Idele Group Research Articles

Orthogonal to Principal Ideles

We describe the orthogonal to the group of principal ideles with respect to the global tame symbol pairing on the group of ideles of a smooth projective algebraic curve over a field.

A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

We construct examples of number fields which are not isomorphic but for which their adele groups, the idele groups, and the idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group G and subgroups H1 and H2 which are not conjugate in G but for which the G-module Z[G/H1] is isomorphic to Z[G/H2].

An adelic resolution for homology sheaves

We propose a generalization of the ordinary idele group by constructing certain adelic complexes for sheaves of -groups on schemes. Such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated with the presheaf of a homology theory with certain natural axioms satisfied, in particular, by -theory. In this case it is proved that the adelic complex provides a flabby resolution for this sheaf on smooth varieties over an infinite perfect field and that the natural morphism to the Gersten complex is a quasi-isomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative. In particular, this enables us to reprove that the intersection in Chow groups coincides (up to a sign) with the natural product in the corresponding -cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in -cohomology groups with certain indices.

Representation type of commutative Noetherian rings. III. Global wildness and tameness

Introduction Preliminaries Dedekind-like rings Wildness Structure of a genus Substitute for conductor squares Isomorphism classes in a genus, idele group action Web of class groups Direct sums Finite normalization Appendix A Appendix B Bibliography.

Automorphic Forms, Definite Quaternion Algebras, and Atkin–Lehner Theory on Trees

Associated to the multiplicative group of a definite quaternion algebra over Q are two notions of automorphic forms. One is a complex-valued function on the idele group. The other, a p-adic-valued function on a p-adic “upper half-plane,” arises from considering differentials on a Mumford curve associated to the quaternion algebra. Explicit connections between these forms are explored. In particular, forms of the second type are seen to arise from certain newforms of the first type. In order to define newforms, the Atkin–Lehner theory is developed for certain spaces of functions on the tree associated to PGL2(Qp). As an application, Schneider's proposed p-adic L-series associated to a differential on a Mumford curve is seen to be better interpreted as an adelic form twisted by a p-adic character.

A note on Rubin’s methods of the use of two Euler systems

Here we show that Rubin's method of the use of two Euler systems to the proofs of lwasawa main conjectures of the rational number field and of an imaginary quadratic field is noly proper to these two kinds of number fields. We mainly use the properties of adele ring and idele group in class field theory to get the result.

On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

LetL/kandT/kbe finite extensions of algebraic number fields. In the present work we introduce the factor group ofk*∩NL/kJLNT/kJTby (k*∩NT/kJT)NL/kL*, whereJLandJTare the idele groups ofLandT, respectively. The main theorem shows that the computation of this factor group can be reduced to the computation in finite group theory, and the computation with Galois groups of local extensions at a finite number of primes of the base fieldk. We then apply the main theorem to establish a number of interesting results on the equality of norm groups as subgroups of the multiplicative group ofk. In particular, we obtain new results on solitary non-Galois extensions of algebraic number fields.

Hilbert cusp forms and special values of Dirichlet series of Rankin type

Let K be a totally real number field of degree nover ℚ and let c be an integral ideal of a maximal order of K. Given a nonnegative integer j and a Hecke character on the group of ideles of K, let denote the space of Hilbert cusp forms of holomorphic type on ℋn of weight j, level c and character ψ where ℋn is the n-th power of the Poincaré upper half plane ℋ.Let g be an element of , where 1 is the trivial character. If u ∈ Sk(c, ψ), then the product gu is an element of Sk+l (c, ψ), and therefore we can consider the linear map sending u to gu. Let be the adjoint of the linear map Φg with respect to the Petersson inner product.

Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields

One of the fundamental theorems of global class field theory states that there is a one-to-one correspondence between finite Abelian extensions of an algebraic number fieldkand the norm groups of the idele class groupCk=Jk/k* ofk. More generally, for finite extensionsKandLofkthere is the following group theoretic interpretation ofNK/kCK⊆NL/kCL. LetEbe a finite Galois extension ofkcontainingKandL, and letG=G(E/k),H=G(E/K), andN=G(E/L) be the corresponding Galois groups. It follows by global class field theory thatNK/kCK⊆NL/kCLiffG′H⊆G′N, whereG′ is the commutator subgroup ofG. In the present work we prove thatNK/kJK⊆NL/kJLiff every element ofHof prime power order is conjugate inGto an element ofN. We also show that the same group theoretic condition is equivalent toN(K/k)⊆N(L/k), whereN(K/k) is the group of elements ofk* that are local norms everywhere fromKtok. We then use this group theoretic criterion to investigate the equality of norm groups as subgroups ofk*.

On a Conjecture of Shimura Concerning Periods of Hilbert Modular Forms

Introduction. In this paper, we shall give an affirmative answer to an essential part of the conjecture of Shimura on P-invariants of Hilbert modular forms. Let F be a totally real algebraic number field of degree n and JF be the set of all isomorphisms of F into C. Let FA (resp. FA) be the adele ring (resp. the idele group) of F and Fx be the archimedean part of FA. Let X be a primitive system of eigenvalues of Hecke operators which occurs in the space of holomorphic Hilbert modular cusp forms on GL(2, FA) of weight k and f be the primitive form which belongs to X. In [S 1], Shimura introduced an invariant u(e, f) E CX for every 6 E (Z/2Z)JF such that

Extra Capitulation and Central Extensions

Let L be a cyclic unramified extension of the number field K, with G ≔ Gal(L/K), and L(1) the Hilbert class field of L. The central object of studying those ideals of K which become principal, i.e., capitulate, has been H1(G, EL), where EL denotes the group of global units of L. However, if one lets CL and UL denote the idele class group of L and the group of unit ideles, respectively, there is an isomorphism Hi+1(G, EL) = Hi(G, UL/EL), and UL/EL has the advantage of being isomorphic to an idele class subgroup of CL; this is our basic tool. In this paper, we study "extra" capitulation, that is, whenever there is more capitulation than one would normally expect. More precisely, we show that there is a nontrivial ramified central extension of L(1)M/K, with M some abelian extension of K, exactly when there is extra capitulation.

On the structure of the idele group of an algebraic number field

The purpose of this paper is to present the results of E. Artin and Furtwängler, with which they proved the principal ideal theorem, as a structure theorem of the idele group of an algebraic number field. Such treatment may be helpful to clarify the Arithmetic nature these results possess.

On Automorphic Forms on GL 2 and Hecke Operators

In [4], Jacquet and Langlands showed that if an irreducible unitary representation of the general linear group of degree 2 over the adele ring of an A-field' is an irreducible constituent of the representation on the space of cuspidal functions, then it is decomposable, i.e., it is a tensor product of irreducible representations of local components of the group. In the present paper, we prove the semi-simplicity of the algebra generated by some Hecke operators operating on a certain space of cusp forms. As a consequence of this, we show that two such irreducible constituents are equivalent if their local factors are equivalent for almost all places including all archimedean places. In more detail, let F be an A-field, and FA (resp. FAX) the adele ring (resp. the idele group) of F. We put GF= GL2(F) and GA = GL2(FA). We know that the center ZA of GA is isomorphic to FAX. For a unitary character X of the idele class group FA /Fs, let L'(GF\GA, X) denote the space of measurable functions on GA satisfying

On Zeta Functions of Quaternion Algebras

The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a totally real algebraic number field 1? of degree m, and let D be an order in A; let S be the idele group of A, and U the group of units in S with respect to D; let p be a representation of ( a/f)*, f being an integral two-sided rD-ideal, and let ki (1 < i < m) be non-negative integers. These being given, we can speak of a space of automorphic functions associated with (p, {ki}) (cf. ? 2.2 and ? 2.5) and of a repre-sentation Z of the Hecke ring 9R(U, @) in this space. Let T(q) be the sum of all integral elements in 9R(U, @ of norm q and let C(s) be the Dirichlet. series defined by

An application of the generalized norm residue symbol

Let k be an algebraic number field of finite degree or an algebraic function field of dimension 1 with finite constant field and let K/k be a finite, separably Galois extension field. We call an algebraic extension field L, not necessarily Galois nor necessarily of finite degree, of K an everywhere locally abelian extension, denoted in the following as EL-abelian extension, for brevity, of K over k, if and only if for every finite extension L' of K in L and every valuation v of L it holds that the completion field Lv' of L' with reference to v is a composition field of KV and an abelian extension Mv of kv in Lv', where we denote by kv and KV the closures of k and K in Lv', respectively. Let Q be an algebraic closure of K, BK/k the maximum separably EL-abelian extension of K/k in Q, CK/k the maximum separably central extension of K/k in Q (i.e. the maximum extension CK/k which is separably normal over k, contains K, and has the Galois group of CK/k/K in the center of the Galois group of CK/k/k), Ik and IK the groups of the ideles of k and K, Ak and AK the maximum separably abelian extensions of k and K in Q, respectively. Let DK/k=BK/k nCKIk. Then, the norm residue symbol 0Jk in Ak/k is, if restricted into NK/kIK, extensible into a homomorphism UK/k of NK/kIK into the Galois group G(DK/k/K). We shall study in the present article arithmetical meanings of the extensibility. In ?1, we shall define oUK/k precisely, in ?2, study the kernel and obtain a principal genus theorem (Theorem 2), from which will follow easily a certain generalized formulation of some fundamental theorems in the class field theory.1 Combining it with a known property2 of the quotient group of the connected component of the unit element of the idele class group by the natural image of the maximal compact subgroup of the connected component of the unit element of the idele group, we shall obtain, in ?3, our main result (Theorem 8), which concerns total norm residues in the principal idele group of k for K/k, and which is, the author thinks, an idele-theoretic reconstruction of a research of Scholz.3

Galois Group of the Maximal Abelian Extension over an Algebraic Number Field

The aim of the present work is to determine the Galois group of the maximal abelian extensionΩAover an algebraic number fieldΩof finite degree, which we fix once for all.Let Z be a continuous character of the Galois group ofΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Zcharacter of Ω. For our purpose, it suffices to determine the groupXlof the characters ofΩwhose orders are powers of a prime numberl.

On a 3-Cohomology Class in Class Field Theory and the Relationship of Algebra- and Idele-Classes

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-