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Jean-Robert Belliard

Publications and source records attributed to Jean-Robert Belliard.

7 recordsLinked to original sources
arXiv2011

Indices isotypiques des éléments cyclotomiques

Given $F$ a real abelian field, $p$ an odd prime and $χ$ any Dirichlet character of $F$ we give a method for computing the $χ$-index $\displaystyle (H^1(G_S,\mathbb{Z}_p(r))^χ: C^F(r)^χ)$ where the Tate twist $r$ is an odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This $χ$-index can now be computed in terms only of elementary arithmetic of finite fields $\FM_\ell$. Our work generalizes previous results by Kurihara who used the assumption that the order of $χ$ divides $p-1$.

math.NT↗
arXiv2009

Sous-modules d'unités en théorie d'Iwasawa

We give a necessary and sufficient "Galois descent" condition to the freeness of the Iwasawa module built from Sinnott's circular units. Then we describe explicit examples for which this condition is not fulfilled.

math.NT↗
arXiv2009

Global Units modulo Circular Units : descent without Iwasawa's Main Conjecture

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\ZM_p$-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants $\la$ and $μ$ attached to the inverse limit $X_\infty=\limpro X_n$. It relies on "good" descent properties satisfied by $X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic it is known that the $p$-parts of the orders of the global units modulo circular units $U_n/C_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n/C_n$, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa's Main Conjecture.

math.NT↗
arXiv2009

On modified circular units and annihilation of real classes

For an abelian totally real number field $F$ and an odd prime number $p$ which splits totally in $F$, we present a functorial approach to special "$p$-units" previously built by D. Solomon using "wild" Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the $p$-class group of $F$ (in the particular context here), as well as related annihilation results and index formulae.

math.NT↗
arXiv2007

Asymptotic cohomology of circular units

Let $F$ be a number field, abelian over the rational field, and fix a odd prime number $p$. Consider the cyclotomic $Z_p$-extension $F_\infty/F$ and denote $F_n$ the ${n}^{\rm th}$ finite subfield and $C_n$ its group of circular units. Then the Galois groups $G_{m,n}=\Gal(F_m/F_n)$ act naturally on the $C_m$'s (for any $m\geq n>> 0$). We compute the Tate cohomology groups $\Hha^i(G_{m,n}, C_m)$ for $i=-1,0$ without assuming anything else neither on $F$ nor on $p$.

math.NT↗
arXiv2006

Global Units modulo Circular Units : descent without Iwasawa's Main Conjecture

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\ZM_p$-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants $\la$ and $μ$ attached to the inverse limit $X_\infty=\limpro X_n$. It relies on "good" descent properties satisfied by $X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic it is known that the $p$-parts of the orders of the global units modulo circular units $U_n/C_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n/C_n$, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa's Main Conjecture.

math.NT↗