Search

This is a research announcement of the joint work with Jyunheong Kim, Takeo Noda and Yuji Terashima [KMNT]. It is motivated by the question, posed by Deninger, on finding out analogies for 3-manifolds of the Hilbert reciprocity law for number fields along the line of arithmetic topology. We introduce local symbols and show the reciprocity law for a 3- dimensional foliated dynamical system., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic fields of degree p, unconditionally for p=3 and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the p=3 case this gives the distribution of the 3-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the 4-torsion in quadratic fields., Comment: 30 pages

Let p ≡ 3 ( mod 4 ) {p\equiv 3\pmod{4}} and l ≡ 5 ( mod 8 ) {l\equiv 5\pmod{8}} be different primes such that p l = 1 {\frac{p}{l}=1} and 2 p = p l 4 {\frac{2}{p}=\frac{p}{l}_{4}} . Put k = ℚ ⁢ ( l ) {k=\mathbb{Q}(\sqrt{l})} , and denote by ϵ its fundamental unit. Set K = k ⁢ ( - 2 ⁢ p ⁢ ϵ ⁢ l ) {K=k(\sqrt{-2p\epsilon\sqrt{l}})} , and let K 2 ( 1 ) {K_{2}^{(1)}} be its Hilbert 2-class field, and let K 2 ( 2 ) {K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type ( 2 , 2 , 2 ) {(2,2,2)} . Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group G = Gal ⁡ ( K 2 ( 2 ) / K ) {G=\operatorname{Gal}(K_{2}^{(2)}/K)} , and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within K 2 ( 1 ) {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.

This is a survey article of several topics on pro-p-extensions of number fields with restricted ramification, particularly concerning explicit presentations of various pro-p Galois groups by generators and relations.

Rewrite of certain parts - Introduction of a new § on capitulation of classes in the tower - New data; Let k be a number field, p≥2 a prime and S any set of finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k) = \Z/p^N\Z and if S non-empty is totally ramified. Using analogues of Chevalley's formula (Gras, Proc. Math. Sci. (127)(1) (2017)),we give an elementary proof of a stability theorem (Theorem 3.1) for generalized p-class groups X_n of the layers k_n < K:let lambda = max(0, #S-1-rho) given in Definition 1.1; then#X_n = X_0 x p^{\lambda n} for all n in [0,N], if and only if #X_1 = #X_0 x p^{lambda}. This improves the case lambda = 0 of Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020),whose techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We deduce capitulation properties of X_0 in the tower. Finally we apply our principles to the torsion groups T_n of abelian p-ramification theory. Numerical examples are given.

International audience; 2 and for every integer m.In passing we will see the notion of geometric Galois representations and the conjecture of Fontaine-Mazur.We will try to make this talk accessible to a general audience.

We correct the faulty formulas given in a previous article and we compute the defect group for the Iwasawa $\lambda$ invariants attached to the S-ramified T-decomposed a belian pro-{\ell}-extensions on the Z{\ell}-cyclotomic extensionof a number field. As a consequence, we extend the results of Itoh, Mizusawa and Ozaki on tamely ramified Iwasawa modules for the cyclotomic Z{\ell}-extension of abelian fields., Comment: in French

A rigid meromorphic cocycle is a class in the first cohomology of the discrete group $\Gamma :={\mathrm{SL}}_{2}(\mathbb{Z}[1/p])$ with values in the multiplicative group of nonzero rigid meromorphic functions on the $p$ -adic upper half-plane $\mathcal{H}_{p}:=\mathbb{P}_{1}(\mathbb{C}_{p})-\mathbb{P}_{1}(\mathbb{Q}_{p})$ . Such a class can be evaluated at the real quadratic irrationalities in $\mathcal{H}_{p}$ , which are referred to as “RM points.” Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of $\Gamma $ -orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the values of modular functions on ${\mathrm{SL}}_{2}(\mathbb{Z})\backslash \mathcal{H}$ at complex multiplication (CM) points: in particular, they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high $p$ -adic accuracy leads to convincing numerical evidence for the algebraicity and factorization of the resulting singular moduli for real quadratic fields.

Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.

Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($\lambda = \mu = 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, by means of images in ${\mathcal T}_k$ of ideal norms from the layers $k_n$ of the cyclotomic tower (Theorem (5.2)). These images are obtained via the formal algorithm computing, by ``unscrewing'', the $p$-class group of~$k_n$. Conjecture (5.4) of equidistribution of these images would show that the number of steps $b_n$ of the algorithms is bounded as $n \to \infty$, so that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true ``with probability $1$''. No assumption is made on $[k : \mathbb{Q}]$, nor on the decomposition of $p$ in $k/\mathbb{Q}$., Comment: New shorter version, new title, improvements and corrections suggested by a colleague whom I warmly thank

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields., in French

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$. Furthermore, using the modular method, we prove Fermat's Last Theorem effectively, over some number fields whose degrees over $\mathbb{Q}$ are $3,4,5,6$ and $8$., Comment: in French

We extend to logarithmic class groups the results on abelian principalization of tame ray class groups of a number field obtained in a previous article. As a consequence, for any extension $K/k$ of number fields which satisfies the Gross-Kuz'min conjecture for the prime $\ell$ and where at least one of the infinite places completely splits, we prove that there exists infinitely many abelian $\ell$-extensions $F/k$ such that the relative subgroup $\mathcal{C}\ell_{K/k} = \Ker (\mathcal{C}\ell_K\to\mathcal{C}\ell_\k)$ of the $\ell$-group of logarithmic classes of $K$ capitulates in the compositum $FK$.

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $\Gamma$ is boundedly generated as an abstract group., Comment: Final version - to appear in `Algebra and Number Theory'

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$ , let $k_{\mu}$ be the ray class field of $k=\mathbb{Q}(e^{2\pi i/\ell})$ modulo $2p^{\mu}$ . We present certain class fields $K_{\mu}$ of $k$ such that $k_{\mu}\subset K_{\mu}\subset k_{\mu+1}$ , and we provide a necessary and sufficient condition for $K_{\mu}=k_{\mu+1}$ . We also construct, in the sense of Hilbert, primitive generators of the field $K_{\mu}$ over $k_{\mu}$ by using Shimura’s reciprocity law and special values of theta constants.

Based on the method of Boston and Leedham-Green et al. for computing the Galois groups of tamely ramified $p$-extensions of number fields, this paper gives a large family of triples of odd prime numbers such that the maximal totally real $2$-extension of the rationals unramified outside the three prime numbers has the Galois group of order $512$ and derived length $3$. This family is characterized arithmetically, and the explicit presentation of the Galois group by generators and relations is also determined completely.