subject Classification.
Primary 03C10,03C60,11R56,11R42,11U05,11U09,Secondary 11S40,03C90.
Global fields
Restricted direct products and measures
Ring of adeles
Idempotents and Boolean values
Real, complex, and non-Archimedean Boolean values
Finite adeles
Defining finite support idempotents
Enriched theories of Boolean algebras and valued fields
Enrichments of Boolean algebras
Enrichments of valued fields
Generalized products and restricted products
Language for restricted products
Quantifier elimination in restricted products
The case of finite index set
An example from algebraic groups: Weil’s conjecture onTamagawa numbers
Definability in adeles
Definable subsets of A mK Measurability of definable subsets
Countable unions and intersections of locally closed sets
Euler products and zeta values at integers
Definable subsets of the set of minimal idempotents
A question of Ax on decidability of all the rings Z /m Z Ax’s question
Reducing Ax’s problem to adelic decidability
Decidability of A Q Decidability of A K Elementary equivalence and isomorphism for adele rings
The number field degree
The case of normal extensions
Splitting types and arithmetical equivalence
Elementary equivalence of adele rings - a rigidity theorem
Axioms for rings elementarily equivalent to restricted directproducts and converse to Feferman-Vaught
The question and connection to nonstandard models ofPeano arithmetic
Axioms for the rings
The ring-theoretic Feferman-Vaught and converse toFeferman-Vaught
Some stability theory
Stable embedding ODEL THEORY OF ADELES AND NUMBER THEORY 3
The tree property of second kind
Stable formulas and definable groups
Adele geometry
Adele spaces of varieties
Tamagawa measures on adele spaces
Boolean Presburger predicates and Hilbert symbol
Imaginaries in the adeles and the quotient of the space ofadele classes by maximal compact subgroup of idele classgroup
Artin reciprocity
Remarks on the idele class group of Q Euler products of p -adic integrals Analytic properties of the Euler products
Conjugacy class zeta functions in algebraic groups
Adelic height zeta functions and rational points
Finite fields with additive characters and continuous logic p -adic fields with additive characters and continuous logic p -adic and adelic multiplicative characters and L -functions Tate’s thesis and zeta integrals for GL Automorphic representations and zeta integrals for GL n Langlands program and Jacquet-Langlands theory
Adelic constructible integrals
On identities between adelic integrals
Adelic Poisson summation
Adelic transfer principles
A completeness problem
Introduction
History of adeles and role in number theory.
Given an algebraic number field K (i.e. a finite extension of Q ) and valuation v of K , let K v denote the completion of K with respect to the absolute valuecorresponding to v . The adeles of K , denoted A K is the subset of the directproduct of K v , over all v , consisting of elements ( a ( v )) ∈ Q v K v such that a ( v ) lies in the ring of integers O v of K v (i.e. has non-negative valuation) for all butfinitely many v . A K is a locally compact ring (with componentwise addition andmultiplication) and the image of K diagonally embedded is a discrete subspacewith compact quotient A K /K . Being locally compact, A K admits an invariantmeasure. J. DERAKHSHAN
The ring of adeles can be defined for any global field, i.e. a number field offunction field in one variable over a finite field, see [13]. In this paper we onlyconsider the case of number fields. It is expected that similar results hold also forglobal fields of positive characteristic.The ring of adeles was first introduced by Weil in a letter to Hasse on 3 November1937. It was for the case of function fields and as "a method to establish (a newproof of) the theorem of Riemann-Roch". Around this time, Chevalley definedthe notion of ideles, also in a letter to Hasse on 20 June 1935, and initially calledthem "élements idéaux" (Ideles are invertible adeles, and form a group denoted I K ). For more details on the early history of adeles see Peter Roquette’s book[79], page 191 and Subsection 11.3.1.The adeles were later independently introduced by Artin and Whaples in 1945based on the ideles. The name adele was invented by Weil as "additive adele" andalready used by him in a 1958 Bourbaki talk on "adeles and algebraic groups " onresults that later appeared in [85] systematically studying Tamagawa measures ofadelic algebraic groups and proving for many classical groups what is now calledWeil conjecture on Tamagawa numbers, a highly influential work and conjecture.Weil’s main motivation for defining adeles was a proof of the Riemann hypothesisfor function fields of curves over finite fields. It was clear from the beginning thatadeles have rich connections with L -functions and zeta functions of number fieldsinvolving characters. Interestingly most of the developments and applications ofadeles have been in continuation of this path. I shall give a short account ofof some key developments below, and later a model-theoretic approach to theseproblems.Dedekind had defined a generalization of the Riemann zeta function for anynumber field and Hecke proved analytic continuation and functional equation forthese zeta functions.At the suggestion of Artin, in 1950, in [82] Tate used harmonic analysis on theadeles to give a new proof of Hecke’s results. Tate’s proof gave more information.In 1952 independently Iwasawa obtained the same results.Tate’s work has greatly influenced several works in number theory. This includesthe Langlands program, starting with work of Jacquet and Langlands [57] on GL and then Godement and Jacquet [48] on GL n , and later work by Langlandsand others, see [66]. These have given applications of adeles to automorphicforms. The classical theory of modular forms and work of Hecke was generalizedadelically. These have given rise to a large number of results within the Langlandsprogram on automorphic representations of adelic groups G ( A K ) , where G is analgebraic group over K . The case of GL being Tate’s thesis, GL the Jacquet-Langlands work which extends results on classical modular forms and Maass forms,and the case of GL n being work of Godement and Jacquet [48]. The case of ageneral reductive group involves the concept of Langlands dual group. For later ODEL THEORY OF ADELES AND NUMBER THEORY 5 developments, especially relating to l -adic representations of elliptic curves andalgebraic varieties see [70].Another important development was Weil’s adelic interpretation of work ofSiegel on quadratic forms (called Siegel Mass formula) as the adelic volume of SO n ( A Q ) /SO n ( Q ) with respect to the Tamagawa measure being for all n . Weil’sconjecture on Tamagawa numbers states that the volume of G ( A K ) /G ( K ) is forany semi-simple simply connected linear algebraic group G over a number field K . This was proved by Weil for many classical groups, Langlands for split semi-simple groups, and Kottwitz in general, (see Kneser’s article in the volume [13]and the book [70]). The conjecture of Birch and Swinnerton-Dyer emerges froman analogue of Weil’s conjecture for elliptic curves. The results in Subsection 4.4and Section 10 relate to these issues.An insight of Paul Cohen (unpublished notes with Peter Sarnak) was that prop-erties of A K /K ∗ , defined as the quotient of the space of adele classes A K by theaction of K ∗ by multiplication, are relevant for the the solution of the RiemannHypothesis. Much work in this direction has been carried out by Alain Connesand co-authors. See [21] where results from Tate’s thesis are generalized to adeleclasses to obtain results on zeros of zeta functions. The results in Section 12 andSections 18-20 are related to these ideas and concepts.1.2. Model theory.
The first work on the model theory of adeles was by Weispfenning [86]. Heproved decidability of the ring of adeles of number fields via his work on modeltheory for lattice products. In 2005 Angus Macintyre told me of his ideas andinsights on a model theory for adeles. This was the start of our long term collab-oration. I shall give a survey of many of our results in Sections 2-12.Like Weispfenning, we use the celebrated work of Feferman and Vaught [43] onproducts of structures but our approach is different. For us the Boolean algebraof subsets of the index set of the product is replaced by the Boolean algebra ofidempotents in A K . We also prove analogues of results of Feferman-Vaught onproducts for the case of restricted products. The interpretation of the Booleanalgebra in the ring of adeles enables studying model theory of A K as an L rings -structure which has been needed to understand issues related to the topology,measure, and analytic structure on the adeles. It enables one to internalize theFeferman-Vaught theorems and obtain quantifier elimination results in the lan-guage of rings and prove that the definable subsets of A mK , for any m ≥ , aremeasurable (with respect to a Haar measure on A K ) and study their measures.We also present the Feferman-Vaught theorems for restricted products in a moregeneral case of many-sorted languages with relation and function symbols. We canalso combine the ring-theoretic approach to A K with the Feferman-Vaught typestructures involving the Boolean algebra of subsets of the index set of normalized J. DERAKHSHAN valuations and consider expansions of the Boolean language and expansions of thelanguage of rings. Any choice for these expansions yields a language for the adeleswhich is an example of what we call a language for restricted products.The approach to A K as a ring via its idempotents has enabled much furtherresults. These include solution to a problem on Ax on decidability of Z /m Z forall m , study of elementary equivalence of adele rings, and converse of Feferman-Vaught theorems which amount to Feferman-Vaught theorems for rings. In thisconnection we have given axioms for adeles and general restricted products. Outline of the Paper
The material in Sections 2-12 are all joint works with Angus Macintyre .In Section 2, I state the basic results and notation on adeles and idempotents.In Section 3, I state results on enriched Boolean algebras and enriched valuedfields that are needed.Section 4 contains the model-theoretic formalism for restricted products andtheir languages and the quantifier elimination results for restricted products andadeles.Section 5 contains results on definable sets and their measurability, and con-nections to values of zeta functions at integers, and a description of the definablesubsets of the set of minimal idempotents in the adeles (analogue of primes) interms of Ax’s Boolean algebra of primes in [1].Section 6 contains a sketch the solution to a problem posed in 1968 by Ax [1]on decidability of the rings Z /m Z for all m > , via decidability of A Q , of whicha short proof is also sketched.Section 7 is concerned with results on elementary equivalence of adele rings andtheir isomorphism, and its relation to arithmetical equivalence of number fieldsand their zeta functions.Section 8 is concerned with axioms for restricted products, and a converse toFeferman-Vaught. The main result states that any commutative unital ring sat-isfying the axioms is elementarily equivalent to a restricted product of connectedrings. This is proved via a Feferman-Vaught theorem for general commutativerings.Section 9 is concerned with the stability-theoretic notions of stable embeddingand tree property of the second kind for the adeles.Section 10 contains results on adele spaces of algebraic varieties introduced byWeil. The results state that these spaces can be put into the formalism of Section4 and are model-theoretic restricted products and admit quantifier elimination ina certain geometric language. Some results are obtained on Tamagawa volumesand uniform definability of convergence factors for adelic measures on adele spacesfor certain varieties.In Section 11 an expansion of the language of rings is considered which is suitablefor studying reciprocity laws in number theory and definability results on Hilbert ODEL THEORY OF ADELES AND NUMBER THEORY 7 symbols using results on certain enrichments of Boolean algebras introduced in[34].Section 12 concerns imaginaries in adeles, the space of adele classes A Q / Q ∗ , andthe quotient ˆ Z ∗ \ A Q / Q ∗ , where ˆ Z ∗ the maximal compact subgroup of the ideleclass group I K /K ∗ which appears in various works by Alain Connes and Connes-Consani on the distribution of zeros of the Riemann zeta function and on thearithmetic site topos, see [21],[24],[23]. The result stated proves that the doublequotient is interpretable in A Q answering a question of Zilber.In Sections 12-18, I introduce a model theoretic approach to certain topics innumber theory and formulate several questions and problems.Section 13 concerns Artin reciprocity from class field theory, the idele classgroup, and a number of problems.In Section 14 I state a general result of myself [30] on analytic properties ofEuler products, over all p , of p -adic integrals (suitably normalized) introducedand studied by Denef, and later Denef, Loeser and others in motivic integration.This result shows that it is possible to analytically continue the Euler productbeyond its abscissa of convergence and give some information on its poles whichis enough to give an asymptotic formula for the partial sums of the coefficients ofthe Dirichlet series the represents the Euler product.This result naturally applies to several global zeta functions that have the formof an Euler product of p -adic integrals. An application is stated in [30] and [32]to a question of Uri Onn on counting counting conjugacy classes in congruencequotients of algebraic groups over number fields.I state a special case of this result for SL n , n ≥ . Let c m denote the number ofconjugacy classes in the congruence quotient SL n ( Z /m Z ) . Then the global con-jugacy class zeta function P i ≥ c m m − s admits meromorphic continuation beyondits abscissa of convergence, and consequently c + · · · + c m ∼ cN α for some c ∈ R > as N → ∞ . This is proved by writing the global conjugacyzeta function as an Euler product of local conjugacy class zeta functions whichcount conjugacy classes in congruence quotients of the group of p -adic points andby my joint work with Mark Berman, Uri Onn, and Pirita Paajanen [8] the localconjugacy class zeta functions can be written as p -adic integrals of Denef type overdefinable sets.I also state a connection of the result on meromorphic continuation of Eulerproducts to counting rational points in orbits of group actions and a problem ofGorodnik and Oh in [49] that is studied in [31].Section 15 is concerned with finite fields with additive character in continuouslogic and a question of Hrushovski is stated. J. DERAKHSHAN
Section 16 is concerned with p -adic fields with additive character in continuouslogic and consequences for the model theory of adeles with additive characters inusual first-order and in continuous logic.Section 17 concerns ideles with multiplicative character and associated L -functions.In Section 18 I discuss Tate’s thesis on analytic continuation and functionalequation for a large class of zeta integrals generalizing zeta functions of numberfields, and formulate model-theoretic questions.Section 19 is concerned with automorphic representations and the Langlandsconjectures. Having considered the basic case of GL in Section 18 on Tate’s thesis,I focus on automorphic representations of GL . I propose a framework to model-theoretically study these issues and formulate a number of notions, problems, andquestions in relation to model-theoretic aspects of the adelic integrals involved.The ideas on model theory of adeles presented in Sections 2-12 naturally connectwith these problems, and it is hoped that the notions and questions can be a firststep towards a systematic investigation on Langlands correspondence.In this regard, I define notions of L -constructible adelic integrals and C -valued L -constructible functions on the adeles, where L is a suitable language. Theseintegrals are mostly Euler products over primes, and their local p -adic factors havesome similarities to the integrals of motivic constructible functions of Cluckers-Denef-Loeser in [20].A proof of Jacquet and Langlands shows that the Jacquet-Langlands globalzeta integrals of cuspidal automorphic representations of GL are L -constructibleof a certain kind (I call it of Whittaker type). This is stated and several relatedproblems are formulated.These are hoped to be a starting point for systematic model-theoretic investiga-tions on the Langlands program beginning with the works of Jacquet-Langlandsand Godement-Jacquet on GL n . A model theory for modular forms and automor-phic forms could follow.The Archimedean completions of number fields need to be considered as well inrelation to the p -adic completions. Related to this, I have stated questions involv-ing the expansion of ( R , + , ., , by restricted real analytic functions with expo-nentiation introduced by van den Dries-Macintyre-Marker [84], and O-minimalityin connection to Hodge theory a la Bakker-Klingler-Tsimerman [5].In Section 20, I formulate questions around Tate’s adelic Poisson summationformula, adelic transfer principles guided by the Ax-Kochen transfer principlefor truth of sentences and Cluckers-Denef-Loeser-Macintyre transfer principles foridentities of local integrals across families of local fields, and a completeness prob-lem for identities between adelic integrals that relates to axioms for adeles.1.3. Acknowledgments.
ODEL THEORY OF ADELES AND NUMBER THEORY 9
I am very grateful and indebted to Angus Macintyre for introducing me tothe subject, and for generously and patiently teaching me and sharing with mehis ideas and insights that have led to the results in this paper, and for ourcollaborations and his continuous inspiration and support.Many thanks to Ehud Hrushovski and Boris Zilber for many discussions and ad-vice, from which I have learned much. Udi introduced me to some of his questionsand results that are stated in Section 16, which influenced Sections 15-16.I am grateful to Peter Sarnak and Nicolas Templier for discussions and askingsome questions that I have stated in Section 13, and for their advice.I am grateful to Ramin Takloo-Bighash for helpful discussions on topics innumber theory, and on work of Jacquet-Langlands that were helpful in my under-standing of the topics.Many thanks to Laurent Lafforgue for many discussions on aspects of the Lang-lands conjectures, and for teaching me some of his insights and recent works onfunctoriality, especially during a visit to IHES. These have influenced Sections18-20. 2.
Adeles, idempotents, and Boolean values
Global fields.
A global field K is either a finite extension of Q (called a number field) or a finiteseparable extension of F q ( t ) where t is transcendental over F q (called a functionfield).In this paper, we shall only be concerned with number fields. See Cassels’ article[11] for basic results and notions around absolute values and valuations on a field.Here we only state that two absolute values | . | and | . | are called equivalent if | . | = | . | c for some c ∈ R > , and this holds exactly when they define the sametopology. We consider absolute values on a number field K up to this equivalence.These absolute values are then of only the following kinds:(1) Discrete non-Archimedean with residue field finite of cardinality q ,(2) Completion of | . | is R ,(3) Completion of | . | is C .Among these absolute values we chose a distinguished one in each kind, thatis called normalized . In Case (1) | . | is normalized if | π | = 1 /q where π is auniformizing element (i.e. v ( π ) is a minimal positive element) of the value groupof K (see below as well). In Case (2) (resp. Case (3)), | . | is normalized if it is theusual absolute value (resp. square of the usual absolute value).We denote by K v the completion of K with respect to v . For details on globalfields see Cassels’ paper [11].In the non-Archimedean cases, the absolute values are given by valuations v ( x ) which are maps from K ∗ into an ordered abelian group Γ called the value group.One extends v ( x ) to K by putting v (0) = ∞ . The non-Archimedean absolute values | . | p on K = Q arise from the standard p -adic valuations v p as | a/b | p = p − v p ( a/b ) , where v p ( z ) = k if k is the largest powerof p that divides z when z ∈ Z , and v p ( a/b ) = v p ( a ) − v p ( b ) , for a, b ∈ Z . Thusthe absolute values of Q correspond to primes of Z and the absolute value givenby the embedding of Q in R .For a general number field K , one replaces the prime numbers p by prime ideals p of the ring of integers O K of K , and similarly the absolute values correspondto these and the real as well as complex embeddings of K . Similarly one has avaluation v p of K (which we again normalize) which restricts to ( p ) , and any ( p ) extends to finitely many p . Furthermore, the real absolute value extends to finitelymany real and complex absolute values.We denote by V finK the set of all normalized valuations on K which give a non-Archimedean absolute value up to equivalence. We denote by Arch( K ) the finiteset of Archimedean normalized absolute values, and will call them Archimedeanvaluations . We put V K = V finK ∪ Arch(K) , and call it the set of normalized valua-tions of K . Γ (or Γ K ) shall denote the value group, O K := { x : | x | ≤ } the valuation ring,and M K := { x : | x | < } the maximal ideal of a valued field K .For a non-Archimedean completion K v of K , we also denote the absolute valueof K v by | . | v , the valuation ring by O v , the maximal ideal by M v , and the residuefield by k v .2.2. Restricted direct products and measures.
Let Λ be an index set and Λ ∞ a fixed finite subset. Suppose we are given, for each λ ∈ Λ , a topological space G λ , and for all λ / ∈ Λ ∞ , a fixed open subset H λ of G λ . Let Q λ ∈ Λ G λ denote theCartesian product of the G λ . We denote an element of this set by x = ( x ( λ )) λ ,where x ( λ ) denotes the λ -component of x which is an element of G λ .The restricted direct product G of G λ with respect to H λ is defined to be theset of all elements ( x ( λ )) λ ∈ Q λ ∈ Λ G λ such that x ( λ ) ∈ H λ for all but finitelymany λ , and denoted Q ′ λ ∈ Λ G λ . It carries the restricted product topology with abasis of open sets consisting of the products Q λ ∈ Λ Γ λ , where Γ λ ⊆ G λ is open forall λ ∈ Λ , and Γ λ = H λ for all but finitely many λ ∈ Λ .Suppose for each λ ∈ Λ , G λ carries a measure µ λ such that µ λ ( H λ ) = 1 for all λ / ∈ Λ ∞ . The measure µ on G induced by the µ λ is the measure with a basis ofmeasurable sets consisting of the sets Q λ ∈ Λ M λ , where M λ ⊆ G λ is µ λ -measurableand M λ = H λ for all but finitely many λ ∈ Λ , and where µ ( Y λ ∈ Λ M λ ) = Y λ ∈ Λ µ λ ( M λ ) . It is denoted by Q λ µ λ .Suppose G λ is a locally compact group for all λ ∈ Λ , and H λ a compact opensubgroup for all λ / ∈ Λ ∞ . Then G λ carries a Haar measure µ λ such that µ λ ( H λ ) = 1 ODEL THEORY OF ADELES AND NUMBER THEORY 11 (for all λ / ∈ Λ ∞ ). For any finite subset S ⊆ Λ containing Λ ∞ , let G S := Y λ ∈ S G λ × Y λ/ ∈ S H λ . This set is locally compact and open in G , and G is the union of the G S over allfinite subsets S of Λ containing Λ ∞ . In particular, G is locally compact.2.3. Ring of adeles.
The ring of adeles of a number field K is the restricted direct product A K = Q ′ v ∈ V K K v of the additive groups of K v with respect to the subgroups O v , with ad-dition and multiplication defined componentwise. We write an adele a as ( a ( v )) v .The ring of finite adeles of K is the restricted direct product A finK = Q ′ v ∈ V finK K v of all non-Archimedean K v with respect to O v . One has A K = ( Y v ∈ Arch(K) K v ) × A finK , an algebraic and topological isomorphism.Let S be a finite subset of V K containing all the Archimedean valuations. Weput A K,S = Y v ∈ S K v × Y v / ∈ S O v . Then A K = S S A K,S , over all finite subsets S ⊆ V K containing V ArchK .There is an embedding of K into A K sending a ∈ K to ( a, a, · · · ) . The imageis the ring of principal adeles, which we identify with K . It is a discrete subspaceof A K with compact quotient A K /K . If K ⊆ L are number fields, then A L isisomorphic to A K ⊗ K L algebraically and topologically. See [13], page 64.The group of ideles I K is the group of units of A K and coincides with therestricted direct product of the multiplicative groups K ∗ v with respect to the unitgroups O ∗ v of O v . It is given the restricted direct product topology. For any a ∈ K ∗ , | a | v = 1 for all except finitely many v . Thus K ∗ is (diagonally) embeddedin I K . One also has the product formula Q v ∈ V K | a | v = 1 . See [13, pp.60].Each K v is a locally compact field, and carries an additive Haar measure dx v .We make the following choice for these Haar measures:(1) The measure on K v such that O v has volume if K v is non-Archimedean(2) The usual Lebesgue measure if K v is R ,(3) The measure dxdy if K v is C .These give measures Q v ∈ V K dx v and Q v ∈ V finK dx v on A K and A finK respectively.The multiplicative groups K ∗ v carry a Haar measure d ∗ x v = dx v / | x v | v invariantunder multiplication, which give the measure Q v ∈ V K d ∗ x v on I K .As in Weil [85], we call the induced measure Q v ∈ V K dx v on A K the canonicalmeasure and denote it by ω A K . Idempotents and Boolean values.
Let L rings = { + , − , ., , } denote the language of rings and L Boolean = {∧ , ∨ , , , ¬} the language of Boolean algebras. The set B K = { a ∈ A K : a = a } of idempotents in A K is a Boolean algebra with the Boolean operations e ∧ f = ef,e ∨ f = 1 − (1 − e )(1 − f ) = e + f − ef, ¬ e = 1 − e. B K is L rings -definable in A K . It carries an order defined by e ≤ f if and only if e = ef , which is L rings -definable. An idempotent e is minimal if it is non-zero andminimal with respect to this order.There is a correspondence between subsets of V K and idempotents e in A K defined by X e X , where given X , e X = ( if v ∈ X if v / ∈ X It is clear that e X ∈ A K . Conversely, if e ∈ A K is idempotent, then we let X = { v : e ( v ) = 1 } . We have e = e X . Under this correspondence a minimal idempotent e corresponds to a normalizedvaluation { v } which we denote by v e . Conversely, v corresponds to e { v } .Note that e A K is an ideal in A K but not a unital subring of A K . It is a unitalring with induced addition and multiplication, and e as the unit element.The map a + (1 − e ) A K ea gives an isomorphism A K / (1 − e ) A K ∼ = e A K . This follows from the isomorphism A K ∼ = e A K ⊕ (1 − e ) A K since e is idempotent.If e is a minimal idempotent, and corresponds to { v e } , we have an isomorphism e A K ∼ = K v e given by ea a ( v e ) . Let Φ( x , . . . , x n ) be an L rings -formula. It is easily seen (see [37]) that there is anassociated L rings -formula Φ Glob ( y, x , . . . , x n ) constructed from Φ and independentof K such that for all a , . . . , a n ∈ A K and idempotents ee A K | = Φ( ea , . . . , ea n ) ⇔ A K | = Φ Glob ( e, a , . . . , a n ) . ODEL THEORY OF ADELES AND NUMBER THEORY 13
For a , . . . , a n ∈ A K , we define [[Φ( a , . . . , a n )]] to be the supremum of all theminimal idempotents e in B K such that e A K | = Φ( ea , . . . , ea n ) . Note that the set of such minimal idempotents is empty precisely when [[Φ( a , . . . , a n )]] = 0 , and the set of such idempotents coincides the set of all minimal idempotentsprecisely when [[Φ( a , . . . , a n )]] = 1 . We remark that [[ ... ]] is an internal version of the Boolean values of Feferman-Vaught (see [43]) defined by [[Φ( x , . . . , x n )]] = { v ∈ V K : K v | = Φ( a ( v ) , . . . , a n ( v )) } . Note that e A K is L rings -definable with the parameter e , and the functions A nK → A K defined by ( a , . . . , a n ) → [[Φ( a , . . . , a n )]] are L rings -definable independently of K , for any Φ .The idempotent-support of a ∈ A K is defined as the Boolean value [[ a = 0]] ,and we denote it by supp( a ) Real, complex, and non-Archimedean Boolean values.
We define the following formulas and idempotents.. Ψ Arch denotes a sentence that holds in R and C but does not hold in anynon-Archimedean local field, for example ∀ x ∃ y ( x = y ∨ − x = y ) . . Ψ R denotes a sentence that holds precisely in the K v which are isomorphic to R , for example Ψ Arch ∧ ¬∃ y ( y = − . . Ψ C denotes a sentence that holds precisely in the K v which are isomorphic to C , for example Ψ Arch ∧ ∃ y ( y = − . . A minimal idempotent e is Archimedean if e A K | = Ψ Arch , and non-Archimedean otherwise.. A minimal idempotent e is real if e A K | = Ψ R , and complex if e A K | = Ψ C . . e R (resp. e C ) denotes the supremum of all the real (resp. complex) minimalidempotents. It is supported precisely on the set of v such that K v is real (resp.complex).. e ∞ denotes the supremum of all the Archimedean minimal idempotents. It issupported precisely on the set Arch( K ) .. e na denotes − e ∞ . It is supported precisely on the set of non-Archimedeanvaluations.Let Φ( x , . . . , x n ) be an L rings -formula and a , . . . , a n ∈ A K . Then. [[Φ( a , . . . , a n )]] real denotes the supremum of all the minimal idempotents e such that e A K | = Ψ R ∧ Φ( ea , . . . , ea n ) . . [[Φ( a , . . . , a n )]] complex denotes the supremum of all the minimal idempotents e such that e A K | = Ψ C ∧ Φ( ea , . . . , ea n ) . . [[Φ( a , . . . , a n )]] na denotes the supremum of all the minimal idempotents e such that e A K | = ¬ Ψ Arch ∧ Φ( ea , · · · , ea n ) . The functions ( a , . . . , a n ) → [[Φ( a , . . . , a n )]] real ( a , . . . , a n ) → [[Φ( a , . . . , a n )]] complex and ( a , . . . , a n ) → [[Φ( a , . . . , a n )]] na are L rings -definable from A nK to A K , uniformly in K .2.6. Finite adeles.
We can identify the ring of finite adeles A finK with the ideal in A K consistingof all adeles a such that a ( v ) = 0 for all v ∈ Arch ( K ) (this will be the case inthe results on von Neumann regularity and definition of valuation in Subsection2.7). However in the results of Section 4 on quantifier elimination and definablesets it is preferable to work with A finK as a restricted direct product and A K as theproduct of A finK with Q v ∈ Arch ( K ) K v .We let B finK denote the Boolean algebra of idempotents in A finK (considered eitheras a restricted product or as an ideal in A K ) with the same Boolean operations as B K . It is definable in B K , A finK , and A K .Given a , · · · , a n ∈ A finK , and an L rings -formula Φ( x , · · · , x n ) , the idempotent [[Φ( a , · · · , a n )]] na can be defined inside A finK as the supremum of all the minimalidempotents e from B finK such that e A finK | = Φ( ea , . . . , ea n ) , which is again definable in L rings , uniformly in K . ODEL THEORY OF ADELES AND NUMBER THEORY 15
Defining finite support idempotents.
An idempotent e has finite support if { v ∈ V K : e ( v ) = 0 } is a finite set.The following theorem is of great importance in our works on the model theoryof adeles. Theorem 2.1 (Derakhshan-Macintyre [37]) . The set of finite support idempotentsin A K is L rings -definable, uniformly in K . We denote the set of finite support idempotents by
F in K . It is an ideal in theBoolean algebra B K . In [37] we give two proofs of this result. The first uses theconcept of a von Neumann regular ring defined as follows.We call an element a of a commutative ring R von Neumann regular if ( a ) theideal generated by a is generated by an idempotent. R is von Neumann regularif every element is von Neumann regular. Examples are direct products of fields.In contrast, A K is not von Neumann regular, however it is shown in [37] that anelement a ∈ A K is von Neumann regular if and only if the set { v ∈ V K : v ( a ( v )) > ∧ a ( v ) = 0 } is finite. This implies that an idempotent e ∈ A K has finite support if and only iffor all a ∈ A K , the element ae is von Neumann regular.So we let φ ( x ) denote the L rings -formula ∃ e ∃ y ∃ z ( e = e ∧ x = ey ∧ e = xz ) . Then the set
F in K of finite support idempotents in A K is defined by the ∀∃ -formula x = x ∧ ∀ y ( φ ( yx )) . This definition is independent of K . It follows that the set of finite supportidempotents in A finK is also definable.The second proof uses uniform definition of valuation rings in the languageof rings which used quite often in the works of Macintyre and myself on themodel theory of adeles. Explicit definitions previous to our work (e.g. for Q p forall p ) used a constant in the language for a uniformizing element. Non-explicitdefinitions had existed as well, using Beth definability type arguments because ofthe uniqueness of the Henselian valuations of the K v . These results were mostlyapplied in connection to decidability and undecidability results on Hilbert 10thproblem (see [60]).In joint work with Cluckers-Leenknegt-Macintyre [18] the following is proved. Theorem 2.2 (Cluckers-Derakhshan-Leenknegt-Macintyre [18, Theorem 2]) . Thereis an existential-universal formula in the language of rings that uniformly definesthe valuation ring of all Henselian valued fields with finite or pseudo-finite residuefield.
This theorem is best possible in terms of quantifier complexity as we show in[18] that there is no existential or universal L rings -formula that uniformly definesthe valuation rings of Q p , or F p (( t )) for almost all p , or the valuation rings of allthe finite extensions of a given F p (( t )) or Q p .We denote by Φ val ( x ) the formula given by Theorem 2.2. For our work on adeleswe apply Theorem 2.2 to all the completions K v , where v is non-Archimedean. Note 2.1.
Note that Theorem 2.2 applies not only to all finite extensions of Q p forall p , but to all local fields of positive characteristic F q (( t )) and their ultraproducts.In fact, the formula Φ v ( x ) is defining the valuation ring of all these fields as wellas the valuation rings of all Henselian valued fields with higher rank value groupsand pseudofinite residue field. In [37] that an idempotent e has finite support if and only if A K | = ∃ x ( e na e = [[ ¬ Φ val ( x )]] na ) which is also independent of K .3. Enriched theories of Boolean algebras and valued fields
Our analysis of the model theory of the adeles and restricted products dependscrucially on the theory of an associated Boolean algebra and the theories of thelocal fields which are the factors or "stalks" of the adeles.In this analysis, various enrichments are of fundamental importance. TheBoolean algebra should be enriched by at least adding a predicate
F in ( x ) to thelanguage with intended interpretation in a Boolean algebras of sets as "finite", andin the Boolean algebra of idempotents in the adeles or in rings as "finite unions ofatoms". We should also enrich the language for the factors. We give some detailsbelow.3.1. Enrichments of Boolean algebras.
As before L Boolean denotes the language of Boolean algebras {∨ , ∧ , ¬ , , } .Tarski proved the classical result that if we enrich L Boolean by natural predicates C n ( x ) for all n ≥ , with the interpretation that there are at least n distinctatoms α with α ≤ x , then in the enriched language the theory of all infiniteatomic Boolean algebras is complete, has quantifier elimination and is decidable.This theory, that we denote by T Bool , is axiomatized by sentences stating thatthe models are infinite Boolean algebras and every nonzero element has an atombelow it. (See [62, Theorem 16,pp.70], a new proof is given by Macintyre andmyself [34] that is uniform for the other enrichments stated below).The main examples are Boolean algebras of subsets of an infinite set I , namely P ( I ) (which denotes the powerset of I ) with the usual set-theoretic Boolean oper-ations. These are clearly not the only models, since no countable model can be apowerset algebra. ODEL THEORY OF ADELES AND NUMBER THEORY 17
Feferman-Vaught [43] and Mostowski added a unary predicate
F in ( x ) to thelanguage of Tarski with the intended interpretation in a Boolean algebra of subsets P ( I ) as " x is finite". Feferman-Vaught [43] proved that this theory is complete,decidable, and has quantifier elimination. A new proof was given Macintyre andmyself in [34]. Note that there are other models where F in ( x ) holds and x is notfinite.Let L finBoolean denote the enrichment of L Boolean got by adding all the unarypredicates C j ( x ) and F in ( x ) , for all j ≥ . Let ♯ ( x ) denote the number of atoms α such that α ≤ x . Note that the C j are definable in L Boolean but
F in ( x ) is not.Let T fin of infinite atomic Boolean algebras in the language L finBoolean . Theorem 3.1 (Feferman-Vaught [43], new proof by Derakhshan-Macintyre [34]) . The theory T fin of infinite atomic Boolean algebras with the set of finite setsdistinguished is complete, decidable and has quantifier elimination with respect toall the C n , ( n ≥ , and F in (i.e. in the language L finBoolean ). The axioms requiredfor completeness and quantifier elimination are the axioms of T Bool together withsentences expressing that
F in is a proper (Boolean) ideal, the sentence ∀ x ( ¬ F in ( x ) ⇒ ( ∃ y )( y < x ∧ ¬ F in ( y ) ∧ ¬ F in ( x ∧ ¬ y ))) . and, for each n < ω , the sentence ∀ x ( ♯ ( x ) ≤ n ⇒ F in ( x )) . This theorem plays a fundamental role in our works on the model theory ofadeles and restricted products. It also applies to a commutative ring via theBoolean algebra of its idempotents (see Section 8). In this way it has been usedin the work in [35] of Macintyre and myself on axioms for rings elementarilyequivalent to restricted products (see Section 8).Motivated by questions in number theory, around reciprocity laws, in [34] weexpanded the language L finBoolean by unary predicates Res ( n, r )( x ) for all n, r ∈ Z , n > , with the intended interpretation, in P ( I ) , that F in ( x ) holds and thecardinal of x is congruent to r modulo n .Let L fin,resBoolean denote the enrichment of L finBoolean by these predicates and T fin,res the theory of all infinite atomic Boolean algebras in the language L fin,resBoolean .In [34], Macintyre and myself gave axioms for T fin,res and prove the following. Theorem 3.2 (Derakhshan-Macintyre [34]) . The theory T fin,res of infinite atomicBoolean algebras in the enriched language with all the C n , ( n ≥ , F in , and all
Res ( r, n ) , ( n, r ∈ Z , n > , is complete, decidable, and has quantifier elimina-tion. The axioms needed to get the completeness and quantifier elimination arethe axioms of T fin together with the Boolean-Presburger axioms as follows: ∀ x ( Res ( n, r )( x ) ⇒ F in ( x )) , ∀ x ( F in ( x ) ∧ ♯ ( x ) = m ∧ m ≡ r (mod n ) ⇒ Res ( n, r )( x )) , ∀ x ( Res ( n, r )( x ) ∧ r ≡ s (mod n ) ⇒ Res ( n, s )( x )) , ∀ x ( Res ( n, r )( x ) ∧ r s (mod n ) ⇒ ¬ Res ( n, s )( x ) , ∀ x ( Res ( m, r )( x ) ∧ n | m ⇒ Res ( n, r )( x )) , ∀ x ( F in ( x ) ⇒ _ ≤ r Res -predicates and have quantifier elimination and decidability. In Section 11 wegive an application of this on Hilbert reciprocity in number theory from [37].3.2. Enrichments of valued fields. As before, for a valued field K , we denote the valuation by v : K → Γ ∪{∞} , thevalue group by Γ (with top element ∞ ), the valuation ring by O K , the maximalideal by M K , and the residue field by k .The quantifier elimination for adeles or generally restricted products takes placein two steps. The first step is a general quantifier elimination that works for allrestricted products, and would depend on a chosen enrichment of infinite atomicBoolean algebras. This is stated in Subsection 4.2. The second step is a finer resultthat requires a quantifier elimination in the factors or "stalks" of the restrictedproduct. This depends on a chosen enrichment of Henselian valued fields. Weshall state some convenient languages in this Subsection for this purpose. Theresulting results are presented in Subsections 4.2 and 5.1.Let L be a language for the factors K v . This means that K v are L -structures. If v is non-Archimedean, then K v is non-trivially valued Henselian whose value groupis Z . One can have a symbol for the valuation in both -sorted and many-sortedsituations. Even though the valuation of the non-Archimedean K v is uniformlydefinable by an L rings -formula that is ∃∀ (see Theorem 2.2), the sorting helps withsome of the finer results and enables us to keep track of residue fields or residuerings where uniformities depend on, as v or K vary.On the other hand for us it is important to keep the basic analysis within thelanguage of rings and regard A K as a ring, which we do in all cases.In the interpretation of K v as an L -structure, we have to take into account theArchimedean K v as well. In this case, in the situations that we add a predicate ODEL THEORY OF ADELES AND NUMBER THEORY 19 for the valuation v of the non-Archimedean K v , we adopt the convention that v is interpreted in the Archimedean K v as the trivial valuation. This means thatin the Archimedean K v , v ( a ) = 0 if and only if a is not zero. This means that K v is the valuation ring and { } the maximal ideal. For the real K v we take thelanguage of ordered rings L rings ∪ { < } and for the K v that are complex, we takethe language of rings. Then the Archimedean K v admit quantifier elimination byresults of Tarski for real closed and algebraically closed fields (see [62]).Note that while the valuation rings of all the non-Archimedean K v and theirunit balls are L rings -definable, the unit ball in R is also L rings -definable, but theunit ball in C is not L rings -definable.The required quantifier elimination for the K v is the following. Q.E. for stalks: For an L -formula ϕ ( x ) , where x is a tuple of variables, thereexits an L -formula ψ ( x ) which is quantifier-free in a distinguished sort of L , suchthat for all but finitely many K v we have K v | = ∀ x ( ϕ ( x ) ⇔ ψ ( x )) , and in each of the exceptional K v , φ (x) is equivalent to a quantifier-free formularelative to a finite set of sorts. Recall the following basic result. Theorem 3.3 (Ax-Kochen-Ershov [15], see also [37]) . For each sentence φ of thelanguage of valued fields effectively there exist a positive integer n and a sentence ψ of the language of rings so that for any Henselian valued field K of characteristic with residue field k of characteristic not dividing n and value group a Z -group, i.e.elementarily equivalent to ( Z , + , , < ) in the language of ordered abelian groups,we have K | = φ ⇔ k | = ψ It is convenient to use the following languages for the non-Archimedean K v . [68] . L Mac = { + , ., , , P n ( x ) } is the expansionof L rings by predicates P n ( x ) interpreted in a field as the set of non-zero n thpowers, for all n ≥ . Macintyre [68] proved that the L Mac -theory of Q p admitselimination of quantifiers. It follows that a definable subset of Q mp , for any m , (in L rings or L M ac ) is a finite union of locally closed sets (i.e. an intersection of anopen and a closed set) in p -adic topology, and is thus measurable. In [77], Prestel and Roquette defined an extension L P R of L Mac for thetheory of p -adically closed fields of rank d , which are defined by the condition that O K / ( p ) has dimension d over F p , by adding of constant symbols for an F p -basis ofthis quotient. They proved that the theory of p -adically closed fields of p -rank d admits elimination of quantifiers in this language. Remark that if K is a finite extension of Q p of degree d , then K is p -adicallyclosed of p -rank d .It follows that an L P R -definable subset of each non-Archimedean K vm , for any m , is a finite union of locally closed sets, and thus measurable, and an infinitedefinable subset of K v has non-empty interior. Note 3.1. For the Archimedean K v , it follows from Tarski’s quantifier eliminationtheorems for R and C that a definable subset of K mv for any m , is a finite unionof locally closed sets and is measurable.3.3.3. In [7], Belair defined an extension of the Macintyre language in whichthe theory of Q p , for all p , admits uniform elimination of quantifiers. He addedconstants for an element of least positive value and for coset representative for thegroup of non-zero n th powers and solvability predicate of Ax for the residue fields Sol m ( x , . . . , x m ) , m ≥ , interpreted in a valued field K by ( ^ ≤ i ≤ m v ( x i ) ≥ ∧ ∃ y ( v ( y ) ≥ ∧ v ( y m + x y m − + · · · + x m ) > . By a result of Kiefe [58] the theory of pseudofinite fields admits quantifier elimi-nation in the language of rings augmented by Sol k , for all k , defined by Sol k ( y , . . . , y n ) ↔ ∃ z ( z m + y z + · · · + y m = 0) . This elimination of quantifiers holds uniformly for all F q , where q is large enough(either fixed or unbounded characteristic). ([73],[74]). The Denef-Pas language L Denef − P as = ( L field , L residue , L group , v, ¯ ac ) is a 3-sorted language with the language of rings for the field sort L field and for theresidue field sort L residue , and the language of ordered abelian groups { + , , ≤ , ∞} with a top element ∞ for the value group sort L group .There is a function symbol v from the field sort to the value group sort in-terpreted in a valued field K as the valuation, and a function symbol from thefield sort to the residue field sort interpreted in K as the angular component mapmodulo M K . This map is defined by the following conditions1) ¯ ac (0) = 0 ,2) The restriction of ¯ ac to K ∗ is a multiplicative map into k ∗ ,3) The restriction of ¯ ac to the group of units of O K coincides with the restrictionof the residue map to the group of units. See [73].After choosing a uniformizing element π v in a valued field K , we can define ¯ ac ( x ) = Res ( xπ − v ( x ) v ) , where Res denotes the residue map O → k modulo M K (extended to K by zero). ODEL THEORY OF ADELES AND NUMBER THEORY 21 Each non-Archimedean K v has an ¯ ac -map defined as above. We can also get an ac -map from a cross section which exists in an ℵ -saturated valued field (see [16]).By Pas’ Theorem [73, Theorem 4.1, pp.155], the theory of Henselian valued fieldsof equicharacteristic zero admits quantifier elimination in the language L Denef − P as for the field sort relative to the other sorts. Since this quantifier elimination holdsfor all ultraproducts of K v of unbounded residue characteristic with respect to anon-principal ultrafilter, it follows that the quantifier elimination holds uniformlyfor all K v of residue characteristic p > N for some N . This elimination is effective,as in Theorem 3.3, i.e. N and the quantifier free formulas can be effectively given.In Pas [74], Pas defined an extension L P as of the Denef-Pas language got byadding the higher ac-maps ¯ ac n , for all n ≥ , and infinitely many sorts L Res ( n ) equipped with the language of rings, and interpreted in a valued field K as O K / M n , with maps Res n interpreted as the residue maps O K → O K / M nK ex-tended to K be zero (together with connecting maps between the residue sorts).An interpretation of ¯ ac n is Res n ( xπ − v ( x ) ) .By [74], each non-Archimedean K v has quantifier elimination for the field sortrelative to the other sorts in L P as .Combining these results we can deduce. Corollary 3.1 (Follows from Pas [73] and [74]) . Let K be a number field withnon-Archimedean completions K v , v ∈ V finK . Given a formula φ ( x , . . . , x n ) fromthe ring language or the language of valued fields, there is by an effective procedure • an integer N ≥ , • an L Denef − P as -formula ψ ( x , . . . , x n ) that has no quantifiers over the fieldsort and has its quantifiers over the value group Γ and residue field k , • for each K v of residue characteristic p < N , an L P as -formula ψ p ( x , . . . , x n ) and finitely many positive integers p , . . . , p l such that ψ p ( x , . . . , x n ) isquantifier-free in the field sort and has its quantifiers ranging from thesorts L Res ( p ) , . . . , L Res ( p l ) involving the maps ¯ ac p , . . . , ¯ ac p l ,such that • if K v has residue characteristic greater than N , then φ and ψ are equivalentin K v , • if K v has residue characteristic p < N , then ψ and ψ p are equivalent in K v .3.2.5. The language of Basarab [6] L Basarab = ( L field , L r : r ≥ has infinitelymany sorts where L r = ( L rings,r , L group,r , L group , v, θ r , v r ) with the language of valued fields for the field sort L field , the language of ringsfor the sorts L rings,r for all r , the language of groups for the sorts L group,r , and thelanguage of ordered abelian groups with top element ∞ for the sort L group . The sort L field is for the field, the sorts L rings,r are for the residue rings O K,r := O K / M K,r , where M K,r = { a ∈ O K : v ( a ) > rv ( p ) } , the sorts L group,r are for the quotients K × / M K,r , and the sort L group is for thevalue group Γ . The symbol v is interpreted as the valuation and θ r is interpretedas the map θ r ( a + M K, r ) = a (1 + M K,r ) defined on the subset O K, r \ ( M K,r / M K, r ) of O K, r with values in K × / M K,r . v r is interpreted as the map induced from the valuation on the disjoint union O K / M K, r ∪ K × / M k into Γ ∪ {∞} . The structure K r = ( O K, r , K × / M K,r , Γ , θ r , v r ) is called the mixed r -structure assigned to K . Note that M K, = M K is themaximal ideal of O K .If K has residue characteristic zero, then O K / M K,r is the residue field k of K and K × / M K,r = K × / M K for all r . So all the mixed r -structures assigned to K become the triple ( k, K × / M K , v ) with the exact sequence → k K × → K × / M K Γ → . By [6] (Theorem B, page 57), the theory of Henselian valued fields of char-acteristic zero with large residue field of fixed characteristic p admits quantifierelimination in L Basarab for the field sort relative to the sorts L r , r ≥ .Basarab’s Theorem B in [6] also applies to residue characteristic zero Henselianfields, hence to ultraproducts of Henselian valued fields of unbounded residuecharacteristic. It follows that given a formula φ ( x , . . . , x n ) there is a formula ψ ( x , . . . , x n ) which is quantifier-free in the field sort and has its quantifiers fromthe sorted language ( k, K × / M K , v ) such that φ and ψ are equivalent in anyHenselian valued field K of residue characteristic greater than some N dependingon φ only, and N can be found effectively.Combining the residue characteristic zero and and fixed residue characteristic p > results of Basarab, we deduce the following. Corollary 3.2 (Follows from Basarab [6, Theorem B]) . Let K be a number fieldwith non-Archimedean completions K v , v ∈ V finK . Given an L Basarab -formula (orin particular a formula from the ring language or the language of valued fields) φ ( x , . . . , x n ) there is (by an effective procedure) • an integer N ≥ , ODEL THEORY OF ADELES AND NUMBER THEORY 23 • an L Basarab -formula ψ ( x , . . . , x n ) that has no quantifiers over the field sortand has its quantifiers from the sorts ( k, K × / M K , v ) , • for each K v of residue characteristic p < N , an L Basarab -formula ψ p ( x , . . . , x n ) and an integer r p ≥ such that ψ p ( x , . . . , x n ) is quantifier free in the fieldsort and has its quantifiers from the sorted language L r p ,such that • if K v has residue characteristic greater than N , then φ and ψ are equivalentin K v , • if K v has residue characteristic p < N , then ψ and ψ p are equivalent in K v . Note: Other many-sorted languages for quantifier elimination in Henselianvalued fields have been introduced by Weispfenning [87] and Kuhlmann [63]. Theseare closely related to L Basarab . [33] . This language has three sorts ( L adeles,ring , L adeles,value , L adeles,residue , v ∗ , ¯ ac ∗ ) , with the language of rings for the sort L adeles,ring , the language of ordered abeliangroups together with a top element ∞ for the sort L adeles,value , and the languageof rings for the sort L adeles,residue . L adeles,ring is for the ring of adeles, L adeles,value for the restricted direct productof the value groups Q ′ v ∈ V fK (Γ v ∪∞ ) , where Γ v is the value group of K v and ∞ a topelement (a restricted product is with respect to the formula x ≥ , cf. Subsection19); and L adeles,residue is for the direct product of the residue fields of K v over all v ∈ V fK .The function symbol v ∗ is interpreted as the product valuation A finK → ′ Y v ∈ V fK (Γ v ∪ ∞ ) onto the lattice-ordered group Q ′ v ∈ V fK (Γ v ∪ ∞ ) defined by v ∗ ( a ) = ( v ( a ( v )) v , for a ∈ A K .The function symbol ¯ ac ∗ is interpreted as the map from A K → Q v ∈ V K k v ontothe product defined by ac ∗ ( a ) = ( ac ( a ( v ))) v .For more details and model-theoretic results on the product valuation, see [33].4. Generalized products and restricted products The model theoretic notions of generalized product of L -structures, for a lan-guage L , were introduced and studied in the works of Feferman-Vaught andMostowski (see [43]). What we call restricted product of L -structures appearsin [43] under the name of weak product. It is a substructure of the generalized product. Most of the analysis of Feferman-Vaught is for the generalized product,however the results can also be proved for restricted products as well.In [37] and [33], Macintyre and myself do this in a more general case of havinga many-sorted language with function symbols and relation symbols. We give anoutline in this Section, especially aimed at results on quantifier elimination forrestricted products.We remark that in [35] Macintyre and myself proved an analogue of the maintheorem of Feferman-Vaught for rings. Interestingly, this gives both a converse toFeferman-Vaught and at the same time axioms for rings elementarily equivalentto restricted products and adeles. The proof is a modification and a ring-theoreticanalogue of [43]. This shall be discussed in Section 8 of this paper.4.1. Language for restricted products. Let L denote a many-sorted first-order language with a set of sorts Sort andsignature Σ with relation and function symbols and equality in each sort. See[42, Section 4.3] for the basic definitions and results on many-sorted languages onwell-formed formulas, substructures. We give a few definitions.An L -embedding F : N → M is a collection of maps F σ : N σ → M σ indexed by the sorts σ , such that for any relation symbol R of sort ( σ , . . . , σ k ) , N | = R ( f , . . . , f k ) ⇔ M | = R ( F σ ( f ) , . . . , F σ k ( f k )) , and for any function symbol G of sort ( σ , . . . , σ k , σ k +1 ) , G ( F σ ( f ) , . . . , F σ k ( f k )) = F σ k +1 ( G ( f , . . . , f k )) , where f , . . . , f k , f k +1 range over elements of sorts σ , . . . , σ k , σ k +1 respectively.Note that each F σ is injective since we have equality as a binary relation oneach sort. N is said to be an L -substructure of M if N σ ⊆ M σ for all σ and the identitymaps are L -embeddings.Now suppose that ( M i ) i ∈ I is a family of L -structures. Let Π := Q i ∈ I M i . Wegive Π an L -structure and make it sorted by the set Sort , and give an interpretationof the signature Σ as follows.If σ ∈ Sort , then the σ -sort of Q i ∈ I M i is the product Q i ∈ I ( M i ) σ , where ( M i ) σ is the σ -sort of M i .The interpretation in Π of a relation symbol R of sort ( σ , . . . , σ r ) is Q i ∈ I R M i ,where R M i is the interpretation of R in M i (a subset of ( M i ) σ × · · · × ( M i ) σ r ).The interpretation in Π of a function symbol of sort ( σ , . . . , σ r , σ r +1 ) is givenby τ (Π) ( f , . . . , f r )( i ) = τ ( M i ) ( f ( i ) , . . . , f r ( i )) for all i ∈ I , where f , . . . , f r range over elements of sorts σ , . . . , σ r respectively. ODEL THEORY OF ADELES AND NUMBER THEORY 25 Let Φ( x , . . . , x r ) be an L -formula. Define [[Φ( f , . . . , f r )]] := { i : M i | = Φ( f ( i ) , . . . , f r ( i )) } , where f , . . . , f r range over elements of sorts σ , . . . , σ r respectively. This is amany-sorted generalization of Feferman-Vaught’s Boolean values.Let L + Boolean denote a given enrichment of the language of Boolean algebras L Boolean that contains the predicate F in ( x ) (e.g. L finBoolean and L fin,resBoolean from Sub-section 3.1).Let P ( I ) denote the Boolean algebra of subsets of I and P ( I ) + its expansion toan L + Boolean -structure. Definition 4.1. For any L + Boolean -formula Ψ( z , . . . , z m ) and L -formulas Φ , . . . , Φ m in the free variables x , . . . , x r of sorts σ , . . . , σ r respectively, let Ψ ◦ < Φ , . . . , Φ m > denote the relation defined by Π | = Ψ ◦ < Φ , . . . , Φ m > ( f , . . . , f r ) ⇔ P ( I ) + | = Ψ([[Φ ( f , . . . , f r )]] , . . . , [[Φ m ( a , . . . , a r )]]) , where a , . . . , a r range over elements of sorts σ , . . . , σ r respectively. Expand L by adding a new relation symbol for each of these relations. Let L + Boolean ( L ) denote the resulting language. This gives Π an L + Boolean ( L ) -structure,generalizing the 1-sorted case in Feferman-Vaught [43]. See also [33].We now define a many-sorted generalization of the Feferman-Vaught notion ofa generalized product.Suppose for each sort σ we have a formula Φ σ ( x ) in a single free variable x ofsort σ . Suppose that for all σ and i the set S σ,i = { a ∈ Sort σ ( M i ) : M i | = Φ σ ( a ) } is an L -substructure of M i . In particular, for any function symbol F of sort ( σ, τ ) ,if a ∈ S σ,i , then F ( a ) ∈ S τ,i for all i . Definition 4.2. With the above assumptions and notation, define the restrictedproduct of M i with respect to the formulas Φ σ ( x ) , denoted by Q (Φ σ ) i ∈ I M i , to bethe structure sorted by Sort , such that for σ ∈ Sort , its σ -sort is the set of all a ∈ Q i ∈ I ( M i ) σ such that [[ ¬ Φ σ ( a )]] is finite. Q (Φ σ ) i ∈ I M i is an L + Boolean ( L ) -substructure of Π . Indeed, if F is a function symbolof sort ( σ, τ ) , and a is in the σ -sort of Q (Φ σ ) i ∈ I M i , then since the sets S σ,i are L -substructures of M i for all i , we have F in ([[ ¬ Φ τ ( F ( a )]]) , so F ( a ) is in τ -sort of Q (Φ σ ) i ∈ I M i . Clearly Q (Φ σ ) i ∈ I M i is L + Boolean ( L ) -definable. Quantifier elimination in restricted products. The following theorem, originally proved by Feferman-Vaught and extended tomany-sorted case by Macintyre and myself in [33] gives quantifier elimination for Q (Φ) i ∈ I M i in the language L + Boolean ( L ) . Theorem 4.1 (Feferman-Vaught [43], Derakhshan-Macintyre[33]) . For any L + Boolean ( L ) -formula Ψ( x , . . . , x n ) , where x , . . . , x n are free variables of sorts σ , . . . , σ n re-spectively, one can effectively construct L -formulas ree Ψ ( x , . . . , x n ) , . . . , Ψ m ( x , . . . , x n ) with the same free variables, and an L + Boolean -formula Θ( X , . . . , X m ) such that forany indexed family ( M i : i ∈ I ) of L -structures and any a , . . . , a n ∈ Q (Φ) i ∈ I M i , (Φ) Y i ∈ I M i | = Ψ( a , . . . , a n ) if and only if P ( I ) + | = Θ([[Ψ ( a , . . . , a n )]] , . . . , [[Ψ m ( a , . . . , a n )]]) . Theorem 4.1 applies to many restricted products. To apply it to the ring ofadeles A K and the ring of finite adeles A finK represent A K (resp. A finK ) as therestricted product of the K v , where v ∈ V K (resp. v ∈ V finK ) with respect to theformula Φ val ( x ) from Theorem 2.2 that uniformly defines the valuation rings of K v for all v . This is uniform for all number fields K . Note 4.1. In Section 10.1 we show that a variant of this theorem holds for theadele spaces of varieties V ( A K ) = Q ′ v ∈ V K V ( K v ) , where V is an algebraic varietyand the restricted product is with respect to V ( O v ) . The space V ( A K ) coincideswith the set of solutions of the defining equations of V in the adeles A K . In applying Theorem 4.1, we choose a language L extending L rings such thatall the completions K v are L -structures. If L is a definitional extension of L rings ,then A K is an L rings -structure. If L is not a definitional extension of L rings ,then we consider A K as an L + Boolean ( L ) -structure. Taking L to be L Mac , L Belair , L Denef − P as , or L Basarab and applying Theorem 4.1 we get quantifier eliminationsfor A K in L + Boolean ( L ) .Taking L to be L rings and applying Theorems 3.1 and 4.1 we get. Corollary 4.1 (Derakhshan-Macintyre [37]) . Let ϕ ( x , . . . , x n ) be an L rings -formula.Then there are L rings -formulas ψ ( x , . . . , x n ) , . . . , ψ l ( x , . . . , x n ) , where l ≥ , and a Boolean combination Ψ( x , . . . , x n ) of F in ([[ ψ k ( x , . . . , x n )]]) and C j ( ψ s ([[ x , . . . , x n ]])) , where k, s ∈ { , . . . , l } , such that A K | = ∀ x . . . ∀ x n ( ϕ ( x , . . . , x n ) ⇔ Ψ( x , . . . , x n )) . ODEL THEORY OF ADELES AND NUMBER THEORY 27 Corollary 4.2 (Derakhshan-Macintyre [37]) . Let n ≥ . A definable subset of A nK in the language of rings is a Boolean combination of sets defined by the L rings -formulas (1) F in ([[ ψ ( x , . . . , x n )]]) , (2) C j ([[ φ ( x , . . . , x n )]]) ,where j ≥ , and ψ and φ are L rings -formulas. To see that F in ([[ ψ ( x , . . . , x n )]]) and C j ( φ ([[ x , . . . , x n ]])) are L rings -formulas,we use Theorem 2.1. For example, F in ([[ ψ ( x , . . . , x n )]]) can be expressed as"there exists an idempotent e such that F in ( e ) holds and e is the supremum ofall the minimal idempotents e such that e A K | = ψ ( x ( e ) , . . . , x n ( e )) . Note 4.2. By Theorem 4.1, the formulas Ψ j do not depend on the choice of thefamily of structures M i , hence the quantifier eliminations in Theorem 4.1 andCorollaries 4.1 and 4.2 are independent of the number field K . The case of finite index set. If the index set I is finite, then Theorem 4.1 becomes the following statement,which is of independent interest and extends results going back to Mostowski forthe 1-sorted case. Theorem 4.2. [37] Consider a finite index set I = { , . . . , s } . Let ψ ( x , . . . , x n ) be an L -formula. Then there are finitely many t -tuples of formulas ( ψ ( x , . . . , x n ) , . . . , ψ t ( x , . . . , x n )) for some t ∈ N , and elements S , . . . , S k , for some k ∈ N , where each S j is in P ( I ) t (where P ( I ) denotes the powerset of I ) such that for arbitrary L -structures M , . . . , M s , and any a , . . . , a n in M × · · · × M s M × · · · × M s | = ψ ( a , . . . , a n ) if and only if for some j the sequence [[ ψ ( a , . . . , a n )]] , . . . , [[ ψ t ( a , . . . , a n )]] is equal to S j .Proof. Follows immediately by Theorem 4.1. (cid:3) Corollary 4.3. [37] Let A ⊂ M × · · · × M s be an L -definable set. Then A is afinite union of rectangles B × · · · × B s , where B i is a definable subset of M i .Proof. Follows immediately by Theorem 4.2. (cid:3) Remark 4.1. For any finite subset S = { v , . . . , v l } of V K containing the allArchimedean valuations, we can view A K as the finite direct product K v × · · · × K v l × A SK , where A SK is the restricted direct product of K v over all v / ∈ S with respect to therings O v . Remark 4.2. For any subset T = { i , . . . , i l } of the index set I , we can view Q (Φ) i ∈ I M i as the finite direct product M i × · · · × M i l × (Φ) Y i/ ∈ T M i , where Q (Φ) i/ ∈ T M i is the restricted direct product of the M i with respect to the for-mula Φ( x ) . In this way Theorem 4.2 and Corollary 4.3 can be applied to A K and Q (Φ) i ∈ I M i .By Corollary 4.3, the definable subsets of Y v ∈ S K v × A SK (resp. Y t ∈ T M t × (Φ) Y t/ ∈ T M t ) , are finite unions of sets of the form X × · · · × X l × Y where X j is a definable subset of K v (resp. M i j ) for v ∈ S (resp. for j = 1 , . . . , l ),and Y is a definable subset of A SK (resp. Q (Φ) i/ ∈ S M i ).As Y is a restricted product, Theorem 4.1 applies to it (and for example givesresults on its definable subsets). This direct product decomposition can be specially useful in calculating mea-sures of definable sets in A K and Q (Φ) i ∈ I M i .Taking S to be the set of all Archimedean valuations, this way one can comparethe measures got from the Archimedean factors (a finite product) with those gotfrom the non-Archimedean factors (an infinite restricted direct product).4.4. An example from algebraic groups: Weil’s conjecture on Tamagawanumbers. For the definition of the adele space of a variety V ( A K ) and of Tamagawanumber see Subsection 10.1. These are naturally definable subsets of A mK for some m . They also have the structure of a model-theoretic restricted product (seeSubsection 10.1), and so the results of Subsection 4.2 are applicable to them. ODEL THEORY OF ADELES AND NUMBER THEORY 29 Let G be an algebraic group over a number field K . Tamagawa proved thatthe volume of SO n ( f )( A K ) /SO n ( f )( K ) with respect to the Tamagawa measure isequal to , where SO n ( f ) denotes the special orthogonal group of a non-degeneratequadratic form f in n -variables with rational coefficients, and proved that thisis equivalent to Siegel’s famous formula for the Mass of a quadratic form (calledSiegel’s Mass formula), thereby giving also a new volume-theoretic proof of Siegel’sformula.Weil gave a more general conjecture that the Tamagawa volume of G ( A K ) /G ( K ) is equal to for all simply connected semi-simple groups and proved it for manyclassical groups. Langlands proved it for all Chevalley groups. Kottwitz provedthe general case. The story of Weil’s conjecture and related results has been quiteinteresting. Eskin-Rudnick-Sarnak gave a new proof of Siegel’s Maas formula usingergodic theory. For details and references see [76, Chapter 5] (and Kneser’s articlein [13] for the early results). See also Subsection 10.2.In each of these, the volume is calculated after first computing the volume ofthe points over the finite adeles A finK , then computing the product of the volumesof the set of points over the Archimedean factors, and then finally comparing thetwo quantities. Mysteriously in all these cases, the product is an integer.As stated above, this method of calculating volumes can be carried out fordefinable sets by Theorems 4.1 and 4.2 and Corollaries 4.2, 4.1, and 4.3. It is apowerful method for calculating adelic volumes.5. Definability in adeles Definable subsets of A mK . Let L be a language for the K v . Given a subset I of V finK , a formula φ ( x , . . . , x n ) from L , and a , . . . , a n ∈ A K , we denote [[ ψ ( a , . . . , a n )]] I = { v ∈ I : K v | = φ ( a ( v ) , . . . , a n ( v )) } . Theorem 5.1 (Derakhshan-Macintyre [37]) . Let K be a number field and n ≥ .Let X be a definable subset of A nK defined by a formula φ ( x , . . . , x n ) that is anyof the following • an L rings -formula or a formula of the language of valued field, • an L finBoolean ( L Basarab ) -formula (resp. an L finBoolean ( L Denef − P as ) -formula oran L finBoolean ( L P as ) -formula),Then there is a finite set S = { v , . . . , v t } of non-Archimedean valuations effec-tively computable from X , an integer N ≥ , and L Basarab (resp. L P as ) formulas ψ v , . . . , ψ v t , such that X is a Boolean combination of the following sets: (1) { ( a , . . . , a n ) ∈ A nK : P ( V K ) + | = Θ( a , . . . , a n ) ∧ V v j ∈ T K v j | = ψ v j ( a ( v j ) , . . . , a n ( v j )) } , (2) { ( a , . . . , a n ) ∈ A nK : P ( V K ) + | = F in ([[ ψ ( a , . . . , a n )]] } , where T ⊆ S , and Θ( a , . . . , a n ) is a conjunction from the following set of condi-tions • C j ([[ ϕ ( a , . . . , a n )]] real ) • C k ([[ ϕ ( a , . . . , a n )]] complex ) • C s ([[ ϕ ( a , . . . , a n )]] V finK \ S ) such that j, k, s ≥ , and the following hold: • ϕ is quantifier-free in the language of ordered rings, • ϕ is quantifier-free in L rings , • ϕ and ψ are L Basarab -formulas (resp. L Denef − P as -formulas) that are quantifier-free in the field sort and have their quantifiers from the residue field sort, • for every j , ψ v j is an L Basarab -formula (resp. an L P as -formula) that isquantifier-free in the field sort and has its quantifiers from the sort L r ( v j ) (resp. the sort L Res r ( vj ) ) for some r ( v j ) ≥ depending on v j .A definable subset of the truncated restricted product Q v ∈ V finK \ S K v defined by aformula φ as above is a Boolean combinations of sets of the types (1) { ( a , . . . , a n ) ∈ A nK : C j ([[ ϕ ( a , . . . , a n )]]) } , (2) { ( a , . . . , a n ) ∈ A nK : F in ([[ ψ ( a , . . . , a n )]] } ,where ψ and ϕ are L Basarab or L Denef − P as formulas that are quantifier-free in thefield sort and have all their quantifiers from the residue field sort. If K = Q since we have uniform quantifier elimination for Q p for all p in L Belair ,we can get a simpler description. Theorem 5.2 (Derakhshan-Macintyre [37]) . Let X be a subset of A n Q in the lan-guage of rings or the language of Belair, where n ≥ . Then X is a Booleancombination of sets of the following types: (1) { ( a , . . . , a n ) ∈ A nK : B + K | = Θ( a , . . . , a n ) } (2) { ( a , . . . , a n ) ∈ A nK : B + K | = F in ([[ ψ ( a , . . . , a n )]]) } ,where Θ( a , . . . , a n ) is a conjunction from the following statements (1) C j ([[ ϕ ( a , . . . , a n )]] real ) (2) C k ([[ ϕ ( a , . . . , a n )]] complex ) (3) C s ([[ ϕ ( a , . . . , a n )]] na ) ,and j, k, s ≥ , ϕ is quantifier-free in the language or ordered rings, ϕ isquantifier-free in L rings , and ϕ and ψ are quantifier-free in L Belair . Remarks 5.1. (1) In Theorems 5.1 and 5.2 a special case of the sets in the clause (1) are setsof the form { ( a , . . . , a n ) ∈ A nK : P ( V K ) + | = [[Θ( a , . . . , a n )]] = 1 } . In this case we call X a definable set of Type I. ODEL THEORY OF ADELES AND NUMBER THEORY 31 (2) If φ is from L rings or the language of valued fields, then P ( V K ) + canbe replaced by the Boolean algebra of idempotents B + K , thus obtaining aquantifier-elimination that takes place within the ring of adeles. (3) The plus + in P ( V K ) + and B + K indicate that the Boolean algebras P ( K ) and B K are enriched with the predicates of the expanded language. Measurability of definable subsets.Theorem 5.3 (Derakhshan-Macintyre [37]) . A definable subset of A nK , n ≥ , inthe language of rings is measurable. Remark that measurability for a subset of A nK , n ≥ , is with respect to theproduct measure induced from a measure on A K (cf. 2.2).To give an idea of the proof suppose X is defined by F in ([[Ψ( x , . . . , x n )]]) .Then X = { ( a , . . . , a n ) ∈ A nK : B + K | = F in ([[Ψ( a , . . . , a n )]])= [ F ( \ v ∈F { ( a , . . . , a n ) : K v | = Ψ( a ( v ) , . . . , a n ( v )) }∩ \ w / ∈F { ( a , . . . , a n ) ∈ A nK : K w | = ¬ Ψ( a ( w ) , . . . , a n ( w )) } ) , where F ranges over all the finite subsets of V K . Then one uses the fact thatdefinable subsets in K v are finite unions of locally closed sets (cf. 3.2) hencemeasurable. Note 5.1. We also have the following strengthening in [37] . Let L be any expansionof the language of rings with the property that the L -definable subsets of K nv , forany n ≥ and v ∈ V K , are measurable. Then any L fin,res ( L ) -definable subset of A nK , where n ≥ , is measurable. Note 5.2. The language L fin,res ( L ) has more expressive power than L rings for theadeles. Countable unions and intersections of locally closed sets. The proof of Theorem 5.3 shows the following. Corollary 5.1 (Derakhshan-Macintyre [37]) . A definable subset of A nK in thelanguage of rings or the language L finBoolean ( L ) , where L is L Denef − P as or L Belair , is acountable union or countable intersection of locally closed sets (in adelic topology). Indeed, let φ be formula from L rings or L finBoolean ( L ) as in Corollary5.1 . Then itis easily seen that, • sets of the form { ¯ a : [[ φ (¯ a )]] = 0 } and { ¯ a : [[ φ (¯ a )]] = 1 } are finite unions oflocally closed sets, • sets of the form { ¯ a : F in ([[ φ (¯ a )]]) } are countable unions of locally closedsets, • set of the form { ¯ a : ¬ F in ([[ φ (¯ a )]]) } are countable intersections of locallyclosed sets,Similarly for the C j ( x ) .This description of definable sets is optimal and can not be improved. Eventhough the definable subsets of K mv , for any v (Archimedean or non-Archimedean)and m ≥ , are finite unions of locally closed sets (by quantifier elimination, cf.3.2), this does not hold for A Q as the following example shows. Example 5.1. [37] Let X = { a ∈ A K : F in ([[ a = a ]]) } . Then X is not a finiteunion of locally closed sets in adelic topology, equivalently, X is not a Booleancombination of open sets (cf. [37] for details). Euler products and zeta values at integers. Measures of definable sets in A mK are closely related to values of zeta functionsat integers. The following is proved in [37]. Theorem 5.4 (Derakhshan-Macintyre [37]) . Let n ≥ be an integer. • ζ ( n ) − , the Euler product Q p ≡ mod (1 − p − n ) , and Euler products of theform Q p ∈ S (1 − p − n ) , where S is a set of primes of the form { p : F p | = σ } and σ is a sentence of the language of rings, are measures of L rings -definablesubsets of A Q . • ζ ( n ) is the measure of an L finBoolean ( L Denef − P as ) -definable subset of A Q . In the proof we show that, for any n ≥ , the number (1 − p − n ) − is the measureof a subset of Q p that is L Denef − P as -definable independently of p (with n as its onlyparameter). The Euler product Q p ≡ mod (1 − p − n ) relates to the zeta functionof the quadratic field Q ( i ) (see [50]). Problem 5.1. [37] Generalize Theorem 5.4 to number fields. Problem 5.2. [37] What can one say about measures of definable subsets of A nK ,where n ≥ ? Definable subsets of the set of minimal idempotents. Recall the correspondence between minimal idempotents in A K and valuationsof K . The following question naturally arises. What are the ∅ -definable subsets ofthe set of minimal idempotents in A K ? Note that the reason to have definabilitywithout parameters in the question is that if we allow parameters then every subsetof the set of minimal idempotents is definable as is easily seen by taking sup andinf of idempotents. ODEL THEORY OF ADELES AND NUMBER THEORY 33 Let g ( x ) be a polynomial over Z in a single variable x . Let P ( g ) denote the setof primes p such that the reduction of g ( x ) modulo p has a root in F p .In [1], Ax proved that if σ is an L rings -sentence, then there are g ( x ) , . . . , g n ( x ) ∈ Z [ x ] such that { p : F p | = σ } is a Boolean combination of the sets P ( g ) , . . . , P ( g n ) .The following gives an answer to the question above in terms of Ax’s Booleanalgebra. Theorem 5.5 (Derakhshan-Macintyre [37]) . Let K be a number field. Let X bea parameter-free L rings -definable subset of the set of minimal idempotents in A K .Then the following hold. • X is a union of sets of the form { v : K v | = σ } , where σ is an L rings -sentence, together with one of the following:i) all Archimedean v ,ii) all real v ,iii) all complex v , • There is a finite subset F of X containing all the minimal idempotentssupported on the Archimedean valuations such that X \ F consists of min-imal idempotents corresponding to valuations from a union of sets of theform { v : k v | = σ } , where σ is an L rings -sentence and k v is the residue fieldof K v .If K = Q , then there is a finite Boolean combination B of P ( g ) . . . . , P ( g n ) for some g ( x ) , . . . , g n ( x ) ∈ Z [ x ] such that X \ F consists of the minimalidempotents corresponding to the primes from B . A question of Ax on decidability of all the rings Z /m Z Ax’s question. In his fundamental paper [1] on the model theory of finite and pseudofinite fields,Ax asked (Problem 5, page 270) if the elementary theory of all the rings Z /m Z ,for all m > , is decidable. In other terms, given a sentence φ of the language ofrings, whether it is possible to decide that φ holds in Z /m Z , for all m > .If we take the m to range over the primes p , then decidability of a sentence inall the F p is proved in [1] as a consequence of the axiomatization and decidabilityof the theory of pseudofinite fields. Such methods do not give decidability resultsfor finite rings beyond fields which is necessary for solving Ax’s problem.In [37] Macintyre and myself gave a positive solution to Ax’s problem by re-duction to the L rings -decidability of A Q using the definability of F in and of theBoolean algebra of idempotents. We give a sketch of this below. We also give asketch of a proof of decidability of the rational adeles A Q which uses besides ourformalism and machinery, only Ax’s main result in [1]. This shows the usefulness of adelic methods (here on finite rings) where previoustechniques would not suffice. We hope that adelic methods can be used in otherdecision problems too.Remark that similar decidability proofs via adeles, are given in works on D’Aquinoand Macintyre related to a question of Zilber and on a model-theoretic analysis ofquotients of non-standard models of Peano arithmetic, see [26].6.2. Reducing Ax’s problem to adelic decidability. Let φ be an L rings -sentence in prenex normal form Q x . . . Q m x m ψ ( x , . . . , x n ) , where Q i is either ∀ or ∃ , and ψ is a disjunction of conjunctions of the form f ( x , . . . , x n ) = 0 ∧ · · · ∧ f k ( x , . . . , x n ) = 0 ∧ g ( x , . . . , x n ) = 0 ∧ · · · ∧ g r ( x , . . . , x n ) = 0 , where f i and g j are polynomials over Z .Let atom ( x ) denote the statement that x is a non-zero minimal idempotent.Then φ holds in Z /m Z for all m > if and only if the following holds:for any z ∈ A K if [[ z ]] Arch = 0 ∧ F in ( supp ( z )) and ∀ e ( atom ( e ) ∧ supp ( e ) ⊆ supp ( z )) ⇒ Φ val ( ez ) ∧ ¬ Φ val (( ez ) − ) , then supp ( z ) A Q | = Q x . . . Q m x m ∃ y ( f ( x , . . . , x n ) = zy ∧ · · · ∧ f k ( x , . . . , x n ) = zy ) ∧¬∃ y ( g ( x , . . . , x n ) = zy ∧ · · · ∧ g r ( x , . . . , x n ) = zy ) . Indeed, supp ( z ) A Q is the product of the Z p where p ranges over the finitelymany primes p , . . . , p r corresponding the minimal idempotents in the support of z , and for every such p , z ( p ) has positive p -adic valuation. So supp ( z ) A Q /z A Q isisomorphic to the product Z p /p k Z p × · · · × Z p r /p k r r Z p , where k j is the p j -adic valuation of z ( p j ) for each ≤ j ≤ r . Thus φ holds in allthe rings Z /m Z for all m > if and only if φ holds in supp ( z ) A Q /z A Q for all such z , which is expressed by the above formula. ODEL THEORY OF ADELES AND NUMBER THEORY 35 Decidability of A Q . We give a sketch of the proof of decidability of the L rings -theory of A Q dueto Macintyre and myself in [37] using only a theorem of Ax [1] (see also [45],or Theorem 31.2.4 (a) in [44]) and Corollary 4.1. The first proof of decidabilityof A K (in the language of generalized products of Feferman-Vaught) is due toWeispfenning [86]. Our proof is simpler.The following fundamental theorem about model theory of finite fields is whatwe need. Theorem 6.1 (Ax [1]) . Let φ be an L rings -sentence. It is possible to decide if φ is true in F p for almost all p , and if so to list the exceptional primes. Now we give the adelic decision procedure. Let ϕ be an L rings -sentence. ByCorollary 4.1, it suffices to decide the following statements: ( I ) F in ([[ ψ ]]) , ( II ) C j ([[ φ ]]) ,where ψ, φ are L rings -sentence. The decision procedure for ( I ) . Since the number of Archimedean normalizedvaluations is finite, it suffices to decide F in ([[ ψ ]] na ) (which says ψ holds in finitelymany Q p ). By Theorem 3.3 (Ax-Kochen-Ershov) there is an L rings -sentence τ andan effectively computable C > such that for any prime p ≥ C Q p | = ψ ⇔ F p | = τ. Thus it suffices to decide whether τ holds in finitely many F p , which follows from6.1. The decision procedure for ( II ) . We want to decide if φ holds in at least j many K v , where j ≥ is given. By ( I ) we can decide if F in ([[ φ ]]) holds or not. If it doesnot hold, then C j ([[ φ ]]) holds for all j . If F in ([ φ ]]) holds, then consider ψ := ¬ φ ,which holds in almost all Q p , and the exceptional primes are exactly the primes p where φ holds in Q p . By Theorem 6.1, we can list this finite set of primes anddecide if this set has cardinality at least j or not.This concludes the proof of decidability of A Q . Remark 6.1. In [37] we show that this proof can modified to prove decidability inthe languages L + ( L ) , where L + is L finBoolean or L fin,resBoolean and L is L rings , L Denef − P as or L Basarab . Note 6.1. We could not directly apply Theorem 4.1 or Feferman-Vaught [43] to therings Z /m Z to solve the problem of Ax. We needed to reduce to the decidability of adeles A Q . For the decidability, to apply Ax, we needed the quantifier-eliminationfor generalized products given by [43] or Theorem 4.1, and quantifier-eliminationof the Boolean theory T fin . Decidability of A K . In [38], decidability is proved by Macintyre and myselffor A K for any number field K , and for A K for all K of bounded degree over Q .The following question arises. Problem 6.1. [37] Is the theory of A K for all number fields K (i.e. the set ofsentences in some given language that hold in A K for all K ) decidable? We can show the following. Theorem 6.2 (Derakhshan-Macintyre [37][38]) . The set of all L rings -sentencesthat hold in A K for all number fields K is decidable if and only if for a given p ,the theory of all finite extensions of Q p is decidable. This raises the question. Problem 6.2. [37] Is there a suitable language L such that given an L -sentencewe can decide if it holds in all finite extensions of Q p ? Kochen [59] proved decidability for the maximal unramified extension Q urp of Q p .In [36], Macintyre and myself prove model-completeness in the language of rings for Q urp and finitely ramified extensions of it, more generally for any Henselian valuedfield with finite ramification whose value group is a Z -group, and we characterizemodel-complete perfect fields with procyclic Galois group.The essence of Problem 6.2 concerns model theory of infinitely ramified exten-sions of Q p . Even the abelian case this is out of reach, i.e. we do not know whetherthe maximal abelian extension Q abp of Q p is decidable, or any model theory for it. Problem 6.3. Use adelic methods (in the spirit of our solution to the Ax problem)combined with suitable Galois theory to approach model theory of infinitely ramifiedextensions of Q p . Elementary equivalence and isomorphism for adele rings In [38] Macintyre and myself consider the question of how the A K , as K varies,are divided into elementary equivalence classes. The main tool used is Theorem2.2 on uniform definition of valuation rings in the non-Archimedean completionsof number fields.Given an adele ring A K , the completions K v , are recoverable as the "stalks" A K / (1 − e ) A K where e is a minimal idempotent, and for any minimal idempotent,the quotient above is isomorphic to some K v (Archimedean or non-Archimedean).By Theorem 2.2 the valuation, maximal ideal, and residue field of the non-Archimedean stalks can be uniformly defined or interpreted using sentences from ODEL THEORY OF ADELES AND NUMBER THEORY 37 the language of rings. Furthermore, we have a uniform definition, independent of K but depending on p , of the collection of stalks with residue characteristic p forany given p .Let p be a prime in Z . Then p lifts to finitely many primes P , . . . , P r in O K ,and we have the decomposition p O K = P e . . . P e r r .e i = e i ( P i /p ) is called the ramification index of P i over p . O K / P i is a finiteextension of F p of dimension f i = f ( P i /p ) over F p which is called the residuedegree of P i over p .If K P i is the completion of K at P i , then Q p ⊆ K P i , as valued fields, and e i and f i are respectively the ramification index and residue field degree of K P i . Wehave the fundamental inequality r X i =1 e i f i = [ K : Q ] . Note that e i , f i ≤ [ K : Q ] . The prime p is said to be unramified if e i = 1 for all i , and ramified otherwise. p splits completely if in addition all f i = 1 .7.1. The number field degree.Theorem 7.1 (Derakhshan-Macintyre [38]) . A K ≡ A K implies that [ K : Q ] =[ K : Q ] . We give an idea of the proof. Let K be a number field K . To detect thedimension of K over Q inside A K in a first-order way we first find a prime p thatsplits completely in K . Then by the fundamental inequality we must have r = n = [ K : Q ] . So we can define [ K : Q ] as the number of minimal idempotents e such that A K / (1 − e ) A K has residue field F p and v ( p ) = 1 , i.e. v ( p ) is the minimal positiveelement of the value group of e A K ∼ = K v e , where v denotes the valuation of K v e .This can be expressed by an L rings -sentence independently of K (but dependingon p ) by Theorem 2.2.To get a prime p that splits completely in K , take the normal closure L of K .By the Chebotarev density theorem (see [70],[71]) there are infinitely many primes p that split completely in L . It follows that p splits completely in K . Example 7.1. Note that the converse of Theorem 7.1 does not hold e.g. K = Q ( √ , K = Q ( √ . The case of normal extensions.Theorem 7.2 (Derakhshan-Macintyre [38]) . Suppose that K is normal over Q .If L is a number field such that A L is elementarily equivalent to A K , then L = K . The proof uses a corollary of the Chebotarev density theorem that states thatif K is a Galois extension of Q , then K is completely determined by the rationalprimes that split completely in K (see [71], Corollary 13.10 page 548).7.3. Splitting types and arithmetical equivalence. Let p be a prime. We do not assume that p is unramified in K . The splittingtype of p in K is a sequence Σ p,K = ( f , . . . , f r ) , where f ≤ · · · ≤ f r is such that p O K = P e . . . P e r r and f j is the residue degreeof P j . Note that there can be repetitions and that the ramification indices e i arenot present.For a splitting type A , define P K ( A ) = { p : Σ p,K = A } . Note that P K ( A ) is empty for all but finitely many A (since P rj =1 f j ≤ [ K : Q ] ).Let K be a number field. The (Dedekind) zeta function of K is defined by ζ K ( s ) = P a ∈ Spec ( O K ) N ( a ) − s , where N ( a ) = [ O K : a ] .In [75, Theorem 1] Perlis proves that if K and K are number fields, then ζ K ( s ) = ζ K ( s ) if and only P K ( A ) = P K ( A ) for all A . In this case K and K are said to be arithmetically equivalent.By [75, Theorem 1], if K and K are arithmetically equivalent, then they havethe same discriminant, the same number of real (resp. complex) absolute values,the same normal closure and unit groups.It follows from Theorem 2.2 that if A K ≡ A L , then for each p , Σ p,K = Σ p,K . Applying Hermit’s theorem that there are only finitely many number fields withdiscriminant bounded by any given positive integer (see [71]), one can deduce thefollowing. Theorem 7.3 (Derakhshan-Macintyre [38]) . For any given number field K , thereare only finitely many number fields L such that are A K and A L are elementarilyequivalent. This raises the question. Problem 7.1. Given a number field K , classify the number fields L such that A L is elementarily equivalent to A K . What are the elementary invariants? ODEL THEORY OF ADELES AND NUMBER THEORY 39 Elementary equivalence of adele rings - a rigidity theorem. The question asking to what extent a number field is determined by its zetafunction has a long history.A number field K that is isomorphic to any number field L such that ζ K ( s ) = ζ L ( s ) is called arithmetically solitary. Examples are any normal extension of Q .The first nonsolitary field was discovered by Gassman in 1925 who gave two fieldsof degree 180 over Q which are arithmetically equivalent but not isomorphic (cf.[75]).By a theorem of Uchida [83], two number fields L and K are isomorphic if andonly if their absolute Galois groups G K and G K are isomorphic, a theorem inthe realm of Grothendieck’s anabelian conjectures.Iwasawa [56] proved that for number fields K and L , if A K is isomorphic to A L , then ζ K ( s ) = ζ L ( s ) . The converse to Iwasawa’s theorem relates to interestingquestions. The converse is not true in general, but is true if the extensions areGalois, see [75].From Perlis’ [75, Theorem 1] and Theorem 2.2 it follows that if A K and A L areelementarily equivalent, then ζ K ( s ) = ζ L ( s ) .In [38] Macintyre and myself prove that elementary equivalence does determinethe adele rings up to isomorphism, giving a converse to Iwasawa’s theorem undera stronger hypothesis. This is a first-order "rigidity theorem " for adeles. Theorem 7.4 (Derakhshan-Macintyre [38]) . Let K an L be number fields. If A K and A L are elementarily equivalent (as rings), then they are isomorphic. The proof uses a theorem of Iwasawa in [56, pages 331-356] that for number fields K and L , the adele rings A K and A L are isomorphic if and only there there is abijection φ : V finK → V finL such that the completions K v and L φ ( v ) are isomorphicfor all v ∈ V finK . This condition is also equivalent to the condition that the finiteadeles A finK and A finL are isomorphic (cf. [38]). Problem 7.2. Find conditions under which adele rings are isomorphic. We also pose. Problem 7.3. Does Theorem 7.4 extend to algebraic groups G ? Find algebraicgroups G over Q such that if G ( A K ) and G ( A L ) are elementarily equivalent in thelanguage of groups, then they are isomorphic. Is this true when G is a Q -splitsemi-simple algebraic group over Q ? We note that one believes that for a Q -split semi-simple algebraic group G , thefield Q p is definable in the group G ( Q p ) . It would be interesting to investigateadelic versions of this and use it to approach Problem 7.3. Axioms for rings elementarily equivalent to restricted directproducts and converse to Feferman-Vaught The question and connection to nonstandard models of Peano arith-metic. The question of finding axioms for the theory of A K is part of the generalquestion of finding axioms under which any commutative unital ring is elementarilyequivalent to a restricted direct product of connected rings (a ring is connected if , are the only idempotents).This problem is solved in joint work with Macintyre in [35] and is based on thework of D’Aquino and Macintyre in [25] solving the case of products, which was inturn used by D’Aquino and Macintyre to answer a question of Zilber’s on modelsof Peano arithmetic. The problem asks for a non-standard model M of PA and k ∈ M , whether M /k M interprets arithmetic. The solution in [26] is that itdoes not interpret arithmetic and much more is proved around its model-theoretictameness.Another ingredient in the Macintyre-D’Aquino solution to Zilber’s problem isthe work D’Aquino-Macintyre and myself in [27] on truncated ordered abeliangroups.In this work we provide axioms for a class of linear orders with addition calledtruncated ordered abelian groups, and prove that any model of these axioms isan initial segment of an ordered abelian group, thus has a semi-group structurearising from a process of truncation. This work applies to quotients of valuationrings with truncated valuations. We remark that Zilber’s question was inspiredby model-theoretic insights into quantum mechanics.8.2. Axioms for the rings. We now discuss the axioms of Macintyre and myself from [35]. We shall thenprove an analogue of the results of Feferman-Vaught [43] and the results in Section4 for commutative unital rings. So we develop the analogue of the required notions(e.g. Boolean values) in the case of rings.Let R be a commutative unital ring. The set B = { x ∈ R : x = x } of idempotents is a Boolean algebra with operations e ∧ f = ef, ¬ e = 1 − e,e ∨ f = 1 − (1 − e )(1 − f ) = e + f − ef. B carries an ordering defined by e ≤ f ⇔ ef = e , which is L rings -definable. The atoms of B are by definition the minimal idempotents that are not equal to , . ODEL THEORY OF ADELES AND NUMBER THEORY 41 For any e in B , R/ (1 − e ) R ∼ = eR ∼ = R e , where R e is the localization of R at { e n : n ≥ } . The first isomorphism isstraightforward and the second isomorphism is shown in Lemma 1 in [25]. R e isthe stalk of R at e . Of special important are the R e for atoms e .We define Boolean values in the case of rings as follows. Definition 8.1. Let Θ( x , . . . , x n ) be a formula of the language of rings, and f , . . . , f n ∈ R . Then [[Θ( f , . . . , f n )]] is defined to be _ e { e : e an atom , R e | = Θ(( f ) e , . . . , ( f n ) e ) } provided W exists in B , where f e is the image of f in R e . Note that f e can be identified with f + (1 − e ) R using the above isomorphism.We augment the language of rings L rings by a unary predicate symbol F in ( x ) that is interpreted in R as a finite support element, i.e. a finite union of atoms.Let F in denote the ideal of finite support elements in R .Let L finrings = L rings ∪ { F in ( x ) } . We fix an L rings -formula ϕ ( x ) in the singlevariable x .Let A ϕ denote the following axioms expressed as L finrings -sentences. As in Sub-section 3.1, T fin denotes the theory of infinite atomic Boolean algebras in thelanguage L finBoolean . Axiom 1. B is atomic. Axiom 2. [[Θ( f , . . . , f n )]] exists (an an element of B ). Axiom 3. For any atomic formula Θ( x , . . . , x n ) of the language of rings, R | = Θ( f , . . . , f n ) ⇔ B | = [[Θ( f , . . . , f n )]] = 1 . Axiom 4. ( B , F in ) | = T fin , and for all L rings -formulas Θ( x , . . . , x n , w ) and f , . . . , f n ∈ R there is a g ∈ R such that if [[ ∃ w Θ( f , . . . , f n , w )]] ∩ ¬ [[ ∃ w ( ϕ ( w ) ∧ Θ( f , . . . , f n , w ))]] ∈ F in, then [[ ∃ w Θ( f , . . . , f n , w )]] ∩ ¬ [[Θ( f , . . . , f n , g )]] ∈ F in. Note 8.1. A special case of Axiom 4 is the following. Axioms 4’. For all Θ( x , . . . , x n , w ) and f , . . . , f n ∈ R , there is a g ∈ R suchthat if [[ ∃ w ( ϕ ( w ) ∧ Θ( f , . . . , f n , w ))]] is cofinite in [[ ∃ w Θ( f , . . . , f n , w )]] , then [[ ∃ w Θ( f , . . . , f n , w )]] is cofinite in [[Θ( f , . . . , f n , g )]] . Here "cofinite" reallymeans cofinite. Axiom 5. ∀ x ( F in ([[ ¬ ϕ ( x )]])) .Let ( M i ) i ∈ I be a family of L rings -structures. Axioms 1-5 hold Q ( ϕ ) i ∈ I M i , therestricted product of M i with respect to ϕ ( x ) (for Axiom 4 use Axiom of Choice).8.3. The ring-theoretic Feferman-Vaught and converse to Feferman-Vaught.Theorem 8.1 (Derakhshan-Macintyre [35]) . Let ϕ (¯ x ) be an L rings -formula. Let R a commutative unital ring satisfying the axioms A ϕ . Then for each L rings -formula Θ( x , . . . , x m ) there is, by an effective procedure, L rings -formulas Θ ( x , . . . , x m ) , . . . , Θ k ( x , . . . , x m ) and an L finBoolean -formula ψ ( y , . . . , y k ) such that for all f , . . . , f m in RR | = Θ( f , . . . , f m ) ⇔ ( B , F in ) | = ψ ([[Θ ( f , . . . , f m )]] , . . . , [[Θ k ( f , . . . , f m )]]) . Since R and the restricted product Q ( ϕ ) e atom of B R e have the same idempotents,the same ideal F in , and same localization R e for all atoms e , the same restrictingformula ϕ , and satisfy the axioms A ϕ , Theorem 8.1 implies the following. Corollary 8.1 (Derakhshan-Macintyre [35]) . Let ϕ (¯ x ) be an L -formula and R acommutative unital ring satisfying the axioms A ϕ . Then R ≡ ( ϕ ) Y e atom of B R e , the restricted direct product with respect to ϕ . This result can be regarded as a converse to Theorem 4.1 and the theorems ofFeferman-Vaught [43]).These axioms for restricted products connect well with the issues on elementaryinvariants for adele rings discussed in Section 7 and in [38]. ODEL THEORY OF ADELES AND NUMBER THEORY 43 Remark 8.1. Consider the L rings -formula Φ val ( x ) that defines the valuation ringof all non-Archimedean K v from Subsection 2.7, and the associated axiom system A Φ val ( x ) . If we augment A Φ val ( x ) by the axioms for p -adically closed fields (in [3] or [77] ) in all the stalks e A Q where e is non-Archimedean, and the axioms for realclosed fields in all the e A Q where e is real, then we get a complete system of axiomsfor adeles A Q . See [35] . Some stability theory Stable embedding. It is known that for many Henselian valued fields, the value group and theresidue field are stably embedded. See [51]. In [33], we show that the localfields K v are stably embedded in the adeles A K (via the identification of K v with e { v } A K ). Theorem 9.1 (Derakhshan-Macintyre [33]) . Let X be a definable subset of A nK with parameters from A K , where n ≥ . Let e be a minimal idempotent. Then X ∩ ( e A K ) n is definable with parameters from e A K . Problem 9.1. [33] Prove a general stable embedding theorem for the factors of arestricted product of structures with respect to a formula (defined in Section 4). In Subsection 3.2 we defined the product valuation Q v from the finite adeles A finK into the restricted product Γ of the lattice-ordered monoids Z ∪ {∞} indexedby the non-Archimedean valuations. Γ is interpretable in the ring A finK , cf. [33]. Theorem 9.2 (Derakhshan-Macintyre [33]) . The value monoid Γ of A fin Q is notstably interpreted via the product valuation map. In the proof we define the subset X of A Q consisting of idempotents which aresupported exactly on the primes p that are congruent to modulo . For this, let Ψ be a sentence that holds in Q p for exactly the primes p that are congruent to modulo , and let Ψ ′ be a sentence that holds in all non-Archimedean local fieldsand fails in all the Archimedean local fields. Then X = { x ∈ A Q : supp ( x ) = [[Ψ ∧ Ψ ′ ]] } . The image of X under the product valuation Q v is the set Y of all g in Y p ( Z ∪ {∞} ) which are at p and ∞ elsewhere. Applying the Feferman-Vaught Theorem orTheorem 4.1 to Γ and using the Presburger quantifier elimination for the factors(cf. [41]), it follows that X is not definable in the value monoid The tree property of second kind. The property of not having the tree property of the second kind N T P is ageneralization of the properties of being simple and N IP (the negation of theindependence property).It is known that ultraproducts of Q p and certain valued difference fields have N T P (cf. [17]).The theory of A K , for K a number field, has the independence property in twodifferent ways, firstly via the residue fields by Duret [40] and [44], and secondlybecause the definable Boolean algebra B K . Theorem 9.3 (Derakhshan-Macintyre [33]) . The theory of finite adeles A finK andthe theory of adeles A K do not have the property N T P . Stable formulas and definable groups. Local stability theory is the study of stability properties of a formula. There ismuch literature on this. Here we only mention that Hrushovski and Pillay in [55]develop a unified approach to local stability for Q p , R , and pseudo-finite fields.The central notion being that of a geometric field . Problem 9.2. Develop local stability theory for adeles or adele spaces of algebraicvarieties (cf. Section 10) using local stability for the fields K v using the notion ofgeometric fields in the sense of Hrushovski-Pillay. In particular, what can one sayabout restricted products of geometric fields? To what extent the model-theoreticproperties of geometric fields are preserved under restricted products? In [55], theorems are proven about groups definable in R and Q p showing theyare related to algebraic groups. Problem 9.3. What can one say about a group that is definable in A K ? Is itrelated (e.g. virtually isogenous) to G ( A K ) for an algebraic group G ? Adele geometry Adele spaces of varieties. Adele spaes of algebraic varieties were defined by Weil (cf. [85]) and are impor-tant in number theory and arithmetic geometry. In joint work with Macintyre [37]we show that the results of Section 4 apply to these spaces and they admit an in-ternal quantifier elimination and Feferman-Vaught theorem in a natural geometriclanguage.For simplicity, let V be an affine variety over a number field K . Noncanonicallychoose m and polynomials f , . . . , f e ∈ Q [ x , . . . , x n ] whose zero set is V . Forconvenience we assume K = Q . ODEL THEORY OF ADELES AND NUMBER THEORY 45 For any valuation v , consider the sets V ( K v ) and V ( O v ) . The adele space V ( A K ) is defined as the restricted direct product of the V ( K v ) with respect to the V ( O v ) . This is the union S S V S , where V S = Y v ∈ S V ( K v ) × Y v / ∈ S V ( O v ) , as S ranges over all finite subsets of V K containing all the Archimedean valuations.Remark that the adele space can be defined for any abstract variety V and isdefined independently of the choice of an affine covering, see Weil’s [85], Chapter1.2.Note that V ( A K ) coincides with the set of solutions of the polynomials f , . . . , f e in A nK , and so is naturally a definable set in A nK in the language of rings. Nev-ertheless, it is important to show that V ( A K ) can be represented as a restricteddirect product in the sense of Section 4 as that will give an internal connectionbetween definability and the measure theory of V ( A K ) , and also yields an internalquantifier-elimination and Feferman-Vaught theorem for V ( A K ) .This is done in [37] as follows. We consider V ( K v ) as a subvariety of K nv ,uniformly in v . Define the relational language L V to consist of predicates R W corresponding to all Q -subvarieties W of V l for all l = 1 , , , . . . . Suppose R W has arity l . We interpret R W in V ( K v ) as W ( K v ) . Note that this is a subset of V ( K v ) l .It is easy to see that every subset of V ( K v ) l defined by an L rings -formula ψ ( x , . . . , x l ) is L V -definable, uniformly in v . By Theorem 2.2 on the uniform L rings -definability of O v in K v , the rings O v are uniformly L V -definable. So thereis an L V -formula Φ V ( x ) that uniformly defines V ( O v ) in V ( K v ) , for all v .Thus V ( A K ) can be represented as the restricted direct product Q (Φ V ) v ∈ V K V ( K v ) relative to Φ V ( x ) . Theorem 4.1 then gives quantifier elimination and a Feferman-Vaught theorem. Theorem 10.1 (Derakhshan-Macintyre [37]) . Let L + Boolean be an extension of L Boolean containing F in ( x ) . For any L + Boolean ( L V ) -formula Ψ( x , . . . , x n ) , one caneffectively construct L V -formulas Ψ ( x , . . . , x n ) , . . . , Ψ m ( x , . . . , x n ) with the same free variables as Ψ , and an L + Boolean -formula Θ( X , . . . , X m ) suchthat for any a , . . . , a n ∈ V ( A K ) , V ( A K ) | = Ψ( a , . . . , a n ) if and only if P ( I ) + | = Θ([[Ψ ( a , . . . , a n )]] , . . . , [[Ψ m ( a , . . . , a n )]]) . Corollary 10.1 (Derakhshan-Macintyre [37]) . An L finBoolean ( L V ) -definable subsetof V ( A K ) is a Boolean combination of sets defined by the formulas • F in ([[ ψ ( x , . . . , x n )]]) , • C j ( φ ([[ x , . . . , x n ]])) ,where j ≥ , and ψ and φ are L V -formulas.Proof. Immediate by Theorem 10.1 and Theorm 3.1. (cid:3) (cid:3) Note 10.1. If W is a subvariety of V , both defined over Q . Then W ( A K ) is an L finBoolean ( L V ) -definable subset of V ( A K ) . using the Boolean condition [[ .. ]] = 1 . Note 10.2. We do not have idempotents in V ( A K ) as we have for the case ofadeles A K as V ( A K ) is merely a locally compact topological space and not a ring. Theorem 10.1 and Corollary 10.1 are suitable for proving results on definablesubsets of V ( A K ) and their measures.10.2. Tamagawa measures on adele spaces. Let V be a smooth algebraic variety defined over K . Let ω an algebraic differ-ential form on V defined over K of top degree n = dim ( V ) . For any valuation v of K , the form ω induces a measure ω v on the topological space V ( K v ) . See Weil’sbook [85], Chapter 2.2.The measure on V ( A K ) may or may not converge. It converges when V is asemisimpe algebraic group (see [85]). In many other cases it diverges, and one hasto use convergence factors. Let µ v ( V ) = Z V ( O v ) ω v . A set of convergence factors for V is defined to be a collection ( λ v ) v of strictlypositive real numbers indexed by the valuations v ∈ V K such that the product Y v ∈ V finK λ − v µ v ( V ) is absolutely convergent.The Tamagawa measure τ V on V ( A K ) derived from the form ω by means of theconvergence factors ( λ v ) v is defined to be the measure on V ( A K ) inducing in eachproduct Y v ∈ S V ( K v ) × Y v / ∈ S V ( A K ) the product measure µ − dim ( V ) K Y v ∈ V K ( λ − v ω v ) , for any finite subset S of V K containing all the Archimedean valuations, where µ k = | ∆ K | / and ∆ K is the discriminant of K . One usually puts λ v = 1 for allArchimedean v . ODEL THEORY OF ADELES AND NUMBER THEORY 47 (i) GL n and SL n . The Tamagawa measure on the additive group G a ( A K ) andon SL m ( A K ) are convergent as R G a ( O v ) ω v = 1 , and Z SL m ( O v ) ω v = (1 − q − v ) . . . (1 − q − mv ) . The Tamagawa measure on the multiplicative group G m ( A K ) and on GL m ( A K ) are divergent since R G m ( O v ) ω v = 1 − q − v . and Z GL m ( O v ) ω v = (1 − q − v ) . . . (1 − q − mv ) . In these divergent cases, we can use the convergence factors λ v = (1 − q − v ) for v ∈ V finK , and λ v = 1 for Archimedean v , to get a convergent adelic measure.Here, as before, q v is the cardinality of the residue field of K v .(ii) Hypersurfaces. Generalizing work of Weil and Tamagawa (see [85]) aroundan adelic interpretation and proof of the Siegal mass formula on quadratic forms,Ono [72] studied adele spaces of hypersurfaces.Let f ( x , . . . , x n ) be a non-constant absolutely irreducible polynomial over K .Let Ω be a universal domain containing K . Let W f = { ( x , . . . , x n ) ∈ Ω n : F ( x , . . . , x n ) = 0 } . Identify this K -open set with the non-singular hypersurface { ( x , . . . , x n , y ) ∈ Ω n +1 : F ( x , . . . , x n ) y = 1 } . Then W f ( A K ) = { ( x , . . . , x n ) ∈ A nK : f ( x , . . . , x n ) ∈ I K } . The K -open subset W f of Ω n , carries a gauge form ω induced from the form dx . . . dx n on Ω n . Ono [72] proved that the numbers defined by λ v = 1 − q − v when v is non-Archimedean, and λ v = 1 if v is Archimedean, form convergencefactors for W f . Let τ W f denote the Tamagawa measure on W f ( A K ) derived from ω by means of λ v .(iii) Zeta integrals. In Tate’s thesis [82] on the analytic continuation andfunctional equation for the zeta-function of a number field and more general zetaintegrals associated to characters of the idele class group I K /K ∗ , the measurethat is used on I K has its local factors of the form (1 − q − v ) − d ∗ x v at each non-Archimedean v where q v is the cardinality of the residue field of K v (with certainnormalizations for the Archimedean factors). Here the convergence factors are (1 − q − v ) .See Subsection 17 for a description of the zeta integrals in Tate’s thesis andmodel-theoretic approach and questions. In [37], we prove that the above normalization factors are uniformly definable. Theorem 10.2 (Derakhshan-Macintyre [37]) . Let K be a number field. There isan L Denef − P as -definable set of convergence factors (defined independently of p andwithout parameters) for the following measures on adelic points: • The Tamagawa measures on G m ( A K ) and GL m ( A K ) , • The Tamagawa measure on W f ( A K ) , where f is an absolutely irreduciblepolynomial over K . • The measure used by Tate on the ideles I K for analytic continuation andfunctional equation for zeta functions and zeta integrals of characters. Uniform definability of the convergence factors enables the use of model-theoretictools to evaluate the local p -adic integrals with respect to measures induced fromdifferential forms, following Denef, Loeser, Cluckers, and others on motivic inte-gration, cf. [28], [29], [20]. Then the results of [30] which are stated in Section14 yield analytic properties of the Euler product of the local integrals as globalintegrals. See Sections 17 and 19 for examples of Euler products in connectionwith Tate’s thesis and the Langlands program.In the model-theoretic approach to p -adic and motivic integration one integratesfunctions of the form | f ( x ) | s over definable sets, where f is a definable functionfrom Q np → Q p (or a finite extension of Q p ). It is important to try to integrateother functions. Some ideas and guiding themes for this can be found in Section19 in the context of automorphic forms. Problem 10.1. [37] What can be said about the numbers that are Tamagawameasures of adelic spaces of varieties? The work of Kontsevich and Zagier in [61] concerns periods which are complexnumbers whose real and imaginary parts are absolutely convergent integrals, overreal semi-algebraic subsets of R n , of rational functions with rational coefficients.Problem 10.1 can be regarded as an adelic version of some questions of Kontsevich-Zagier on numbers that arise as periods. One expects the Tamagawa measures ofdefinable sets to be related to L -functions, cf. [70].11. Boolean Presburger predicates and Hilbert symbol As in 3.1 L fin,resBoolean denotes the extension of L finBoolean got by adding the Presburgerpredicates Res ( n, r )( x ) for all n, r . We consider A K as a structure for the language L fin,resBoolean ( L rings ) . Theorem 11.1 (Derakhshan-Macintyre [37]) . The L fin,resBoolean ( L rings ) -theory of A K is decidable and has quantifier elimination. This follows from the decidability of the theory of infinite atomic Boolean alge-bras T fin,res in the language L fin,resBoolean proved by Macintyre and myself in [34], seealso [37] and Subsection 3.1. ODEL THEORY OF ADELES AND NUMBER THEORY 49 Even though the rational field Q is undecidable, it turns out that there is an L fin,resBoolean ( L rings ) -definable subset of A Q that contains Q ∗ . Proposition 11.1 (Derakhshan-Macintyre [37]) . Any a ∈ Q ∗ is a non-square in Q p only for an even number of p ’s. This follows from basic properties of the Hilbert symbol. Let p be a prime or p = ∞ , where Q ∞ = R . The Hilbert symbol ( a, b ) p , for a, b ∈ Q p is defined asfollows. ( a, b ) p = ( if ax + by − z has a non-zero solution ( x, y, z ) = (0 , , − otherwise.If p = 2 and | a | p = | b | p = 1 then ( a, b ) p = 1 . If a, b ∈ Q , then, ( a, b ) p = 1 for large p . The product formula for the Hilbert symbol states that for a, b ∈ Q ∗ , Y p ∈{ Primes }∪{∞} ( a, b ) p = 1 . (see [12], pp. 46).It further follows that Q p ∈{ Primes }∪{∞} ( a, b ) p = 1 if and only if the number of ( a, b ) ∈ ( Q ∗ ) such that ( a, b ) p = 1 is even.Now we can define the set of all adeles a ∈ A Q such that a ( p ) is a square at aneven number of p by the formula B + Q | = Res (2 , ¬ P ( x )]]) , where B + Q is the expansion of B Q to the language L fin,resBoolean and P ( x ) is the formulathat x is a square.Similarly, let θ ( a, b ) be the formula ∃ x ∃ y ∃ z ( ax + by − z = 0) . Let K = { ( a, b ) ∈ A Q : Res (2 , θ ( a, b )]]) } , We call this set the adelic kernel of the Hilbert symbol.By the product formula for the Hilbert symbol ( Q ∗ ) ⊆ K where the inclusion of ( Q ∗ ) ⊆ ( A Q ) is induced from the diagonal inclusion ofeach factor.We have shown the following. Proposition 11.2 (Derakhshan-Macintyre [37]) . (1) The set of all adeles a ∈ A Q such that a ( p ) is a non-square in Q p for aneven number of p is an L fin,resBoolean ( L rings ) -definable subset of A Q containing Q ∗ . (2) The adelic kernel of the Hilbert symbol K is an L fin,resBoolean ( L rings ) -definablesubset of A K containing ( Q ∗ ) . Problem 11.1. Extend Proposition 11.2 to general number fields. This should be compared to the undefinability of K in A K (true by the unde-cidability of K proved by Julia Robinson).12. Imaginaries in the adeles and the quotient of the space of adeleclasses by maximal compact subgroup of idele class group Paul Cohen (unpublished notes) and Alain Connes (see for example [21]) definedthe space of adele classes A K /K ∗ . It is the quotient of A K by the action of K ∗ bymultiplication by right. Their motivation was the Riemann hypothesis,Connes and Consani (cf. [23], [24],[22]) studied the space ˆ Z ∗ \ A Q / Q ∗ which isthe quotient of A Q / Q ∗ by the action of the maximal compact subgroup ˆ Z ∗ of theidele class group I Q / Q ∗ acting on A Q / Q ∗ from the left. ˆ Z ∗ \ A Q / Q ∗ is related tothe arithmetic site topos in [23].Boris Zilber asked whether ˆ Z ∗ \ A Q / Q ∗ is interpretable in A fin Q (i.e. its elementsare equivalence classes of a definable equivalence relation). This was proved in[39]. Theorem 12.1 (Derakhshan-Macintyre [37]) . The set ˆ Z ∗ \ A Q / Q ∗ is interpretablein A Q . This raises the question of describing the imaginaries in A K . Problem 12.1. [37] Describe the imaginaries in the theory of A finK . Hrushovski-Martin-Rideau [54] have proved that Q p admits uniform eliminationof imaginaries for all p relative to the "geometric sorts". These sorts are the spacesof lattices GL n ( Q p ) /G n ( Z p ) , for all n ≥ . Problem 12.2. [37] Prove an analogue of the Hrushovski-Martin-Rideau theoremfor the finite adeles A finK . Problem 12.3. [37] Study the model theory of the space of adele classes A K /K ∗ . A K /K ∗ is a hyperring in the sense of Krasner, i.e. a structure with multipli-cation and a multi-valued addition. See Connes [22],[24] and their references forhyperrings.In [33] Macintyre and myself prove quantifier elimination and related model-theoretic results for the Krasner hyperrings K v / M nv for given n , and theirrestricted products Q ′ v ∈ V finK K v / M nv in a language suitable for hyperrings. Problem 12.4. Prove results analogous to those in [33] on the restricted productof the Krasner hyperrings for the case of the adele class hyperring A K /K ∗ . ODEL THEORY OF ADELES AND NUMBER THEORY 51 Artin reciprocity We shall only state Artin reciprocity for a global field. There is also a versionfor local fields which does deserve model-theoretic analysis, but we will not dealwith that here.Let K ⊆ F be an extension of number fields. For any valuation v ∈ V K , consider K v , and for a valuation u of F lying over v consider F u . We have the norm map N F u /K v : F u → K v . This defines the norm map on ideles N F/K : I F → I K where N F/K ( x ( u )) is the idele in I K whose v th component is Q u | v N K u /F v ( x ( u )) .Let C K = I K /K × denote the idele class group of K . Theorem 13.1 (Artin Reciprocity) . Let K be a global field. • There exists a homomorphism called the Artin map θ K : C K → Gal ( ¯ K/K ) ab such that for every finite abelian extension F/K , the composition θ K/F of θ K with the projection Gal ( ¯ K/K ) ab → Gal ( F/K ) is surjective with kernelequal to N F/K ( C F ) . Conversely, any open subgroup N of C K of finite in-dex has the form Ker ( θ F/K ) for some finite abelian extension F/K , and C K /N ∼ = Gal ( F/K ) . • Let F/K be a finite abelian extension. Let p be a prime in K that isunramified in F . Let x p denote the idele (1 , . . . , , π v , , . . . , , where π v isa uniformizing element of K × v and v corresponds to p . Then the map θ F/K is induced (modulo K × ) from a surjective group homomorphism θ F/K : I K → Gal ( F/K ) which sends x p to F rob p .Proof. See [78]. (cid:3) Problem 13.1. (1) Give a model-theoretic analysis and interpretation of Artin reciprocity for-mulated for a restricted direct product with respect to a suitable formula (seeSection 4) of suitable structures which are definable or interpretable in thenon-Archimedean completions K v in a language with predicates for Artinsymbols in the residue fields of K v for v corresponding to an unramifiedprime in K . (2) Use the methods of Galois stratification (see [44] ) and study functorialityand uniformity in the number field K . (3) What generalizations of Artin reciprocity can be obtained this way? The term functoriality in the problem refers to the functoriality in K in Artinreciprocity (see [78]).The following question was asked by Nicolas Templier after a talk I gave inPrinceton. Problem 13.2 (Templier) . Let F/K be a finite extension of number fields. Is theimage of the norm map N F/K ( I F ) definable in I K or A K in some language? Howdoes this definition depend on the number field K ? It seems plausible that N F/K ( I F ) is definable in A K in the language of rings(using results of Section 4 on L rings -definability of Boolean values and F in ) becauseof the following. Example 13.1. If F is the extension Q p ( √ , then for any p > , one has N F/ Q p ( F ∗ ) = { z ∈ Q ∗ p : ∃ x ∃ y ∈ Q p ( z = x − y ) } . In Problem 13.2 one has to investigate what language to have for the K v or K ∗ v so that the induced restricted product language for A K or I K is suitable for therequired definability.The languages of Macintyre and Belair are well-suited to study the idelic normgroups since if L is an extension of Q p of degree n , then N L/ Q p ( L ∗ ) is an opensubgroup of Q ∗ p containing the group of nonzero n th powers ( Q ∗ p ) n and in L Belair there are constants for coset representatives for the groups P n of non-zero n thpowers in Q ∗ p for all n and all p (note: there is a bound independently of p on theindex of P n ). Problem 13.3. Study definability properties of the images of norm maps N F/K ( I F ) for all number fields F and K , uniformly in the number field, in the language forrestricted products (cf. 4) induced from the languages of Macintyre and Belair forthe factors (cf. 3.2). Templier suggested that Problem 13.2 and Sarnak suggested that definabilityin adele rings may be useful in some problems on families of L -functions in workof Sarnak-Shin-Templier [80].Generalizations of Artin Reciprocity for non-abelian extensions is one of theaspects of the Langlands Program where the approach is via representations ofadelic groups G ( A K ) , where G is a suitable algebraic group. For more on this seeSection 19.It would be interesting to have a model-theoretic approach to non-abelian ex-tensions and Artin reciprocity. Problem 13.4. (1) Let L be any of the languages in Subsection 3.2. For a finite extension F/K of number fields, is there an L fin,resBoolean ( L ) -definable subset S of A nK , for some n , or an L fin,resBoolean ( L ) -definable subset S of a restricted product Q ϕv ∈ V K M v for some L -structures M v and L -formula ϕ ( x ) , and a definable map ˆ σ F/K from S to Gal ( F/K ) generalizing the Artin map? (2) Let Gal ( ¯ K/K ) m be the maximal m -step solvable quotient of K . Is there ahomomorphism from S to Gal ( ¯ K/K ) m that gives σ F/K by composing withthe natural projection map? (3) How much this would be true beyond the solvable case? Note that S is allowed to be definable by means of predicates related to Hilbertsymbols as we have allowed L fin,resBoolean -definability. ODEL THEORY OF ADELES AND NUMBER THEORY 53 Remarks on the idele class group of Q . One can relate the idele class group to definability in adeles. Theorem 13.2. The idele class group C Q is a definable subgroup of the adeles A Q .Proof. By Proposition 6-12 page 23 in [78], I Q = Q × × R × + × Y p Z × p . (This is used in the adelic proof of the Kronecker-Weber theorem on the maximalabelian extension of Q ). Thus I Q / Q × ∼ = R × + × Q p Z × p . Clearly this is L rings -definable in A Q using the L rings -definability of Boolean values (cf. Section 2). (cid:3) Problem 13.5. Does a similar definability result hold for a general number field K ? If so how does the definition depend on the number field? Euler products of p -adic integrals Analytic properties of the Euler products. Let K be a finite extension of Q p with residue field of cardinality q . Let dx be a normalized Haar measure on K giving the valuation ring O K volume . Let ϕ ( x , . . . , x n ) be a formula of the language of rings. Let f : K n → K be an L rings -definable function. Let X = ϕ ( K ) = { ( a , . . . , a n ) ∈ K n : K | = ϕ ( a , . . . , a n ) } be the set defined by ϕ in K .In [28], Denef initiated the study of p -adic integrals of the form Z ( s, p ) = Z X | f ( x , . . . , x n ) | s dx and proved they are rational functions of q − s . This generalizes work of Igusa forthe case when f ∈ Z [ x , . . . , x n ] and X is Z np or a Zariski closed subset of it.Denef’s result gave a solution to a conjecture of Serre on rationality of p -adicPoincare series of the form P n ≥ c k T k where c k is the number of roots of f modulo p k that lift to a root in Z p . See [28]. We refer to Z ( s, p ) as a definable p -adicintegral.Pas [73] and Macintyre [ ? ] independently proved that there are uniformitiesin the shape of these rational functions as p varies if φ and f are over Q . Thesubject of motivic integration extends this uniformity and gives it a geometricmeaning. It has been developed by Denef-Loeser [29], Cluckers-Loeser [20], andHrushovski-Kazhdan [53], and has had several applications to algebraic geometry,number theory and algebra. In [30], inspired by results and problems in group theory (on zeta functionscounting subgroups of a group) and number theory (on height zeta functionscounting rational points of algebraic varieties), I considered Euler products overall primes p of such definable p -adic integrals. These Euler products are of a globalnature and relate to arithmetical questions on number fields, while the p -adic in-tegrals are of a local nature. But the uniformities that are true over all p of theshape of the rational functions can be used together with some results on algebraicgeometry and model theory of finite and p -adic fields, together with combinatorialarguments, to prove the following result. Theorem 14.1 (Derakhshan [30]) . Let Z ( s, p ) be as above. Let a p, be the constantcoefficient of Z ( s, p ) when expanded as a power series in q − s . Then the Eulerproduct over all primes p Y p a − p, Z ( s, p ) has rational abscissa of convergence α and meromorphic continuation to the half-plane { s : Re ( s ) > α − δ } for some δ > . The continued function is holomorphicon the line Re ( s ) = α except for a pole at s = α . Tauberian theorems of analytic number theory then yield. Corollary 14.1 (Derakhshan [30]) . Suppose that the Euler product Q p a − p, Z ( s, p ) can be written as the Dirichlet series P n ≥ a n n − s , then for some real numbers c, c ′ ∈ R , a + a + · · · + a N ∼ cN α ( logN ) w − a + a − α + · · · + a N N − α ∼ c ′ ( logN ) w as N → ∞ , where w is the order of the pole of Z ( s, p ) at α . Problem 14.1. Formulate an adelic version of Theorem 14.1. For this, add"Archimedean factors" to the Euler products, and write the Euler product as anadelic integral of a suitable function over a definable subset of A mK for some m ≥ . Once this is done, would the "completed Euler product" have meromorphiccontinuation to the whole complex plane? Would it satisfy a functional equation? Problem 14.1 relates to Definitions 19.3, 19.3, 19.4, Remark 19.3, Example19.1, and Problems 19.9 and 19.10. (The conjectural connections to O-minimalstructures and Hodge theory is challenging).14.2. Conjugacy class zeta functions in algebraic groups. Theorem 14.1 applies to zeta functions counting conjugacy classes in Chevalleygroups over a number field. An example of such a result is the following resultfrom [30]. Let c m denote the number of conjugacy classes in SL n ( Z /m Z ) . Thenthe global conjugacy class zeta function P m ≥ c m m − s has rational abscissa of ODEL THEORY OF ADELES AND NUMBER THEORY 55 convergence α and meromorphic continuation the half-plane { s : Re ( s ) > α − δ } for some δ > . It follows that c + · · · + c N ∼ cN α ( logN ) w − for some c ∈ R > as N → ∞ ( w as in Theorem 14.1).To be able to apply 14.1 one must show that the above zeta function is anEuler product of p -adic integrals of Denef-type over definable sets, uniformly in p . This is done in joint work with Mark Berman, Uri Onn, and Pirita Paajanenin [8]. There it is proved that the local factors of the Euler product, which areof the form P m ≥ c m q − ms where c m denotes the number of conjugacy classes inthe congruence quotient SL n ( Z p /p m Z p ) , are definable p -adic integrals and dependonly on the residue field for large p . See [8] and the survey [32] for details. Problem 14.2. Formulate the results on global conjugacy class zeta functionsadelically. Adelic height zeta functions and rational points. The adelic height zeta function is a very useful guiding example for an approachto Problem 14.1, particularly on its Archimedean factors and meromorphic prop-erties.On a general variety X over Q , one can associate a height function H L to everyline bundle L on X via Weil’s height machine. On P n , the height H = H O P n (1) associated to the line bundle O P n (1) of a hyperplane is defined by H ( x ) = q x + · · · + x n , where ( x , . . . , x n ) is a primitive integral vector representing x ∈ P n ( Q ) . Schanuelproved that as T → ∞ , card ( { x ∈ P n ( Q ) : H O P n ( Q ) ( x ) < T } ) ∼ cT n − for an explicit c ∈ R >o . See [70].For a general variety X over Q and an ample line bundle L on X , there is aheight function on X ( Q ) defined via an embedding of X into P n and pulling backthe height function on P n ( Q ) . For a subset U ⊂ X let N U ( T ) = card ( { x ∈ X ( Q ) ∩ U : H L ( x ) < T } ) . Manin has given a conjecture on the asymptotic of the numbers N U ( T ) as T → ∞ .See [70],[49]. In [49], Gorodnik and Oh prove new cases of Manin’s conjecture fororbits of group actions and for compactifications of affine homogeneous varietiesusing an ergodic theoretic approach of Duke-Rudnick-Sarnak.They consider an algebraic group G over a number field K with a representation ρ : G → GL n +1 . Then G acts on P n via the canonical map GL n +1 → P GL n +1 .Let U = u G , where u ∈ P n ( Q ) . Let X be the Zariski closure of U , and H the height function on X ( Q ) obtained by the pull back of H O P n (1) . For simplicity weonly consider the case K = Q of their results.Manin conjecture type estimate state that N T := card ( { x ∈ U ( Q ) H ( x ) < T } ) ∼ cT a . (log T ) b − for c ∈ R > and a, b ∈ Z , a > , b ≥ .In [49], Gorodnik-Oh prove this under conditions on G and the stabilizer of u ,and other conditions. Their beautiful approach is to consider U ( Q ) as a discretesubset of the adelic space U ( A Q ) (defined as a restricted product as in Section 10).The height function H ( x ) can be extended to a hight function on U ( A Q ) , de-noted by H A Q ( x ) , so that B T := { x ∈ U ( A Q ) : H A Q ( x ) < T } is compact. We have that { x ∈ U ( Q ) : H ( x ) < T } = U ( Q ) ∩ B T . Then under certain conditions the asymptotic of the numbers N T follows from anasymptotic for the volumes of B T , and for this they can make use of the ergodic-theoretic work of Duke-Rudnick-Sarnak (cf. [49]).One can ask the following question, whose positive solution would extend someresults in [49]. Problem 14.3. Prove an analogue of Theorem 14.1 on meromorphic continuationbeyond abscissa of convergence for the integrals Z U ( A K ) H A Q ( x ) − s τ, where τ is a suitable measure on U ( A Q ) . Note 14.1. Note that U ( A Q ) can be partitioned into finitely many pieces each ofwhich is a definable subset of Type I of A m Q for some m (using the usual covering ofprojective space by affine pieces). The adelic integral factors as an Euler product bystandard properties of the adelic height. [31] contains work in progress on Problem14.3. Finite fields with additive characters and continuous logic Ax’s results in [1] on decidability of the theory of F p , for all (and all but finitelymany) p , and F q , for all (and all but finitely many) prime powers (both for single p and all but finitely many p ) were proved as a result of the decidability of thetheory of pseudo-finite fields. These are defined as perfect pseudo-algebraicallyclosed fields that have exactly one extension of each degree inside their algebraicclosure. ODEL THEORY OF ADELES AND NUMBER THEORY 57 It is easy to see that infinite models of the theory of finite fields are pseudo-finite.Work of Ax [1] implies the converse statement. Kiefe [ ? ] gave a quantifier elimi-nation for the theory of pseudo-finite fields in an expansion of the ring languageby the solvability predicates stated in Section 3.2.In [14], Chatzidakis, van den Dries, and Macintyre revisited the model theoryof finite fields, and proved, among other results, generalizations of the Lang-Weilestimates for the number of F q -points of an absolutely irreducible variety definedover Q to definable sets. They also introduced a pseudo-finite measure.In [52], Hrushovski added additive characters to the language, and studied thecontinuous logic theory of pseudo-finite fields with an additive character. Firstly,he proves that the usual first-order theory is undecidable.In continuous logic, a structure M is a set together with a function R M : M n → V R for each n -ary relation R , where V R ⊆ C is a compact set called the set of valuesof φ . This gives an interpretation φ M of a formula φ within its set V φ of values asa function φ M : M n → C with compact image. See [52] for decidability, quantifierelimination, and related notions in continuous logic.If the image of φ M is the set { , } , then the pullback of is called a discretelydefinable set. For example the graphs of addition and multiplication are discretelydefinable.Let Ψ p ( n + p Z ) = exp πin/p be an additive character on the field with p elements F p , and Ψ q ( x ) = Ψ p ( tr F q / F p ( x )) , where tr F q / F p ( x ) is the trace map from F q to F p ,an additive character on the finite field F q . Add a unary function symbol to L rings to be interpreted as the additive character in the standard models F q , and let F + q = ( F q , + , ., Ψ q ) be the finite field with Ψ q in continuous logic. We note thatany other additive character on F q has the form Ψ q ( ax ) for a unique a ∈ F ∗ q , thusthe additive characters are all uniformly definable.Let T = T h ( { F + q , q prime power } ) , the theory of all finite fields with additivecharacter. In [52] Hrushovski proves that T is decidable, admits quantifier elim-ination to the level of algebraically bounded quantifiers, and is simple. He alsoproves that the pseudo-finite measure, introduced by Chatzidakis-van den Dries-Macintyre [14] is definable in this setting and its Fourier transform is also definable,and the discretely definable sets are exactly the sets definable in Ax’s theory. Healso proves that the asymptotic first-order theory of F + q with an additive character Ψ q where the characteristic is unbounded, is undecidable.These results generalize the results of Ax [1] and Chatzidakis-van den Dries-Macintyre on pseudo-finite fields to T . We remark that [52] contains applicationsto exponential sums over definable sets in finite fields. p -adic fields with additive characters and continuous logic Given a p -adic number x = P j ≥− N c j p j , where − N = v p ( x ) , the p -adic frac-tional part of x is defined by { x } = X − N ≤ j ≤− c j p j . The map ψ p ( x ) = e πi { x } is an additive character on Q p that is trivial on Z p . Givena finite extension K of Q p , the map ψ p ( tr K/ Q p ( x )) is an additive character on K that is trivial on the ring of integers O K .Enrich the language of rings by a 1-place predicate to be interpreted as thecharacter. Again the additive characters on K are uniformly definable.Hrushovski [52] proves that the first-order theory of ( Q p , ψ p ) and the asymptoticfirst-order theory of ( Q p , ψ p , p prime ) are undecidable.In analogy to [52], one can ask. Problem 16.1 (Hrushovski [52]) . Is the continuous logic theory of ( Q p , ψ p ) de-cidable? Is integration with respect to p -adic measure definable, both for single ( Q p , ψ p ) and asymptotically? Problem 16.1 is related to defining suitable Fourier transform operators on de-finable sets in p -adic fields and adeles which is related to the problems in Section19, especially Problems 19.12 and 20.2.In [52], Hrushovski gives axioms for the theory of pseudo-finite fields with anadditive character. In conversations with him, the first author learned of thefollowing question. Problem 16.2 (Hrushovski) . What are axioms for the continuous logic theory of ( Q p , ψ p ) both for single p and asymptotically? Set ψ ∞ ( x ) = e − πix for x ∈ R . Then the map ψ ( x ) = Y p ≤∞ ψ v ( x ( v )) , where x ∈ A Q , is an additive character on A Q that is trivial on Q . Let tr denotethe trace map from A K of a number field K to A Q , then ψ K ( x ) = ψ ( tr ( x )) is anadditive character on A K . Theorem 16.1. The first-order theory of ( A Q , ψ ) , where ψ ( x ) is an additive char-acter, is undecidable.Proof. Fix a prime p . Let e p denote the supremum of all the minimal idempotents e such that e A Q has residue field equal to F p . Then e A Q is isomorphic to Q p andthe p -adic additive character on Q p is the p th component of ψ ( x ) . This gives anadditive character ψ e on e A Q such that the theory of ( e A Q , ψ e ) is undecidable byHrushovski’s theorem [52] on the undecidability of ( Q p , ψ p ( x )) . Since ( e A Q , ψ e ) isdefinable in ( A Q , ψ ) , we are done. (cid:3) ODEL THEORY OF ADELES AND NUMBER THEORY 59 Problem 16.3. (1) What can one say about the continuous logic theory of A K with an additivecharacter ψ ? (2) What are axioms for this theory? Is it decidable? (3) What can one say about definability of integration and Fourier transformon A K in continuous logic? p -adic and adelic multiplicative characters and L -functions An L -function is generally defined as a Dirichlet series P n ≥ a n n − s that canbe written as an Euler product Q p P p ( s ) over primes p , where P p ( t ) is a rationalfunction. They are initially defined in a right half-plane, but admit meromorphiccontinuation beyond their half plane of convergence. Examples are the Riemannzeta function ζ ( s ) , where a n = 1 for all n , the Dedekind zeta function ζ K ( s ) ofa number field K , and L -functions associated to algebraic varieties and Galoisrepresentations. See Serre’s book [81].Suppose G = Q i ∈ I G i is a restricted product of locally compact abelian groupswith respect to open subgroups H i . If χ is a character on G , then it follows thatfor any x in G , χ ( x ( i )) = 1 for all but finitely many i ∈ I , and χ ( x ) = Q i χ ( x ( i )) (see [78]).Let K be a number field. A multiplicative character of K ∗ v is called unramifiedif χ | O ∗ v = 1 . A multiplicative character χ on the idele class group I K /K ∗ is calledan idele class character. By the above, χ = Q v χ v , where χ v is a character of K ∗ v that is unramified for all but finitely many v . It can be shown that if χ is character of K ∗ v , then χ = µ || . || s , where µ is a character of O ∗ v and Re ( s ) isuniquely determined. See [78].Given an idele class character χ , write it as µ | . | s , where µ is unitary. For each v we get character of K ∗ v defined by χ v ( t ) = χ (1 , . . . , , t, , . . . , , where t is in the v th component, hence χ ( x ) = Q v χ v ( x ) , a product that makessense as the restriction of χ v to the the units O ∗ v is trivial for all but finitely many v . The Hecke L -function attached to χ is the Euler product L ( s, χ ) = Y v ∈ V K L ( s, χ v ) , where for Archimedean v , L ( s, χ v ) is defined using Γ -functions (cf. [78]). If v isnon-Archimedean, corresponding to a prime p , and χ v is trivial on the units O ∗ v (which holds for all but finitely many v ), then one defines L ( s, χ v ) = 11 − χ v ( π v ) , where π v is a uniformizing element of K v . The function L ( s, χ ) can be analyticallycontinued to all of C and has a functional equation, as proved by Tate [82]. These L -functions generalize zeta functions of number fields and L -functions of Dirichletcharacters [78].On the other hand, an Artin L -function is an L -function that is associated to afinite-dimensional representation ρ of the Galois group Gal ( F/K ) , where F/K isa finite extension. It is defined by L ( s, ρ ) = Y v ∈ V K L ( s, ρ v ) where ρ v is the restriction of ρ to the decomposition group (cf. [78]).For the v that are associated to a prime p that is unramified in F (which is for allbut finitely many v ), one has the Frobenius conjugacy class F rob p in Gal ( F/K ) ,and L ( s, ρ v ) = 1 det ( I − ρ ( F rob p N p − s )) = Y ≤ i ≤ d − β i ( p ) N p − s , where β ( p ) , . . . , β d ( p ) are the eigenvalues of ρ ( F rob p ) . L ( s, ρ ) has meromorphic continuation to all of C . Artin conjectured that itis an entire function if ρ is irreducible and non-trivial, and proved this for one-dimensional ρ via proving the following Correspondence between Artin L -functions and Hecke L -functions : An L ( s, ρ ) attached to a one-dimensional ρ has the form L ( s, χ ) for a character χ = χ ( ρ ) of I K vanishing on K ∗ . This follows from Artin’s reciprocity law. See [78] andSection 13. Other cases of Artin’s conjecture have been proved by Langlands andothers (see [66],[67]). Problem 17.1. Is there a language (extending an appropriate language for re-stricted products) where one can express or give a model-theoretic interpretationof the correspondence between Artin L -functions and Hecke L -functions? These questions would have implications for a model-theoretic understanding of automorphic representations and automorphic forms on adele groups , which wouldbe related to a model theory for classical modular forms . See Section 19 Problem 17.2. Let L be a language for the adeles A K augmented by a unarypredicate for a multiplicative character χ defined on the ideles and trivial on K ∗ .What can be said about the L -theory of ( A K , χ ) ? Tate’s thesis and zeta integrals for GL Let K be a local field. A C -valued function f on K is called smooth if it is C ∞ when K is Archimedean, and locally constant when K is non-Archimedean. A ODEL THEORY OF ADELES AND NUMBER THEORY 61 smooth function on K is a Schwartz-Bruhat function if it goes to zero rapidly at in-finity if K is Archimedean and if it has compact support if K is non-Archimedean. S ( F ) denotes the class of Schwartz-Bruhat functions on K . See [78].Tate’s thesis [82] was a beautiful and fundamental work that influenced a widerange of topics in modern number theory and arithmetic geometry. The main goalwas a generalization, via different proofs, of Hecke’s results on meromorphic contin-uation and functional equation for Hecke L -functions for number fields. Howeverthe theory has had far reaching influence and applications.Concerning the Hecke L -functions, it gave more information on the functionalequation and so-called epsilon factors, gamma factors, and root numbers, andvarious quantities acquire an interpretation in terms of volumes of subsets in theadeles or ideles. For example the class number formula can be given a new volume-theoretic proof, see [78].In a similar vein, an important formula of Siegel on quadratic forms was givenan interpretation by Weil in terms of volumes of adelic spaces, leading to Weil’sconjecture that the Tamagawa volume of G ( A K ) /G ( K ) is equal to for a semi-simple simply connected algebraic group G over a number field K . This has beena great influence in the conjecture of Birch and Swinnerton Dyer.In a work, started by Paul Cohen (unpublished) and pursued by Alain Connesand others (see [21]), proving an analogue of Tate’s thesis for the space of adeleclasses A K /K ∗ is considered a key step for a proof of the Riemann hypothesis.Tate’s thesis has also played a central role in the development of the Lang-lands program and in the Langlands conjectures. It has been a starting point forthis theory. See Section 19 for more on these and a suggested model-theoreticframework and questions.Tate’s results on L -functions are deduced from results on local zeta functionsattached to characters. Given a Schwartz-Bruhat function f ∈ S ( K ) on a localfield K with residue field of cardinality q , and multiplicative character χ ∈ X ( K ∗ ) ,the local zeta function is defined by Z ( f, χ ) = Z K ∗ f ( x ) χ ( x ) d ∗ x, where dx ∗ is the measure (1 − q − ) − dx/ | x | , where dx is an additive Haar measureon K . These satisfy functional equations relating Z ( f, χ ) with Z ( f, ˆ χ ) , where ˆ χ is the dual character, cf. [82].Let K be a number field. The class of adelic Schwartz-Bruhat functions on A K is defined as the restricted tensor product S ( A K ) = ⊗ ′ v S ( K v ) consisting of functions of the form f = ⊗ f v ∈ V K , where f v ∈ S ( K v ) and f v = 1 forall but finitely many v . Let χ be a character on I K that is trivial on K ∗ (i.e. an idele class character).Let f ∈ S ( A K ) be an adelic Schwartz-Bruhat function. The global zeta functionof Tate is defined as Z ( f, χ ) = Z I k f ( x ) χ ( x ) d ∗ x. Tate proves meromorphic continuation of Z ( f, χ ) and a functional equation relat-ing Z ( f, χ ) with Z ( ˆ f , ˇ χ ) , where ˆ f is the adelic Fourier transform of f and ˇ χ thedual character. cf [82].There is an Euler product factorization Z ( f.χ ) = Y v ∈ V K Z K ∗ v f v ( x ) χ v ( x ) d ∗ x where χ = Q v χ v . As stated in Section 17, for all but finitely many v , χ isunramified, so has the form | . | s , and integration of this function is well-understoodover Q ∗ p and I K . For the finitely many v where ramification occurs, the integralsare evaluated via the properties of the characters (see [82]).The local factors of these integrals are special cases of motivic integrals of Denef,Cluckers, Loeser and Hrushovski-Kazhdan (see [28], [29], [19], [20], [53]).This raises the general question what would be a generalization of Tate’s thesiswhere the local zeta integrals are replaced with the Denef-Loeser type motivicintegrals. We shall formulate several question in this regard. As a first step onecan ask. Problem 18.1. Use model theory to generalize Tate’s global zeta integrals to in-tegrals of definable functions over definable subsets of A mK for m ≥ , and provemeromorphic continuation results. We propose a form for these "definable integrals" in the Section 19 for the case of GL and G n . We shall propose a generalization of p -adic specialization of motivicintegrals and study their Euler products. One can then apply Theorem 14.1 tostudy these Euler products.19. Automorphic representations and zeta integrals for GL n Tate’s thesis naturally lead to various questions beyond GL . These relateto a wide range of problems and topics including class field theory, non-abelianextensions of Q , non-abelian generalization of Artin reciprocity, mysteries of zetaand L -functions, Langlands’ conjectures, problems in Diophantine geometry ofinteger and rational points on varieties and homogeneous spaces, and arithmeticaspects of algebraic groups. ODEL THEORY OF ADELES AND NUMBER THEORY 63 Langlands program and Jacquet-Langlands theory. A fascinating program and set of conjectures were given by Langlands whichturned out to be related to several of the above topics at the interactions of algebra,geometry, analysis, and representation theory. Here one works with reductivealgebraic groups and the Langlands functoriality conjecture is one of the strongestof the conjectures (see [66]). For an introduction to the Langlands conjectures andprogram see [66], [67], [10], [9].It had been known that an approach to the Langlands program is to start bygeneralizing Tate’s thesis to more general groups. Jacquet and Langlands [57]did this for GL . Godement and Jacquet [48] did it for GL n . The GL -theorycaptures results on modular and Maass forms originated by Hecke and Maass (see[10]). The general case beyond GL n concerns a reductive group and its Langlandsdual group, which we will not consider in this paper, and leave for a future work.In this Subsection I shall propose a model-theoretic framework and pose somequestions on the Jacquet-Langlands theory. This is a first attempt to develop amodel-theoretic study in the Langlands program. At the end I will comment onmore general situations.I formulate the basic definitions in the case of GL n . Let χ be a unitary characterof I K /K ∗ . Let L ( GL n ( K ) \ GL n ( A K ) , χ ) be the space of C -valued measurablefunctions f on GL n ( A K ) such that for all z ∈ I K f (( zI ) g ) = χ ( z ) f ( g ) and Z Z A GL n ( K ) \ GL n ( A K ) | f ( g ) | dg < ∞ , where dg is a Haar measure on GL n ( A K ) , and Z A the group of scalar matriceswith entries in I K .Let L ( GL n ( K ) \ GL n ( A K ) , χ ) be the subspace L ( GL n ( K ) \ GL n ( A K ) , χ ) con-sisting of functions that satisfy the condition Z ( A K /K ) r ( n − r ) f ( (cid:18) I r x I n − r (cid:19) g ) dx = 0 for almost all g ∈ GL n ( A K ) and all ≤ r < n , where x is an r × ( n − blockmatrix. These are called cusp forms .Consider the right regular representation ρ : GL n ( A K ) → End ( L ( GL n ( K ) \ GL n ( A K ) , χ )) defined by ( ρ ( g ) f )( x ) = f ( xg ) . Automorphic representations are irreducible representations of GL n ( A K ) that oc-cur in L ( GL n ( K ) \ GL n ( A K ) , χ ) . The space of cusp forms is invariant under this representation and has theconvenient property that it decomposes as an infinite direct sum of irreducibleinvariant subspaces where each factor appears at most once. This "multiplicityone theorem" was proved by Jacquet-Langlands for n = 2 , and Piatetski-Shapiroand Shalika independently for n > (see [9] and the references there).If π is a representation of GL n ( A K ) that is isomorphic to the representation onone of these invariant subspaces (for some χ ), then π is called an automorphiccuspidal representation with central character χ . When n = 1 , an automorphiccuspidal representation is nothing but an idele class character of I K /K ∗ . For n = 2 , they arise from modular forms and Maass forms. See [9].An instance of the functoriality conjecture of Langlands is the following. Conjecture 19.1 (Langlands) . Let E/F be a finite extension and ρ : Gal ( E/F ) → GL n ( C ) be a representation. Then there is an automorphic representation π cor-responding to ρ such that L ( ρ, s ) = L ( π, s ) . This gives a non-abelian extension of Artin reciprocity. See [66],[9],[67], andimpies Artin’s conjecture that L ( ρ, s ) is entire as one knows the poles of automor-phic L -functions.Interestingly, this conjecture (and more general functoriality conjectures ofLanglands) follow from properties of automorphic representations (involving noGalois group at all!). The strongest result in this direction is due to L. Lafforgue(see [64], [65]), where the full Langlands functoriality conjecture is proved to beequivalent to a non-abelian generalization of the adelic Poisson summation formulaof Tate in [82]. These involve only adelic zeta integrals. See Subection 20.1.The L -functions are defined for GL n generalizing Tate’s method. Their analyticproperties are proved via global zeta integrals defined by Jacquet-Langlands for GL in [57] and Godement-Jacquet in [48]. For simplicity, we restrict ourselves tothe case K = Q .For GL ( A Q ) the notion of an automorphic cuspidal representation is a naturalgeneralization of the notion of a cusp form f ( z ) = ∞ X i =1 a n e πinz on SL ( Z ) , and the work of Jacquet-Langlands leads to a new point of view onthe L -functions L ( s, f ) = (2 π ) − s Γ( s )( ∞ X i =1 a n n − s ) = Z ∞ f ( iy ) y s − dy attached to f by Hecke. See [9],[46].Similarly, the L -functions attached to a Dirichlet character L ( s, χ ) (defined byDirichlet in the proof of his celebrated theorem on arithmetic progressions) can ODEL THEORY OF ADELES AND NUMBER THEORY 65 be written as a Mellin transform Z ∞ ϕ χ ( t ) t s/ dt/t of a so-called θ -series ϕ χ , and can be generalized adelically. See [9].The study of the L -functions proceeds via analytic properties of zeta integralsfollowing Tate’s thesis. The Jacquet-Langlands global zeta integrals have the form Z I Q / Q ∗ ϕ ( (cid:18) a 00 1 (cid:19) ) | a | s − / A Q da, where ϕ is any function in the subspace H π of L ( GL n ( K ) \ GL n ( A K ) , χ ) realizing π , and | . | s A Q is the adelic absolute value | x | A Q = Y p ∈ Primes ∪{∞} | x ( p ) | p , for x ∈ A K . One assumes that ϕ ( x ) is K -finite where K = Q p ∈ Primes ∪{∞} S p , and S p is the maximal compact subgroup of GL n ( Q p ) , in the sense that the right-translates of ϕ by elements k in K span a finite-dimensional space of functions.19.2. Adelic constructible integrals. To capture the essential required model-theoretic properties of these global zetaintegrals and propose a framework to study them and pose questions, we definethe following integrals. As before K is a number field with completions K v . Definition 19.1. Let L be a language extending L rings containing a unary predi-cate for a multiplicative character χ . Let x be an n -tuple of variables and ψ ( x ) , ψ ( x ) , . . . , ψ k ( x ) L -formulas which give definable functions from K nv into K v for all v ∈ V finK . Let ϕ ( x ) be an L -formula.An L -constructible integral of product type on the finite adeles A finK is an Eulerproduct of the form Y v ∈ V finK Z ϕ ( K v ) Ψ( x ) v Φ v ( x ) | ψ ( x ) | sv | ψ ( x ) | v . . . | ψ k ( x ) | v dx where (1) dx is the normalized Haar measure on K nv such that R O nv dx = 1 , (2) Φ v is a Schwartz-Bruhat function on ϕ ( K nv ) such that Φ v = 1 for all butfinitely many v . (3) Ψ( x , . . . , x n ) is a function from ϕ ( A K ) into C that is an Euler product of L -definable functions Ψ v from ϕ ( K v ) in C , (4) For every a ∈ ϕ ( A K ) one has Ψ v ( a ( v )) = 1 for all but finitely many v , (5) χ is interpreted as a multiplicative character on K ∗ v or in case v is non-Archimedean also as a multiplicative character on the residue field k ∗ v . Note. When Ψ v is the term χ ( x ) , we see that Tate’s zeta integrals are L -constructible. Definition 19.2. Let x be an n -tuple of variables and ϕ ( x ) an L -formula. Let L be a language extending L rings containing a unary predicate for a multiplicativecharacter χ . Let ψ ( x ) , ψ ( x ) , . . . , ψ k ( x ) be L -definable functions from ϕ ( A nK ) intothe finite ideles I finK . Let Φ be a Schwartz-Bruhat function as in Def 19.1.An L -constructible integral on the finite adeles A finK is a function of the form Z ϕ ( A finK ) Ψ( x ) | ψ ( x ) | s A finK | ψ ( x ) | A finK . . . | ψ k ( x ) | A finK dx, where (1) | . | A finK = Q v ∈ V finK | . | v , (2) dx is a normalized Haar measure on ( A finK ) n , (3) Ψ( x , . . . , x n ) is a function from ϕ ( A finK ) into C that is an Euler productof L -definable functions Ψ v from ϕ ( K v ) into C , (4) For every a ∈ ϕ ( A finK ) , for all but finitely many v , Ψ v ( a ( v )) = 1 .If ϕ ( A K ) is a definable set of Type I in the sense of Remark 5.1, then we addSchwartz-Bruhat functions to the integrals, to get Z ϕ ( A finK ) Φ( x )Ψ( x ) | ψ ( x ) | s A finK | ψ ( x ) | A finK . . . | ψ k ( x ) | A finK dx, where Φ = ⊗ v ∈ V finK Φ v where Φ v is a Schwartz-Bruhat function on ϕ ( K v ) and Φ v = 1 for all but finitely many v ∈ V finK . Recall that definable sets of Type I in the adeles are sets of the form X = { a ∈ A nK : [[ ψ ( a )]] = 1 } for an L -formula ψ , where L is a language for all the K v .A definable set X of Type I can be written as a restricted product of definablesets in K v with respect to ϕ ( O v ) . In this case an integral (resp. Schwartz-Bruhatfunction) on A mK decomposes as an Euler product of integrals (resp. Schwartz-Bruhat functions) over K mv .This is the reason that to add Schwartz-Bruhat functions in Definition 19.2we assume ϕ ( A K ) is of Type I since for non-Archimedean K v a construction ofSchwartz-Bruhat functions for definable sets can be given using quantifier elimi-nation (a construction is given by Cluckers-Loeser in [20]). Problem 19.1. Define Schwartz-Bruhat functions for definable sets in R n . By quantifier elimination this reduces to real semi-algebraic sets. One can usereal cell decomposition to reduce it to cells. ODEL THEORY OF ADELES AND NUMBER THEORY 67 Note 19.1. Solving Problem 19.1 would extend Definition 19.2 to the adeles A K . Problem 19.2. Define Schwartz-Bruhat functions for definable sets in A mK for m ≥ which are not of Type I. Remark 19.1. The problem here is that when the definable set is defined by con-ditions of the from F in ([[ ... ]]) or C j ([[ ... ]]) , we do not know that it is a restrictedproduct and we do not know if the Schwartz-Bruhat functions are Euler productsof local functions. Theorem 19.1 (Berman-Derakhshan-Onn-Paajanen [8, Theorem A]) . Suppose ϕ defines a Chevalley group G . Suppose µ is a Haar measure on G ( F ) for a non-Archimedean local field F . Then there is a finite partition of G ( F ) into L rings -definable sets such that on a given piece an integral with respect to µ can be writtenan integral with respect to | φ ( x ) | dx , where φ is an L rings -definable function and dx an additive Haar measure on F dim ( G ) . If F vary over the non-Archimedean localfields K v , then this "change of measure" is uniform in v , i.e. the definable partitionof the domain and the definable functions φ ( x ) can be chosen independently of v . We would need to add Archimedean factors to the constructible integrals sothat they become related to L -functions and have good analytic properties (mero-morphic property, Fourier transform). Problem 19.3. Complete the constructible integrals in Definitions 19.1 and 19.2by adding factors for the Archimedean places v . We propose such a definition for the case K = Q . We let Q ∞ = R . Definition 19.3. Let L = ( L real , L na , ψ ( x )) be a language were ψ ( x ) is a unarypredicate interpreted as a multiplicative character on Q ∗ p for p ≤ ∞ , and L real and L na extend L rings . Let x be an n -tuple of variables. Let ϕ ( x ) be an L -formula. Let φ i ( x ) , φ i ( x ) , . . . , φ ik ( x ) , for i ∈ { , } , be L -formulas which when i = 0 are L real -formulas and give L real -definable functions from ϕ ( R ) into R , and when i = 1 are L na -formulas and give L na -definable functions from ϕ ( Q p ) into Q p for p < ∞ .A special adelic L -constructible integral is an Euler product of the form Y p ∈{ Primes }∪{∞} λ − p Z ϕ ( Q p ) Ψ p ( x ) | φ ( x ) | sp | φ ( x ) | p . . . | φ k ( x ) | p dx where (1) dx is an additive Haar measure on Q np normalized such that R Z np dx = 1 for p < ∞ , (2) λ p ∈ C have the form of a product of convergence factors (to make the Eulerproduct converge) and normalizing factors (to give it a special intendedvalue), (3) For all p ≤ ∞ the following hold,(3.1). Ψ p are functions from ϕ ( Q p ) into C ,(3.2). There are L rings -formulas θ ti ( x ) , for i ∈ { , . . . , n } and t ∈{ , , } that give definable functions from ϕ ( Q p ) into Q p such that θ i ( a ) =0 for all a ∈ ϕ ( Q p ) and all i ,(3.3). There are multiplicative characters χ , . . . , χ N on Q ∗ p and Schwartz-Bruhat functions Φ , . . . , Φ N on Q p such that(i) For p < ∞ and a ∈ ϕ ( Q p ) , Ψ p ( a ) = X ≤ i ≤ N p Φ i ( θ i ( a )) v ( θ i ( a )) χ i ( θ i ( a )) q , where the notation p . . . q means that in the summand, one of more of theterms Φ i ( θ i ( a )) , v ( θ i ( a )) , or χ i ( θ i ( a )) may not be present.(ii) For p = ∞ and a ∈ ϕ ( R ) , there is some element γ ∈ R (possibly γ = 0 ) such that as θ ( a ) → γ , one has Ψ p ( a ) ∼ X ≤ i ≤ N p Φ i ( θ i ( a )) log | θ i ( a ) | χ i ( θ i ( a )) . q The notation p . . . q has the same meaning as in part ( i ) . Note 19.2. Since | . | s is a character of R ∗ , the Archimedean factor of a specialadelic constructible function can include terms of the form | θ ( x ) | s where θ is anon-vanishing definable function on a definable set. This is a real analogue of theintegrals of Denef and Loeser (see Section 14) Definition 19.4. A special adelic L -constructible function is of Whittaker type if Ψ v ( a ) = 0 when | θ ( a ) | ∞ is large, and one can choose γ = 0 in ( ii ) . Remark 19.2. Definitions 19.1 and 19.3 give a family of definitions, one foreach choice for the language L . An important choice is when L real is the lan-guage of restricted analytic functions with exponentiation defined by van den Dries-Macintyre-Marker [84] and L na is any of the languages of Belair [7] , Basarab [6] ,or Denef-Pas [73] from Subsection 3.2. Remark 19.3. One could also formulate the integrals in Definitions 19.2 in termsof a volume form ω on ϕ ( A K ) . For this one starts with a volume form on ϕ ( K v ) which can be constructed as in [19] and [20] . This gives a volume form on ϕ ( A K ) as in Section 10.2.If ϕ ( x ) defines an algebraic group G ( K v ) for each v , then one can use Theorem19.1 to reduce integration with respect to a Haar measure on G ( K v ) to integrationwith respect to a suitable measure for integrating definable functions. This enablesus to reduce integrals of definable functions on G ( A K ) with respect to a Haar mea-sure on G ( A K ) (e.g. Tamagawa measure) to an Euler product of local componentswhich are integrals of function of the form | ψ ( x ) | s , where ψ is a definable functionfrom K mv into K v . ODEL THEORY OF ADELES AND NUMBER THEORY 69 For non-Archimedean K v , these local factors can be evaluated using methodsof Denef (see Section 14) and the Euler product an be understood by applyingTheorem 14.1.Note that G is a definable set of Type I and functions that are at almost allplaces factorize into local functions. Note 19.3. In Definitions 19.1, 19.2, and 19.3, we can require that Ψ is an L fin,resBoolean -definable function. This is more general than being an Euler product of"definable factors". Note 19.4. We can generalize the constructible integrals by replacing L rings bythe restricted product language L fin,resBoolean ( L ) where L is any of the languages for thefactors. This works over definable sets of Type I which are restricted products. Example 19.1. The characters of the group R ∗ are of the form | . | s or sgn | . | s ,where s ∈ C ∗ and sgn is the sign character x → x/ | x | . Let χ = | . | s . Let f ( x ) = e − πx . f is a Schwartz-Bruhat function on R . The function f ( x ) χ ( x ) is part of aspecial adelic constructible integral. Integrating this function gives the Γ -function.Proof. The local zeta function in this case becomes Z ( f, χ ) = Z R ∗ e − πx | x | s d ∗ x = 2 Z ∞ e − πx x s − dx = π − s/ Z ∞ e − u u s/ − du = π − s/ Γ( s/ , where in the second line we have put u = πx . This calculation is in Tate’s thesis[82]. (cid:3) Theorem 19.2. Let π be an automorphic cuspidal representation with centralcharacter χ Let ϕ be any function in the subspace H π of L ( GL n ( K ) \ GL n ( A K ) , χ ) realizing π . The Jacquet-Langlands integrals Z JL ( ϕ, s ) = Z I Q / Q ∗ ϕ ( (cid:18) x 00 1 (cid:19) ) | x | s − / A Q d ∗ x are adelic L Ψ rings -constructible of Whittaker type.Proof. This follows from formulas and calculations of Jacquet and Langlands [57].see [57] and [47]. We sketch it below to give a slight model-theoretic interpretation.Write the Fourier expansion of ϕ : ϕ ( g ) = X ξ ∈ Q ∗ W ψϕ ( (cid:18) ξ 00 1 (cid:19) g ) . Here ψ is a fixed additive character on the adeles A Q that is trivial on the globalfield Q and W ψϕ is the ψ th-Fourier coefficient of ϕ given by W ψϕ ( g ) = Z A Q / Q ϕ ( (cid:18) x (cid:19) g ) ψ ( x ) dx. It follows that Z JL ( ϕ, s ) = Z I Q W ψϕ ( (cid:18) x (cid:19) ) | x | s − / A Q d ∗ x. Now X = { ( x, , , 1) : x ∈ I Q } is a definable subset of A Q isomorphic to I Q . There is a volume form on it whichis the pull-back of the volume form dx ∗ = Q v d ∗ x v on I Q . One has the followingproperties. For proofs of these see [57], [47], and [46, pp.11]. ( i ) . The function W ψϕ (a so-called Whittaker function) is an Euler product W ψϕ ( g ) = Y p ≤∞ W p ( g ( p )) where W p ( g ( p )) are local Whittaker functions and W p ( x ) = 1 for x ∈ GL ( Z p ) . ( ii ) . The local integrals Z Q ∗ p W p ( (cid:18) x (cid:19) ) | x | s − / p d ∗ x are absolutely convergent for Re( s ) large, and one has Z ( ϕ, s ) = Y p ≤∞ Z ( W p , s ) . As stated in [46, page 11], ( iii ) . For a given p , and any x ∈ X , if | det( x ) | is large, then W p ( x ) = 0 , ( iv ) . There are finitely many Schwartz-Bruhat functions Φ , . . . , Φ N and func-tions c , . . . , c N on Q ∗ p such that for any a ∈ XW p ( x ) = X ≤ i ≤ N c i (det( x ))Φ i (det( x )) , where c i are defined to be Q -linear combinations of products of characters χ ( x ) with functions of the form v ( x ) , where m ∈ Z , when p < ∞ , and log | x | when p = ∞ .Since det is a definable function from X into I Q , the proof is complete. (cid:3) Problem 19.4. The proof of Theorem 19.2 suggests constructibility of Fouriercoefficients of automorphic cusp form. Explore this phenomenon. Problem 19.5. Generalize special adelic constructible functions to A K for a gen-eral number field K . Problem 19.6. Give examples of zeta integrals for GL n ( A K ) from the workof Godement-Jacquet [48] that are L -constructible or special adelic constructible(resp. of Whittaker type) for some L . What if the group GL n is replaced by areductive algebraic group? Can we characterize such π ? ODEL THEORY OF ADELES AND NUMBER THEORY 71 Problem 19.7. Formulate a notion of automorphic representation for a definablesubgroup of A mK , m ≥ , generalizing the case of algebraic groups. Define corre-sponding zeta integrals in analogy with the case of GL n or a reductive group, andexplore whether they are L -constructible for some L . Is there any notion similarto that of an automorphic representation for a definable set in A mK ? We would like the zeta integrals of an automorphic representation for a definablegroup G = ϕ ( A K ) to have the form Z (Φ , s, ψ, φ , . . . , φ k , π ) = Z ϕ ( A K ) Φ( x ) π ( x ) | ψ ( x ) | s A K | φ ( x ) | A K . . . | φ k ( x ) | A K dx where ϕ is an L -formula, ψ is an L -definable function from A mK into I K , Φ( x ) is aSchwartz-Bruhat function, and dx is a Haar measure on A mK .If G is Type I definable, then G ( A K ) is a restricted product of definable groupsover K v and Z (Φ , s, ψ, φ , . . . , φ k , π ) has an Euler product factorization into local"definable integrals" Z (Φ v , ψ, φ , . . . , φ k , π v ) = Y v Z ϕ ( K v ) Φ v ( x ) π ( x ) | ψ ( x ) | sv | φ ( x ) | v . . . | φ k ( x ) | v dx where Φ v ( x ) is a local Schwartz-Bruhat function on ϕ ( K v ) and π v ( x ) is a localfactor of π (that should also be defined together with π ). Problem 19.8. Suppose π is defined for a definable set of Type I. Use methodsof model theory including cell decomposition, p-adic integration, and resolutionof singularities, following Denef-Cluckers-Loeser [29] , [19] , [20] to evaluate the localzeta integrals Z (Φ v , ψ, φ , . . . , φ k , π v ) . This would give meromorphic continuationbeyond abscissa of convergence by Theorem 14.1. Note that we assume the set is of Type I to get an Euler product factorizationof the zeta integral. It is an open problem if zeta integrals of general reductivegroups have meromorphic continuation (cf. [66]). Problem 19.8 would give apartial solution using Theorem 14.1. Problem 19.9. • Define L -functions corresponding to the "definable zeta integrals" above.Define local factors at the real places using Γ -functions. We would likethe Archimedean local factors to be definable in some tame extension of R possibly related to issues in Hodge theory and the theory of O-minimalityrelated to the work of Bakker-Klingler-Tsimerman [5] . • Find a product formula connecting the Archimedean and non-Archimedeanlocal factors. Problem 19.10. • Use constructible integrals to develop a model theory for modular forms,Maass forms and automorphic forms via adele groups (or more generaldefinable sets) over A K . • In the case K = Q , we would like to have a -sorted language which hasthe language of van den Dries-Macintyre-Marker [84] for restricted analyticfunctions with exponentiation for the real sort, and a suitable extension onthe language of rings or one of the languages in Subsection 3.2 for thenon-Archimedean sort. • Explore connections to O -minimality and Hodge theory, via [5] by consid-ering adelic versions of the real manifolds and homogeneous space in [5] using results on definability of fundamental domains. • If K is a general number field, explore the factors which are C . We can also ask a basic question. Problem 19.11. Let π be an automorphic cuspidal representation of GL n ( A K ) and Φ( x ) a Schwartz-Bruhat function for GL n ( A K ) . Is there an extension L of L rings and L -formulas ϕ ( x ) and ψ ( x ) , φ ( x ) , . . . , φ k ( x ) which give definable func-tion from ϕ ( A K ) into I K such that Z ϕ ( A K ) π ( x )Φ( x ) | ψ ( x ) | s A K | φ ( x ) | A K . . . | φ k ( x ) | A K dx is L -adelic constructible? Example 19.2. The main example for which Problem 19.11 has a positive solutionis GL . The proof of Theorem 19.2 gives the required definable sets. We define adelic constructible functions. Definition 19.5. A function of the form π ( x )Φ( x ) | ψ ( x ) | s A K | φ ( x ) | A K . . . | φ k ( x ) | A K in Problem 19.11 is called adelic L -constructible. One the main results in the theory of motivic integration is that the class ofmotivic constructible functions is closed under integration and Fourier transform,see [20]. Our class of adelic constructible functions are closely related. At leastover Type I definable sets, they are Euler products whose Euler factors are anextension of the p -adic specializations of motivic constructible functions. We canthus ask. Problem 19.12. Is the class of adelic constructible functions (resp. special/ofWhittaker type) closed under adelic integration and adelic Fourier transform? Note 19.5. Note that in Definition 19.5 and Problem 19.12, a special case iswhen we replace A K by A finK . This is seen by taking π, Φ , ψ, φ j to be trivial at theArchimedean factors. Once one has defined constructible functions for every number field, from amodel-theoretic prospective, it is natural to ask. ODEL THEORY OF ADELES AND NUMBER THEORY 73 Problem 19.13. Given an automorphic representation, to what extent its adelicconstructible integral representation given by positive solution of Problem 19.11depends on the number field? (e.g. on its degree?). If one varies the number fieldhow do they (i.e. their definable functions and formulas) vary? On identities between adelic integrals Adelic Poisson summation. In this section as before K is a number field. Recall that the space of adelicSchwartz-Bruhat functions S ( A K ) is defined as the restricted product of the Schwartz-Bruhat spaces S ( K v ) over all v ∈ V K . Given f ∈ S ( A K ) one writes f ( x ) = Q v ∈ V K f v ( x ( v )) for x ∈ A K .Fix a non-trivial additive character Ψ on A K such that Ψ | K = 1 . See [78,Section 7] for existence of this. Then the adelic Fourier transform of a function f ∈ S ( A K ) is defined by ˆ f ( y ) = Z A K f ( x )Ψ( xy ) dx. (usually one has a certain normalization of dx , see [78]).For any φ ∈ S ( A K ) , to get a function that is invariant under translation byelements of K , define ˜ φ ( x ) = X γ ∈ K φ ( γ + x ) . If this is convergent for all x , then ˜ φ ( x ) = ˜ φ ( x + δ ) for all δ ∈ K .A C -valued function on A K is called admissible if ˜ f and ˜ˆ f are both normallyconvergent on compact sets. Every adelic Schwartz-Bruhat function is admissible(see [78, Lemma 7-6]).The adelic Poisson summation formula of Tate (see [82], [78, Theorem 7-7])states that for any f ∈ S ( A K ) , one has ˜ f = ˜ˆ f , i.e. X γ ∈ K f ( γ + x ) = X γ ∈ K ˆ f ( γ + x ) for every x ∈ A K . Problem 20.1. Give examples of definable admissible functions in a natural lan-guage. Define the notion of admissible for a definable function from A mK into C ). From the Poisson summation formula, Tate derives a Riemann-Roch theoremwhich states that for an idele x and f ∈ S ( A K ) X γ ∈ K f ( γx ) = 1 | x | X γ ∈ K ˆ f ( γx − ) . See [82] and [78, Theorem 7-10]. Tate proves both the Poisson summation formulaand the Riemann-Roch theorem for global fields of positive characteristic as well. In this case the adelic Riemann-Roch theorem implies the usual Riemann-Rochtheorem of algebraic geometry. See [78, Section 7].To get a model-theoretic hold on the adelic Poisson summation, one would needto know the following which seems plausible. Problem 20.2. Can the adelic Poisson summation formula or the adelic Riemann-Roch theorem of Tate be interpreted as an identity between adelic L -constructiblefunctions for some L ? If so, what would be a model-theoretic generalization? Laurent Lafforgue has formulated a conjectural non-abelian generalization ofTate’s Poisson summation formula on G ( A K ) , where G is a reductive algebraicgroup, and has proved that it is equivalent to the Langlands functoriality con-jecture (which implies that above conjecture of Langlands). See [64] and [65]. Itwould be interesting to study connections to this work.20.2. Adelic transfer principles. Model-theoretic transfer principles go back to Tarski who proved that a sentenceof the language of rings holds in an algebraically closed field of characteristic zeroif and only if it holds in every algebraically closed field of characteristic p , forlarge p . See [15], [62]. Ax and Kochen gave axiomatization and completeness fortheories of p -adic fields and Henselian valued fields with characteristic zero residuefield [2],[3],[4],[15]. From this they deduced that a sentence holds in Q p if and onlyif it holds in F p (( t )) , for large p . There are also such results for finite extensionsof these fields.After Denef’s work on rationality of "definable" p -adic integrals in [28], uniform(in p ) rationality results were proved by Pas [73] and Macintyre [69]. In [69]Macintyre, as a result of his uniform rationality theorem, proved that an identityof "definable integrals" Z ϕ ( K ) | f ( x ) | s | g ( x ) | dx = Z ϕ ( K ) | f ( x ) | s | g ( x ) | dx holds for K = Q p when f i , g i are interpreted in Q p if and only if it holds for K = F p (( t )) when f i , g i are interpreted in F p (( t )) , for large p (larger than somefunction of ϕ, f i , g i ).This was generalized by Denef-Loeser [29] and Cluckers-Loeser [20] to motivicintegrals. In [20] the motivic integrals are extended to have Schwartz-Bruhatfunctions and additive characters, including integrals of the form Z Q np f ( x )Ψ( g ( x )) dx, where f ( x ) is a p -adic constructible function in the sense of [19],[20], g ( x ) is a Q p -valued definable function on Q np , and Ψ is a non-trivial additive character on Q p , and it is proved that an identity of two such integrals has a transfer principle ODEL THEORY OF ADELES AND NUMBER THEORY 75 between Q p and F p (( t )) for large p . We note that a p -adic constructible functionis constructed from functions of the form v ( h ( x )) , | s ( x ) | s (cf. [20] for details).The question arises as to whether such a phenomenon is true for the adeles.One way to formulate a question in this connection is the following. Problem 20.3. How does the truth of statements or identities between adelicconstructible integrals transfer between different models of the theory of A K ? orbetween different adele rings A K and their ultraproducts (what we call pseudo-adelic rings)? Or between adeles of number fields and adeles of function fields? A completeness problem. The following problem relates to the completeness of the axioms for adeles in[35] and in Section 8. It would be interesting to investigate if it follows fromcompleteness of a theory related to that of A K .From another perspective, it can be regarded as an adelic version of a ques-tion of Kontsevich and Zagier in [61] on periods. 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