 There has been a lot of news lately about Adele, which I found very exciting because adeles are important in mathematics.  In number theory, it is helpful to complete the rational numbers (in the sense of topology), since then you can do analysis.  The standard completion yields the real numbers.  However there is extra arithmetical information to be found in the $p$-adic completions.  But how do you know which $p$ to pick?  Why not pick all of them at once!  The adeles give you a way to do that (and to throw in the real numbers, too.)  They play a central role in class field theory.  Also, in the study of algebraic groups, introducing the adeles leads to adelic algebraic groups.  (The previous link gives some history on the naming of the adeles.)   A few key people who used adeles, as well as the ideles, early on are Claude Chevalley, Armand Borel, André Weil, John Tate, and Kenkichi Iwasawa.  (They used different names for the ring until the world settled on Weil’s name for it.)  The next time you hear “Hello”, followed by a pause, I hope you will join me in saying, “It’s $\mathbb{R}\times \prod’_{p}\mathbb{Q}_p$”.

Here are some reviews of papers on adeles.

MR0670072 (83m:10032)
Weil, André
With appendices by M. Demazure and Takashi Ono. Progress in Mathematics, 23. Birkhäuser, Boston, Mass., 1982. iii+126 pp. ISBN: 3-7643-3092-9
10C30 (12A82 12A85 20G35)

This important work when it first “appeared” in 1961 introduced the adèles into the study of arithmetic problems in algebraic groups. The question of Tamagawa numbers of algebraic groups was first systematically formulated here. The book is divided into two parts. The first part—Chapters I and II—deals with the geometry and measure on the space of adelic points of an algebraic variety. These materials are not easily available elsewhere. In the second part—Chapters III and IV—the author studies the zeta function of division algebras and central simple algebras and then uses the Poisson summation formula to calculate the Tamagawa numbers of “most” classical groups (for complete results on classical groups see articles by J. G. M. Mars [Ann. of Math. (2) 89 (1969), 557–574; MR0263828 (41 #8427)], the author [Acta Math. 113 (1965), 1–87; MR0223373 (36 #6421)] and R. P. Langlands [Algebraic groups and discontinuous subgroups (Boulder, Colo., 1965), pp. 143–148, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1966; MR0213362 (35 #4226)]). The book ends with an appendix by Demazure proving that the Tamagawa number of the exceptional group $G_2$ is 1, and the appendix by Ono on recent results for groups over number fields has a useful bibliography. The reader who is acquainted with the author’s book [Basic number theory, third edition, Springer, New York, 1974; MR0427267 (55 #302)] will find some familiar material here. It is worth pointing out that the author’s conjecture (Appendix 2, Section 2) on Tamagawa numbers has not yet been completely solved.

Reviewed by K. F. Lai

MR0141670 (25 #5067)
Borel, Armand; Harish-Chandra
Arithmetic subgroups of algebraic groups.
Bull. Amer. Math. Soc. 67 1961 579–583.
22.55 (20.65)

The authors bring to light all the basic qualitative theorems on the arithmetic of algebraic groups which stimulate and support further quantitative investigations. These theorems cover the qualitative results of Siegel, Ramanathan and Weil for classical groups and that of the reviewer for solvable groups. Actually, these known ideas or results, in particular, the notion of Siegel domains, are used and generalized by the techniques of the Lie theory, e.g., Iwasawa’s KAN-decomposition of a reductive group compatible with a Cartan involution. Let $G$ be a connected complex algebraic group defined over ${bf Q}$. For a subring $B$ of ${bf C}$, one denotes by$G_B$ the subgroup of elements of $G$ which have coefficients in $B$, and whose determinant is a unit of $B$. One calls an open subset $U$ of $G_{{bf R}}$fundamental if:
(i) $G_{{bf R}}=UG_{{bf Z}}$;
(ii) $KU=U$ for suitable maximal compact subgroup $K$ of $G_{{bf R}}$;
(iii) $U^{-1}Ucap xG_{{bf Z}}y$is finite for any $x,yin G_{{bf Q}}$.
Theorem 1 states the existence of such $U$. The construction of $U$ is a generalization of Hermite’s procedure in the case of indefinite quadratic forms, thereby a lemma on the rational representation of reductive groups (Lemma 2) is crucial. (iii) implies the finite generation of $G_{{bf Z}}$. Theorem 2, which the reviewer will not state explicitly, is a generalization of the classical fact that the number of classes of integral forms with a given non-zero determinant is finite. Theorem 3 gives a criterion for the compactness of the homogeneous space $G_{{bf R}}/G_{{bf Z}}colon G_{{bf R}}/G_{{bf Z}}$ is compact if and only if the identity component of $G$ has no nontrivial characters defined over ${bf Q}$ and every unipotent element of $G_{{bf Q}}$ belongs to the radical of $G_{{bf Q}}$ [see also Mostow and Tamagawa, #5069]. The details of this paper have been published in Ann. of Math. (2) 75 (1962), 485–535.

Reviewed by T. Ono

MR0141671 (25 #5068)
Borel, Armand
Some properties of adele groups attached to algebraic groups.
Bull. Amer. Math. Soc. 67 1961 583–585.
22.55 (20.65)

This is the adelized version of the theorems reviewed above [#5067]. Let $G$ be a connected algebraic group defined over ${bf Q}$ and denote by $G_A$ the adelization of $G$. Using the terminologies introduced by the reviewer [Ann. of Math. (2) 70 (1959), 266–290; MR0114817 (22 #5635)], the author proves that:
(i) $(G)$ is of type $(F)$ and of type $(M)$;
(ii) $G$ is of type $(C)$ if and only if every unipotent element of $G_{{bf Q}}$ belongs to the radical of $G_{{bf Q}}$;
(iii) $G_A/G_{{bf Q}}$ is compact if and only if $G$ is of type $(C)$ and $X_{{bf Q}}(G)=1$ (Theorem 1).
Theorem 2 is the adelization of Theorem 2 in the paper reviewed above [loc. cit.]: Assuming $G$ reductive, let $picolon Grightarrowtext{GL},(m,{bf C})$ be a rational representation, defined over ${bf Q}$, and $vin{bf Q}^m$ a point whose orbit is closed. Then $vpi_A(G_A)cap{bf Q}^m$ is the union of a finite number of orbits of $G_{{bf Q}}$. Theorem 2 implies the following Theorem 3: Assuming $G$ reductive, the principal homogeneous spaces over ${bf Q}$ which have rational points in all ${bf Q}_p$(including $p=infty$) form a finite number of isomorphism classes. Details of proofs of these theorems are not given.

Reviewed by T. Ono

MR1213914 (95e:14016)
Iwasawa, Kenkichi
Algebraic functions. (English summary)
Translated from the 1973 Japanese edition by Goro Kato. Translations of Mathematical Monographs, 118. American Mathematical Society, Providence, RI, 1993. xxii+287 pp. ISBN: 0-8218-4595-0
14H05 (14-01 30F10 33E05)

This book contains an exposition of the classical theory of algebraic functions in one variable (or equivalently, the theory of compact Riemann surfaces). The author divides the book into two parts: in the first part he gives an exposition of the algebraic theory and in the second part presents the analytic theory.

The aim of the algebraic part of the book is to give the proof of the Riemann-Roch (RR) theorem. The notion of valuation is introduced in Chapter 1 and the proof of the RR theorem is given in the second chapter using the notion of adèles. The proof presented by Iwasawa is based on A. Weil’s proof [J. Reine Angew. Math. 179(1938), 129–133; Zbl 19, 247]. Iwasawa gives two definitions of differentials: the definition of Weil in terms of adèles and the definition of Hasse in terms of power series. The equivalence between these definitions is proved. The final appendix of the book is devoted to explaining the theory of residues of meromorphic functions as developed by J. T. Tate [Ann. Sci. École Norm. Sup. (4) 1 (1968), 149–159; MR0227171 (37 #2756)].

In Chapters 3, 4 and 5 Iwasawa gives an exposition of the analytical theory of compact Riemann surfaces and the theory of abelian integrals and abelian functions. The proof of the existence of analytic functions and differentials with given singularities is presented following H. Weyl’s method [Die Idee der Riemannschen Fläche, Teubner, Leipzig, 1913; Jbuch 44, 492]. In Chapter 5 Iwasawa gives an elementary introduction to the theory of abelian integrals, Jacobian varieties and theta functions, proving the classical theorems of Abel and Jacobi. The preface of the book contains very interesting historical remarks on the theory of algebraic functions. For a modern reader this is a useful book which can help in understanding the historical developement of the theory of algebraic curves or Riemann surfaces, both from the algebraic and the analytical point of view. The book is written in an elementary way, and is accessible to graduate students.

Reviewed by José M. Muñoz Porras

MR0049950 (14,252b) Reviewed
Tate, John
The higher dimensional cohomology groups of class field theory.
Ann. of Math. (2) 56, (1952). 294–297.
10.0X
Journal Article

It is shown that the Galois cohomology group $H^r(G,A)$ in the idèle-class group $A$ of an algebraic number field (or the multiplicative group of a $p$-adic number field) is canonically isomorphic to $H^{r-2}(G,Z)$ where $Z$ is the additive group of rational integers $(r>2)$. Let first $G$ be a finite group and $A$ any abelian $G$-group. Let $alpha$ be a 2-cohomology class of $G$ in $A$, and $overline A$ be the Artin splitting group for $alpha$. It is proved that the following two axioms are equivalent:
(1) $H^1(U,A)=0$, and $H^2(U,A)$ is cyclic of the same order as $U$, generated by the restriction of $alpha$ to $U$, for all subgroups $U$of $G$;
(2) $H^1(U,overline A)=H^2(U,overline A)=0$ for all subgroups $U$ of $G$.
The proof depends on two exact sequences $overline A/Asimeq I$$R/Isimeq Z$, where $R$ is the group ring of $G$ over $Z$, considered as a $G$-module, and $I$ is its ideal generated by the elements $sigma-1 (sigmain G)$. These exact sequences entail exact sequences for 0-, 1-, 2-cohomology groups in $I,A,overline A$ and in $Z,I,R$, respectively. They together lead to the theorem. Combined with the fact that (2) implies the vanishing of higher cohomology groups, due to Serre, Lyndon, Hochschild and the reviewer [cf. Hochschild and Nakayama, Ann. of Math. (2) 55, 348–366 (1952); MR0047699 (13,916d)], it leads to the fact that $H^r(G,A)$ is isomorphic with $H^{r-2}(G,Z)$; the isomorphism is given by cup product with $alpha$. As the fundamental (or canonical) Galois 2-cohomology class $alpha$ in the idèle-class group satisfies (1) [Hochschild and Nakayama, loc. cit.], the statement at the opening of the review follows. The vanishing of $H^3(G,A)$ in this class field theory case, proved independently by Hochschild, is a particular instance. The author promises a subsequent paper in which negative-dimensional cohomology groups will be introduced and the reciprocity law will be regarded as another special case of the result.

Reviewed by T. Nakayama

MR0002357 (2,38c)
Chevalley, C.
La théorie du corps de classes. (French)
Ann. of Math. (2) 41, (1940). 394–418.
10.0X
Journal Article

The words “class field theory” have by now obtained a very general meaning. They are used whenever an extension field of an algebraic number field is determined by the original field. Class field theory in the original sense dealt with all finite abelian extension fields of a given field and their connections with the class groups in the original field. The author deals with all finite or infinite abelian extension fields and their connections with the original field; the words “class group,” however, hardly occur in the whole paper. Another striking difference of this paper from earlier work on class field theory [e.g., the author’s thesis, J. Fac. Sci. Univ. Tokyo 2, 365–476 (1933)] is the exclusion of analytical methods. The theory of the $zeta$-functions which for so long a time seemed necessary can now be omitted. On the other hand, topological methods play an important part in this new presentation of class field theory. [The use of topological methods in this connection was also suggested by D. van Dantzig, Ann. École Norm. (3) 53, 275–307 (1936), especially 281.] A topology is introduced into two groups: (1) The Galois group $G$ of the maximum abelian extension field $A$ of the given field $K$ (that is, the field composed of all abelian extension fields of $K$). This group is made into a topological group following the method of Krull, who noticed that the Galois groups of $A$ with respect to all finite abelian extension fields of $K$ form a complete system of neighborhoods for the unit element of $G$. This topological group is compact and therefore the results of Pontrjagin concerning abelian compact groups can be applied to it. (2) The fundamental group $I$ consisting of the so-called “idèles” of $K$. The introduction of the idèles enables the author to dispose entirely of the congruence groups [see also C. Chevalley, J. Math. Pures Appl. (9) 15, 359–371 (1936)]. An idèle is a sequence of non-zero $frak p$-adic numbers (where $frak p$ runs through all different finite and infinite prime ideals of $K$ taken in some fixed order), of which only a finite number are not $frak p$-adic units. The product of two idèles is obtained by multiplying the coordinates. If $alpha$ is any non-zero number of $K$, the sequence all of whose elements are $alpha$ is an idèle since there is only a finite number of $frak p$’s for which $alpha$ is not a $frak p$-adic unit. These special idèles form a subgroup $P$ of all idèles, the principal idèles. It is isomorphic with the multiplicative group of all non-zero numbers of $K$. For the idèles a norm is defined which satisfies the usual axioms and which agrees with the definition of norm for numbers. Furthermore, a topology is introduced in $I$ which, though weaker than the $p$-adic topology, is more suitable for the present purposes. The character group of $I$ is of great importance. In particular, a character of $I$ which is identically equal to 1 on $P$ is called a differential in analogy to the differentials in the case of algebraic function-fields [see A. Weil, J. Reine Angew. Math. 179, 129–133 (1938)]. The main result is the proof of the existence of an isomorphism between the character group of $G$ and the group of differentials of $K$. This isomorphism can be constructed explicitly. Since, according to Pontrjagin, the character group of the compact group $G$ determines the group $G$ itself, the group $G$ therefore also is determined by $K$ and can indeed be constructed explicitly. This last fact can be considered as the content of Artin’s general law of reciprocity.

Reviewed by O. Todd-Taussky 