Explicit Reciprocity Law Research Articles

Generalized Artin-Hasse and Iwasawa formulas for the Hilbert symbol in a higher local field. II

In this paper, we consider the generalized Hilbert symbol in a higher local field of charactersitic 0 with first residue field of characteristic 0 as well and with perfect last residue field of positive characteristic p, which originates from higher local p-class field theory developed by I. Fesenko. Using the descent to a subfield of mixed characteristic, from the generalized Artin-Hasse and Iwasawa formulas proved in a previous paper we deduce respective Artin-Hasse and Iwasawa explicit reciprocity laws in the case under consideration. Bibliography: 6 titles.

Explicit Reciprocity Law for Lubin–Tate Formal Groups

In this article, using Fontaine’s ΦΓ–module theory, we give a new proof of Coleman’s explicit reciprocity law, which generalizes that of Artin–Hasse, Iwasawa andWiles, by giving a complete formula for the norm residue symbol on Lubin–Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.

A generalization of the Contou-Carrère symbol

Using techniques of Algebraic Geometry, the aim of this paper is to give a generalized definition of the Contou-Carrere symbol as a morphism of schemes. In fact, from formal schemes and Heisenberg groups, we provide a new definition of the Contou-Carrere symbol and a generalization of it associated with a separable extension . Moreover, a reciprocity law is proved and classical explicit reciprocity laws are recovered from it.

Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring#

The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

Explicit reciprocity law and formal moduli for Lubin-Tate formal groups

Explicit reciprocity law and formal moduli for Lubin-Tate formal groups

On explicit reciprocity law over formal groups

Let 𝔣 be a one‐dimensional Lubin‐Tate formal group over ℤp. Colmez (1998) and Perrin‐Riou (1994) proved an explicit reciprocity law for tempered distributions over the formal group 𝔾m. In this paper, the general explicit reciprocity law over the formal group 𝔣 is proved.

Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.

On Explicit Reciprocity Laws for the Local Carlitz–Kummer Symbols

In this article, we prove Wiles and Coates–Wiles explicit reciprocity laws for the local symbols attached to cyclotomic functions fields.

Iwasawa theory and explicit reciprocity law. A remake of an article of P. Colmez

Let V be a crystalline p -adic representation of the absolute Galois group of \mathbb Q_p . The author has built the Iwasawa theory of such a representation in Invent. Math (1994) and conjectured a reciprocity law which has been proved by P. Colmez. In this text, we write the initial construction with simplification and the proof of P. Colmez in a different language. This point of view will allow us to study the universal norms in the geometric cohomology classes associated to V by Bloch and Kato in a forthcoming article.

Théorie d’Iwasawa des représentations 𝑝-adiques d’un corps local

This paper takes up the study of Iwasawa theory of p p -adic representations of a local field via the theory of ( φ , Γ ) (\varphi ,\Gamma ) -modules. In particular, a vast generalisation of Coleman’s isomorphism is given as well as an explicit reciprocity law relating this isomorphism to Bloch-Kato’s exponential maps in the case of a de Rham representation.

The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law

The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law

Explicit formulae for the Hilbert symbol of a formal group over the Witt vectors

In this paper an explicit formula is obtained for a generalisation of the Hilbert symbol, associated with an arbitrary formal group of finite height, defined over the ring of Witt vectors with coefficients in a perfect field of characteristic p>0. This formula becomes the Bruckner-Vostokov formula in the case of a multiplicative formal group. The proof is based on an application of Fontaine's theory of p-adic periods of formal groups, the Fontaine-Wintenberg field-of-norms functor, and Witt's explicit reciprocity law in characteristic p.

On Jacobi sum Hecke characters ramified only at 2

Let K be the cyclotomic field of the mth roots of unity in some fixed algebraic closure of Q. Weil has shown in [Jacobi sums as Großencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487–495] that Jacobi sums induce Hecke characters on K modulo m 2, or equivalently homomorphisms of the idèle class group of K into K ∗ . Given such a Hecke character ϱ, we then obtain for every place p of K a continuous homomorphism ϱ p (the local component of ϱ at p), from ( K p ) ∗ into K ∗ . When p does not divide m, ϱ p is completely determined. When p divides m the determination of ϱ p is much more difficult. Coleman and McCallum in [Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101] have given formulas and computed the conductors of the local components of ϱ at p for p∥ m and m odd. Hasse, in [Zetafunktion und L-Funktionen zu einem arithmetischen Funktionenkorper vom Fermatschen Typus, Abh. Deut. Akad. Wiss. Berlin Kl. Math. Nat. 1954 4 (1955)] has computed the local component and the conductor when m is prime, while in [C. Jensen, Uber die Führer einer Klasse Heckescher Großencharaktere, Math. Scand. 8 (1960), 81–96; D. Rohrlich, Jacobi sums and explicit reciprocity laws, Compositio Math. 60 (1986), 97–110; C.-G. Schmidt, Uber die Führer von Gauss'schen Summen als Großencharaktere, J. Number Theory 12, No. 3 (1980), 283–310] the authors have given estimates and in some cases have determined the conductor of the local component. Here we deal with the case where m = 2 n , and we give an explicit formula for the local component in terms of Hilbert symbols, similar to the one in [R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101]. In [D. Prapavessi, On the conductor of 2-adic Hilbert norm residue symbols, to appear in J. Algebra] we use this result to compute the conductor of ϱ when m = 2 n .

Explicit reciprocity laws on relative Lubin-Tate groups

Explicit reciprocity laws on relative Lubin-Tate groups

An explicit 2nth reciprocity law.

An explicit 2nth reciprocity law.

The explicit reciprocity law in local class field theory

The explicit reciprocity law in local class field theory

On explicit reciprocity laws. II.

On explicit reciprocity laws. II.

On explicit reciprocity laws.

On explicit reciprocity laws.

On the Field of Definition for a Field of Automorphic Functions: II

The purpose of this paper is to complement the main theorems of the previous papers with the same title, referred to hereafter as [FI] and [FII], and to simplify the proofs. In [FI, II], it was shown that a certain abelian extension k of a certain algebraic number field K' can be determined by a family 1(&2) of abelian varieties belonging to a given type 12 of structures. The structures consist of endomorphism algebra, polarization and points of finite order, of an abelian variety. The case of structures without points of finite order was treated in [FI]. We observed that k is unramified over K', and the law of reciprocity for the extension k/K' can be described in terms of the interrelation between 12 and its transform. One of the aims of [FII] was to extend this result to the general case involving points of finite order and ramified abelian extensions. However, we determined only the ideal group in K' corresponding to k, but did not give an explicit reciprocity law in such a general case. This will be done in the present paper. We shall also reformulate the results in a more comprehensible manner, and give a simplified proof by means of the characterization of the field k provided by [2, 5.1]. This characterization enables us to dispense with artificial considerations which were necessary in the original proofs. We devote one section to clarify a point concerning the existence of a special member of the family Y(f2), of which the previous proof in [FI] was not with complete clarity. List of symbols. They were introduced in [FI, II], occasionally with a somewhat general meaning.