It is supported precisely on the set of v such that Kv is real resp. It is supported precisely on the set Arch K. It is supported precisely on the set of non-Archimedean valuations. The functions a1 ,. Finite adeles. Defining finite support idempotents. The following theorem is of great importance in our works on the model theory of adeles. Theorem 2. The set of finite support idempotents in AK is Lrings -definable, uniformly in K. We denote the set of finite support idempotents by F inK. It is an ideal in the Boolean algebra BK.
In [37] we give two proofs of this result. The first uses the concept of a von Neumann regular ring defined as follows. We call an element a of a commutative ring R von Neumann regular if a the ideal generated by a is generated by an idempotent. R is von Neumann regular if every element is von Neumann regular.
Examples are direct products of fields. This definition is independent of K. It follows that the set of finite support idempotents in AfKin is also definable. The second proof uses uniform definition of valuation rings in the language of rings which used quite often in the works of Macintyre and myself on the model theory of adeles. Explicit definitions previous to our work e. Non-explicit definitions had existed as well, using Beth definability type arguments because of the uniqueness of the Henselian valuations of the Kv.
These results were mostly applied in connection to decidability and undecidability results on Hilbert 10th problem see [60]. In joint work with Cluckers-Leenknegt-Macintyre [18] the following is proved.
There is an existential-universal formula in the language of rings that uniformly defines the valuation ring of all Henselian valued fields with finite or pseudo-finite residue field. For our work on adeles we apply Theorem 2.
Note 2. Note that Theorem 2. Enriched theories of Boolean algebras and valued fields Our analysis of the model theory of the adeles and restricted products depends crucially on the theory of an associated Boolean algebra and the theories of the local fields which are the factors or "stalks" of the adeles.
In this analysis, various enrichments are of fundamental importance. The Boolean algebra should be enriched by at least adding a predicate F in x to the language with intended interpretation in a Boolean algebras of sets as "finite", and in the Boolean algebra of idempotents in the adeles or in rings as "finite unions of atoms".
We should also enrich the language for the factors. We give some details below. Enrichments of Boolean algebras. This theory, that we denote by T Bool , is axiomatized by sentences stating that the models are infinite Boolean algebras and every nonzero element has an atom below it. See [62, Theorem 16,pp. 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 a powerset algebra. Feferman-Vaught [43] proved that this theory is complete, decidable, and has quantifier elimination. A new proof was given Macintyre and myself in [34]. Note that there are other models where F in x holds and x is not finite.
Note that the Cj are definable in LBoolean but F in x is not. Let T f in of infinite atomic Boolean algebras in the language LfBoolean in.
Theorem 3. This theorem plays a fundamental role in our works on the model theory of adeles and restricted products. It also applies to a commutative ring via the Boolean algebra of its idempotents see Section 8. In this way it has been used in the work in [35] of Macintyre and myself on axioms for rings elementarily equivalent to restricted products see Section 8.
Let LfBoolean in,res denote the enrichment of LfBoolean in by these predicates and T f in,res the theory of all infinite atomic Boolean algebras in the language LfBoolean in,res. We can apply this theorem to the expansion of the Boolean algebra of idem- potents of AK or of more general commutative rings with adding the F in and Res-predicates and have quantifier elimination and decidability.
In Section 11 we give an application of this on Hilbert reciprocity in number theory from [37]. Enrichments of valued fields. The quantifier elimination for adeles or generally restricted products takes place in two steps. The first step is a general quantifier elimination that works for all restricted products, and would depend on a chosen enrichment of infinite atomic Boolean algebras.
This is stated in Subsection 4. The second step is a finer result that requires a quantifier elimination in the factors or "stalks" of the restricted product. This depends on a chosen enrichment of Henselian valued fields. We shall state some convenient languages in this Subsection for this purpose. The resulting results are presented in Subsections 4. Let L be a language for the factors Kv. This means that Kv are L-structures. If v is non-Archimedean, then Kv is non-trivially valued Henselian whose value group is Z.
One can have a symbol for the valuation in both 1-sorted and many-sorted situations. On the other hand for us it is important to keep the basic analysis within the language of rings and regard AK as a ring, which we do in all cases.
In the interpretation of Kv as an L-structure, we have to take into account the Archimedean Kv as well. Then the Archimedean Kv admit quantifier elimination by results of Tarski for real closed and algebraically closed fields see [62].
Note that while the valuation rings of all the non-Archimedean Kv and their unit balls are Lrings -definable, the unit ball in R is also Lrings -definable, but the unit ball in C is not Lrings -definable. The required quantifier elimination for the Kv is the following. Recall the following basic result.
The Macintyre language [68]. Macintyre [68] proved that the LM ac -theory of Qp admits elimination of quantifiers. It follows that a definable subset of Qmp , for any m, in Lrings or LM ac is a finite union of locally closed sets i. They proved that the theory of p-adically closed fields of p-rank d admits elimination of quantifiers in this language. It follows that an LP R -definable subset of each non-Archimedean Kv m , for any m, is a finite union of locally closed sets, and thus measurable, and an infinite definable subset of Kv has non-empty interior.
Note 3. In [7], Belair defined an extension of the Macintyre language in which the theory of Qp , for all p, admits uniform elimination of quantifiers. He added constants for an element of least positive value and for coset representative for the group of non-zero nth powers and solvability predicate of Ax for the residue fields Solm x1 ,.
This elimination of quantifiers holds uniformly for all Fq , where q is large enough either fixed or unbounded characteristic. The languages of Denef-Pas and Pas [73],[74]. 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 the field sort to the residue field sort interpreted in K as the angular component map modulo MK. See [73]. This elimination is effective, as in Theorem 3.
N and the quantifier free formulas can be effectively given. By [74], each non-Archimedean Kv has quantifier elimination for the field sort relative to the other sorts in LP as. Combining these results we can deduce. Corollary 3. Note: Other many-sorted languages for quantifier elimination in Henselian valued fields have been introduced by Weispfenning [87] and Kuhlmann [63].
These are closely related to LBasarab. Adeles with product valuation [33]. For more details and model-theoretic results on the product valuation, see [33]. 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 and Mostowski see [43].
What we call restricted product of L-structures appears in [43] under the name of weak 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 having a many-sorted language with function symbols and relation symbols.
We give an outline in this Section, especially aimed at results on quantifier elimination for restricted products. We remark that in [35] Macintyre and myself proved an analogue of the main theorem of Feferman-Vaught for rings. Interestingly, this gives both a converse to Feferman-Vaught and at the same time axioms for rings elementarily equivalent to restricted products and adeles. The proof is a modification and a ring-theoretic analogue of [43]. This shall be discussed in Section 8 of this paper.
Language for restricted products. See [42, Section 4. We give a few definitions. LfBoolean in and LfBoolean in,res from Sub- section 3. Definition 4. Expand L by adding a new relation symbol for each of these relations. See also [33]. We now define a many-sorted generalization of the Feferman-Vaught notion of a generalized product. Quantifier elimination in restricted products. Theorem 4. This is uniform for all number fields K. Note 4. In Section In applying Theorem 4. Taking L to be Lrings and applying Theorems 3.
Corollary 4. By Theorem 4. The case of finite index set. If the index set I is finite, then Theorem 4. Follows immediately by Theorem 4. Remark 4. In this way Theorem 4. By Corollary 4. As Y is a restricted product, Theorem 4. Taking S to be the set of all Archimedean valuations, this way one can compare the measures got from the Archimedean factors a finite product with those got from the non-Archimedean factors an infinite restricted direct product. These are naturally definable subsets of Am K for some m.
They also have the structure of a model-theoretic restricted product see Subsection Langlands proved it for all Chevalley groups. Kottwitz proved the general case. See also Subsection In each of these, the volume is calculated after first computing the volume of the points over the finite adeles AfKin , then computing the product of the volumes of the set of points over the Archimedean factors, and then finally comparing the two quantities.
Mysteriously in all these cases, the product is an integer. As stated above, this method of calculating volumes can be carried out for definable sets by Theorems 4. It is a powerful method for calculating adelic volumes.
Definability in adeles 5. Definable subsets of Am K. Let L be a language for the Kv. Theorem 5. Remarks 5. In this case we call X a definable set of Type I. Measurability of definable subsets. Then one uses the fact that definable subsets in Kv are finite unions of locally closed sets cf. Note 5. We also have the following strengthening in [37]. The language Lf in,res L has more expressive power than Lrings for the adeles.
Countable unions and intersections of locally closed sets. The proof of Theorem 5. Corollary 5. This description of definable sets is optimal and can not be improved. Example 5. Then X is not a finite union of locally closed sets in adelic topology, equivalently, X is not a Boolean combination of open sets cf. Euler products and zeta values at integers. Measures of definable sets in AmK are closely related to values of zeta functions at integers.
The following is proved in [37]. Problem 5. Definable subsets of the set of minimal idempotents. Recall the correspondence between minimal idempotents in AK and valuations of K. The following question naturally arises. Note that the reason to have definability without parameters in the question is that if we allow parameters then every subset of the set of minimal idempotents is definable as is easily seen by taking sup and inf of idempotents.
Let P g denote the set of primes p such that the reduction of g x modulo p has a root in Fp. Let K be a number field. Let X be a parameter-free Lrings -definable subset of the set of minimal idempotents in AK. Then the following hold. If we take the m to range over the primes p, then decidability of a sentence in all the Fp is proved in [1] as a consequence of the axiomatization and decidability of the theory of pseudofinite fields.
We give a sketch of this below. We hope that adelic methods can be used in other decision problems too. Let atom x denote the statement that x is a non-zero minimal idempotent.
Indeed, supp z AQ is the product of the Zp where p ranges over the finitely many primes p1 ,. Decidability of AQ. We give a sketch of the proof of decidability of the Lrings -theory of AQ due to Macintyre and myself in [37] using only a theorem of Ax [1] see also [45], or Theorem The first proof of decidability of AK in the language of generalized products of Feferman-Vaught is due to Weispfenning [86]. Our proof is simpler. The following fundamental theorem about model theory of finite fields is what we need.
Theorem 6. Now we give the adelic decision procedure. The decision procedure for I. By Theorem 3. The decision procedure for II. By Theorem 6. This concludes the proof of decidability of AQ. Remark 6. Note 6. We could not directly apply Theorem 4. For the decidability, to apply Ax, we needed the quantifier-elimination for generalized products given by [43] or Theorem 4.
Decidability of AK. The following question arises. Problem 6. We can show the following. The set of all Lrings -sentences that hold in AK for all number fields K is decidable if and only if for a given p, the theory of all finite extensions of Qp is decidable. This raises the question. Kochen [59] proved decidability for the maximal unramified extension Qur p of Qp. In [36], Macintyre and myself prove model-completeness in the language of rings for Qurp and finitely ramified extensions of it, more generally for any Henselian valued field with finite ramification whose value group is a Z-group, and we characterize model-complete perfect fields with procyclic Galois group.
The essence of Problem 6. Even the abelian case this is out of reach, i. Use adelic methods in the spirit of our solution to the Ax problem combined with suitable Galois theory to approach model theory of infinitely ramified extensions of Qp. Elementary equivalence and isomorphism for adele rings In [38] Macintyre and myself consider the question of how the AK , as K varies, are divided into elementary equivalence classes. The main tool used is Theorem 2.
By Theorem 2. Furthermore, we have a uniform definition, independent of K but depending on p, of the collection of stalks with residue characteristic p for any given p. Let p be a prime in Z. Then p lifts to finitely many primes P1 ,. The number field degree. Theorem 7. We give an idea of the proof. Let K be a number field K. To detect the dimension of K over Q inside AK in a first-order way we first find a prime p that splits completely in K. This can be expressed by an Lrings -sentence independently of K but depending on p by Theorem 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. The case of normal extensions. Suppose that K is normal over Q. The proof uses a corollary of the Chebotarev density theorem that states that if K is a Galois extension of Q, then K is completely determined by the rational primes that split completely in K see [71], Corollary Splitting types and arithmetical equivalence.
Let p be a prime. We do not assume that p is unramified in K. Prer and fj is the residue degree of Pj. Note that there can be repetitions and that the ramification indices ei are not present. P Let K be a number field. In this case K1 and K2 are said to be arithmetically equivalent. By [75, Theorem 1], if K1 and K2 are arithmetically equivalent, then they have the same discriminant, the same number of real resp. It follows from Theorem 2. For any given number field K, there are only finitely many number fields L such that are AK and AL are elementarily equivalent.
Problem 7. What are the elementary invariants? Elementary equivalence of adele rings - a rigidity theorem. The question asking to what extent a number field is determined by its zeta function has a long history.
Examples are any normal extension of Q. The first nonsolitary field was discovered by Gassman in who gave two fields of degree over Q which are arithmetically equivalent but not isomorphic cf.
The converse is not true in general, but is true if the extensions are Galois, see [75]. This is a first-order "rigidity theorem " for adeles. Let K an L be number fields.
If AK and AL are elementarily equivalent as rings , then they are isomorphic. This condition is also equivalent to the condition that the finite adeles AfKin and AfLin are isomorphic cf. Find conditions under which adele rings are isomorphic. We also pose. Does Theorem 7. Is this true when G is a Q-split semi-simple algebraic group over Q?
We note that one believes that for a Q-split semi-simple algebraic group G, the field Qp is definable in the group G Qp. It would be interesting to investigate adelic versions of this and use it to approach Problem 7. Axioms for rings elementarily equivalent to restricted direct products and converse to Feferman-Vaught 8.
The question and connection to nonstandard models of Peano arith- metic. The question of finding axioms for the theory of AK is part of the general question of finding axioms under which any commutative unital ring is elementarily equivalent to a restricted direct product of connected rings a ring is connected if 0, 1 are the only idempotents.
The solution in [26] is that it does not interpret arithmetic and much more is proved around its model-theoretic tameness. In this work we provide axioms for a class of linear orders with addition called truncated ordered abelian groups, and prove that any model of these axioms is an initial segment of an ordered abelian group, thus has a semi-group structure arising from a process of truncation.
This work applies to quotients of valuation rings with truncated valuations. Axioms for the rings. We now discuss the axioms of Macintyre and myself from [35]. We shall then prove an analogue of the results of Feferman-Vaught [43] and the results in Section 4 for commutative unital rings. So we develop the analogue of the required notions e. Boolean values in the case of rings. Let R be a commutative unital ring.
The atoms of B are by definition the minimal idempotents that are not equal to 0, 1. The first isomorphism is straightforward and the second isomorphism is shown in Lemma 1 in [25]. Re is the stalk of R at e. Of special important are the Re for atoms e. We define Boolean values in the case of rings as follows. Definition 8. We augment the language of rings Lrings by a unary predicate symbol F in x that is interpreted in R as a finite support element, i. Let F in denote the ideal of finite support elements in R.
As in Sub- f in section 3. Axiom 1. B is atomic. Axiom 2. Axiom 3. Axiom 4. Note 8. A special case of Axiom 4 is the following. Here "cofinite" really means cofinite. Axiom 5. The ring-theoretic Feferman-Vaught and converse to Feferman-Vaught. Theorem 8. Corollary 8. This result can be regarded as a converse to Theorem 4. These axioms for restricted products connect well with the issues on elementary invariants for adele rings discussed in Section 7 and in [38]. See [35]. Some stability theory 9.
Stable embedding. It is known that for many Henselian valued fields, the value group and the residue field are stably embedded. See [51]. Theorem 9. Let e be a minimal idempotent. Problem 9. Q In Subsection 3. In the proof we define the subset X of AQ consisting of idempotents which are supported exactly on the primes p that are congruent to 1 modulo 4.
Applying the Feferman-Vaught Theorem or Theorem 4. The tree property of second kind. The property of not having the tree property of the second kind NT P2 is a generalization of the properties of being simple and NIP the negation of the independence property.
It is known that ultraproducts of Qp and certain valued difference fields have NT P2 cf. The theory of AK , for K a number field, has the independence property in two different ways, firstly via the residue fields by Duret [40] and [44], and secondly because the definable Boolean algebra BK.
Stable formulas and definable groups. Local stability theory is the study of stability properties of a formula. There is much literature on this. Here we only mention that Hrushovski and Pillay in [55] develop a unified approach to local stability for Qp , R, and pseudo-finite fields.
The central notion being that of a geometric field. Develop local stability theory for adeles or adele spaces of algebraic varieties cf. Section 10 using local stability for the fields Kv using the notion of geometric fields in the sense of Hrushovski-Pillay. In particular, what can one say about restricted products of geometric fields?
To what extent the model-theoretic properties of geometric fields are preserved under restricted products? In [55], theorems are proven about groups definable in R and Qp showing they are related to algebraic groups. What can one say about a group that is definable in AK? Is it related e. Adele geometry Adele spaces of varieties. Adele spaes of algebraic varieties were defined by Weil cf. 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 geometric language.
For simplicity, let V be an affine variety over a number field K. Noncanonically choose m and polynomials f1 ,. Note that V AK coincides with the set of solutions of the polynomials f1 ,. Nev- ertheless, it is important to show that V AK can be represented as a restricted direct product in the sense of Section 4 as that will give an internal connection between definability and the measure theory of V AK , and also yields an internal quantifier-elimination and Feferman-Vaught theorem for V AK.
This is done in [37] as follows. We consider V Kv as a subvariety of Kvn , uniformly in v. Suppose RW has arity l. Note that this is a subset of V Kv l. Theorem Corollary Immediate by Theorem If W is a subvariety of V , both defined over Q. Note Tamagawa measures on adele spaces. Let V be a smooth algebraic variety defined over K. The measure on V AK may or may not converge. It converges when V is a semisimpe algebraic group see [85]. In many other cases it diverges, and one has to use convergence factors.
Here, as before, qv is the cardinality of the residue field of Kv. Generalizing work of Weil and Tamagawa see [85] around an adelic interpretation and proof of the Siegal mass formula on quadratic forms, Ono [72] studied adele spaces of hypersurfaces.
Let f x1 ,. Uniform definability of the convergence factors enables the use of model-theoretic tools to evaluate the local p-adic integrals with respect to measures induced from differential forms, following Denef, Loeser, Cluckers, and others on motivic inte- gration, cf.
Then the results of [30] which are stated in Section 14 yield analytic properties of the Euler product of the local integrals as global integrals. It is important to try to integrate other functions. Some ideas and guiding themes for this can be found in Section 19 in the context of automorphic forms. Problem The work of Kontsevich and Zagier in [61] concerns periods which are complex numbers whose real and imaginary parts are absolutely convergent integrals, over real semi-algebraic subsets of Rn , of rational functions with rational coefficients.
One expects the Tamagawa measures of definable sets to be related to L-functions, cf. Boolean Presburger predicates and Hilbert symbol As in 3.
This follows from the decidability of the theory of infinite atomic Boolean alge- bras T f in,res in the language LfBoolean in,res proved by Macintyre and myself in [34], see also [37] and Subsection 3.
Lrings -definable Proposition This follows from basic properties of the Hilbert symbol. We have shown the following. Proposition Extend Proposition Their motivation was the Riemann hypothesis, Connes and Consani cf. This was proved in [39]. This raises the question of describing the imaginaries in AK. Hrushovski-Martin-Rideau [54] have proved that Qp admits uniform elimination of imaginaries for all p relative to the "geometric sorts".
See Connes [22],[24] and their references for hyperrings. K Problem Artin reciprocity We shall only state Artin reciprocity for a global field. There is also a version for local fields which does deserve model-theoretic analysis, but we will not deal with that here. Let K be a global field. Let p be a prime in K that is unramified in F. Let xp denote the idele 1,.
See [78]. The term functoriality in the problem refers to the functoriality in K in Artin reciprocity see [78]. The following question was asked by Nicolas Templier after a talk I gave in Princeton. How does this definition depend on the number field K? In Problem Templier suggested that Problem Generalizations of Artin Reciprocity for non-abelian extensions is one of the aspects of the Langlands Program where the approach is via representations of adelic groups G AK , where G is a suitable algebraic group.
For more on this see Section It would be interesting to have a model-theoretic approach to non-abelian ex- tensions and Artin reciprocity. Note that S is allowed to be definable by means of predicates related to Hilbert symbols as we have allowed LfBoolean in,res -definability. Remarks on the idele class group of Q. One can relate the idele class group to definability in adeles.
The idele class group CQ is a definable subgroup of the adeles AQ. Section 2. 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 Qp with residue field of cardinality q. Let dx be a normalized Haar measure on K giving the valuation ring OK volume 1.
See [28]. We refer to Z s, p as a definable p-adic integral. Pas [73] and Macintyre [? The subject of motivic integration extends this uniformity and gives it a geometric meaning. It has been developed by Denef-Loeser [29], Cluckers-Loeser [20], and Hrushovski-Kazhdan [53], and has had several applications to algebraic geometry, number theory and algebra.
These Euler products are of a global nature 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 the shape of the rational functions can be used together with some results on algebraic geometry and model theory of finite and p-adic fields, together with combinatorial arguments, to prove the following result.
Let Z s, p be as above. Tauberian theorems of analytic number theory then yield. Formulate an adelic version of Theorem Once this is done, would the "completed Euler product" have meromorphic continuation to the whole complex plane? Would it satisfy a functional equation?
The conjectural connections to O-minimal structures and Hodge theory is challenging. Conjugacy class zeta functions in algebraic groups. An example of such a result is the following result from [30]. To be able to apply See [8] and the survey [32] for details. Formulate the results on global conjugacy class zeta functions adelically. Adelic height zeta functions and rational points.
The adelic height zeta function is a very useful guiding example for an approach to Problem See [70]. See [70],[49]. In [49], Gorodnik-Oh prove this under conditions on G and the stabilizer of u0 , and other conditions. Their beautiful approach is to consider U Q as a discrete subset of the adelic space U AQ defined as a restricted product as in Section Then under certain conditions the asymptotic of the numbers NT follows from an asymptotic for the volumes of BT , and for this they can make use of the ergodic- theoretic work of Duke-Rudnick-Sarnak cf.
One can ask the following question, whose positive solution would extend some results in [49]. Prove an analogue of Theorem Note that U AQ can be partitioned into finitely many pieces each of which is a definable subset of Type I of Am Q for some m using the usual covering of projective space by affine pieces.
The adelic integral factors as an Euler product by standard properties of the adelic height. These are defined as perfect pseudo-algebraically closed fields that have exactly one extension of each degree inside their algebraic closure. Work of Ax [1] implies the converse statement. Kiefe [? In [14], Chatzidakis, van den Dries, and Macintyre revisited the model theory of finite fields, and proved, among other results, generalizations of the Lang-Weil estimates for the number of Fq -points of an absolutely irreducible variety defined over Q to definable sets.
They also introduced a pseudo-finite measure. In [52], Hrushovski added additive characters to the language, and studied the continuous logic theory of pseudo-finite fields with an additive character. Firstly, he proves that the usual first-order theory is undecidable.
See [52] for decidability, quantifier elimination, and related notions in continuous logic. For example the graphs of addition and multiplication are discretely definable. In [52] Hrushovski proves that T is decidable, admits quantifier elim- ination to the level of algebraically bounded quantifiers, and is simple. 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 applications to exponential sums over definable sets in finite fields.
Enrich the language of rings by a 1-place predicate to be interpreted as the character. Again the additive characters on K are uniformly definable. In analogy to [52], one can ask. In [52], Hrushovski gives axioms for the theory of pseudo-finite fields with an additive character.
In conversations with him, the first author learned of the following question. Fix a prime p. Let ep denote the supremum of all the minimal idempotents e such that eAQ has residue field equal to Fp. Is it decidable? They are initially defined in a right half-plane, but admit meromorphic continuation beyond their half plane of convergence.
These L-functions generalize zeta functions of number fields and L-functions of Dirichlet characters [78]. See [78] and Section Is there a language extending an appropriate language for re- stricted products where one can express or give a model-theoretic interpretation of 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 would be related to a model theory for classical modular forms.
See Section 19 Problem S F denotes the class of Schwartz-Bruhat functions on K. However the theory has had far reaching influence and applications. Concerning the Hecke L-functions, it gave more information on the functional equation and so-called epsilon factors, gamma factors, and root numbers, and various quantities acquire an interpretation in terms of volumes of subsets in the adeles or ideles. For example the class number formula can be given a new volume- theoretic proof, see [78].
This has been a great influence in the conjecture of Birch and Swinnerton Dyer. It has been a starting point for this theory. See Section 19 for more on these and a suggested model-theoretic framework and questions. There is an Euler product factorization Y Z Z f. For the finitely many v where ramification occurs, the integrals are 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]. We shall formulate several question in this regard.
As a first step one can ask. We propose a form for these "definable integrals" in the Section 19 for the case of GL2 and Gn. We shall propose a generalization of p-adic specialization of motivic integrals and study their Euler products. One can then apply Theorem Langlands program and Jacquet-Langlands theory. A fascinating program and set of conjectures were given by Langlands which turned out to be related to several of the above topics at the interactions of algebra, geometry, analysis, and representation theory.
Here one works with reductive algebraic groups and the Langlands functoriality conjecture is one of the strongest of the conjectures see [66]. For an introduction to the Langlands conjectures and program see [66], [67], [10], [9]. Jacquet and Langlands [57] did this for GL2.
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