The subject of the talk is the recent work of Mihăilescu, who proved that the equation $x^p-y^q=1$ has no solutions in non-zero integers $x,y$ and odd primes $p,q$. Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to $x^u-y^v=1$ in integers $x,y\>0$ and $u,v\>1$ is $3^2-2^3=1$. Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p=2$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $\chi$-parts of $p$-class groups of abelian number fields: first for relative class groups of real fields (again including the case $p=2$). As a consequence, a generalization of the Gras conjecture is stated...

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order $q$-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing...

Soit $K$ une extension cyclique réelle de degré 4 de $\mathbf{Q}$ de sous-corps quadratique $k$. Nous déterminons le nombre de classes et les unités de $K$ puis nous montrons que le problème de la “capitulation” de classes de $k$ dans $K$ est caractérisé par des propriétés élémentaires des unités de $K$. Nous avons obtenu une table numérique du nombre de classes, des unités ainsi que de l’éventuelle “capitulation” d’une classe, pour tous les corps $K$ de conducteur $f\<4000$ ; nous en publions ici un extrait.

Soient $d$ est un entier sans facteurs carrés, $\mathbf{K}=\mathbf{Q}(\sqrt{d},i)$, $i=\sqrt{-1}$, ${\mathbf{K}}_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de ${\mathbf{K}}_2^{\left(1\right)}$ et $G=\mathrm{Gal}({\mathbf{K}}_2^{\left(2\right)}/\mathbf{K})$ le groupe de Galois de ${\mathbf{K}}_2^{\left(2\right)}/\mathbf{K}$. Notre but est de montrer qu’il existe une forme de $d$ tel que le $2$-groupe $G$ est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe $G$ soit métacyclique dans le cas où $d=2p$ avec $p$ un nombre premier tel que $p\equiv 1(mod4)$.

From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field $K$, and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra involve very complicated products of high powers of singular values of the Klein form defined below and singular values of the discriminant $\Delta$. It is the aim of this paper to show, that in many cases a generator...