We indicate the place of Shimura’s reciprocity law in class field theory and give a formulation of the law that reduces the technical prerequisites to a minimum. We then illustrate its practical use by dealing with a number of classical problems from the theory of complex multiplication that have been the subject of recent research. Among them are the construction of class invariants and the explicit generation of ring class fields.

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shimura curves.

The elliptic curve with Weierstrass equation $Y^2 + \sqrt{6}XY - Y = X^3 - (2 + \sqrt{6})X^2$ has good reduction modulo every prime of the ring of integers of $\mathbf{Q}(\sqrt{6})$. We show that every abelian variety over $\mathbf{Q}(\sqrt{6})$ that has good reduction everywhere is isogenous to a power of this elliptic curve.

Let $k$ be a totally real field and $p$ an odd prime number. We assume that $p$ splits completely in $k$ and also that Leopoldt's conjecture is valid for $k$ and $p$. In this note, focusing on Greenberg's conjecture, we will report on our recent results concerning $p$-adic special functions and ideal class groups in the cyclotomic ${\mathbb{Z}}_p$-extension of $k$.

Let $k/F$ and $k'/F$ be CM-extensions, $K = kk'$ and $K_+$ the maximal totally real subfield of $K$. It holds $h^-(k)\;|\; h^-(K)$ if $k/F$ is unramified at all finite primes or $K_+/F$ is unramified. Hence, $h^-(k)$ is an obstacle that prevents $h^-(K)$ from being 1. On the other hand, analytic class number formula implies the class number relation $h^-(K) = h^-(k)h^-(k')/c(K/F)$, where $c(K/F)$ is an integer determined by units. Consistency of the first assertion and the class number relation is guaranteed by $h^-(k')$. As a reason of the consistency, the parity equality $h^-(k) \equiv h^-(k')$ (mod 2) is formulated under the situation of the first assertion. (See Theorems 1 and 2.) Non-trivial Examples (§§3.2) and a proof of the parity equality are given. The tool behind the first assertion and indices related with the class number relation are discussed in detail.

Suppose that $p = tn + r$ is a prime and that $h$ is the class number of the imaginary quadratic field, $\mathbb{Q}(\sqrt{-t})$. If $t \equiv 3$ (mod 4) is a prime, just $r$ is a quadratic residue modulo $t$ and the order of $r$ modulo $t$ is $\frac{t-1}{2}$, then $4p^h$ can be written in the form $a^2 + tb^2$ for some integers $a$ and $b$. And if $t = 4k$ where $k \equiv 1$ (mod 4), $r \equiv 3$ (mod 4), $r$ is a quadratic non-residue modulo $t$ and the order of $r$ modulo $t$ is $k - 1$, then $p^h = a^2 + kb^2$ for some integers $a$ and $b$. Our result is that $a$ or $2a$ is congruent modulo $p$ to a product of certain binomial coefficients modulo sign. As an example, we give explicit formulas for $t = 11$, $19$, $20$ and $23$.

We study the capitulation problem for certain number fields of degree 3, 4, and 6.

(I) Capitulation of the 2-ideal classes of $\mathbb{Q}(\sqrt{d}, i)$ (by A. AZIZI)

Let $d \in \mathbb{N}$, $i = \sqrt{-1}$, $\mathbf{k} = \mathbb{Q}(\sqrt{d}, i)$, ${\mathbf{k}}_1^{(2)}$ be the Hilbert 2-class field of $\mathbf{k}$, ${\mathbf{k}}_2^{(2)}$ be the Hilbert 2-class field of ${\mathbf{k}}_1^{(2)}$, $C_{\mathbf{k},2}$ be the 2-component of the ideal class group of $\mathbf{k}$ and $G_2$ the Galois group of ${\mathbf{k}}_2^{(2)}/\mathbf{k}$. We suppose that $C_{\mathbf{k},2}$ is of type (2, 2); then ${\mathbf{k}}_1^{(2)}$ contains three extensions $F_i/\mathbf{k}$, $i = 1, 2, 3$. The aim of this section is to study the capitulation of the 2-ideal classes in $F_i$, $i = 1, 2, 3$, and to determine the structure of $G_2$.

(II) On the capitulation of the 3-ideal classes of a cubic cyclic field (by M. AYADI)

Let $k$ be a cubic cyclic field over $\mathbb{Q}$, and ${\mathbf{k}}_1^{(3)}$ the Hilbert 3-class field of $\mathbf{k}$. If the class number of $\mathbf{k}$ is exactly divisible by 9, then its 3-ideal class group is of type (3, 3), and ${\mathbf{k}}_1^{(3)}$ contains four cubic extensions ${\mathbf{K}}_i/\mathbf{k}$ in which we study the capitulation problem for the 3-ideal classes of $\mathbf{k}$.

(III) On the capitulation of the 3-ideal classes of the normal closure of a pure cubic field (by M. C. ISMAILI)

Let $\Gamma = \mathbb{Q}(\sqrt[3]{n})$ be a pure cubic field, $\mathbf{k} = \mathbb{Q}(\sqrt[3]{n}, j)$ its normal closure ($j = e^{\frac{2i\pi}{3}}$), ${\mathbf{k}}_1^{(3)}$ the Hilbert 3-class field of $\mathbf{k}$, and let $S_{\mathbf{k}}$ be the 3-ideal class group of $\mathbf{k}$. When $S_{\mathbf{k}}$ is of type (3, 3), we study the capitulation of the 3-ideal classes of $S_{\mathbf{k}}$ in the four intermediate extensions of ${\mathbf{k}}_1^{(3)}/\mathbf{k}$, and we show that if the class number of $\Gamma$ is divisible by 9, then we have some necessary conditions on $n$. We have also some informations about the unit group of $\mathbf{k}$ in some cases.

Today’s notion of “global field” comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E.Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of the 20th century, with emphasis on the emergence of class field theory for function fields. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.