Let $f$ be a modular eigenform of even weight $k\ge 2$ and new at a prime $p$ dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to $f$ a monodromy module $D_f^{FM}$ and an $ℒ$-invariant ${ℒ}_f^{FM}$. The first goal of this paper is building a suitable $p$-adic integration theory that allows us to construct a new monodromy module $D_f$ and $ℒ$-invariant ${ℒ}_f$, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $ℒ$-invariants are equal....

Nous étudions d’abord le foncteur cohomologique local. Ensuite, nous introduisons la notion de $𝒟$-modules arithmétiques surcohérents. Nous prouvons que les $F$- isocristaux unités sont surcohérents et surtout que la surcohérence est stable par images directes, images inverses extraordinaires et foncteurs cohomologiques locaux. On obtient, via cette stabilité, une formule cohomologique pour les fonctions $L$ associées aux complexes duaux de complexes surcohérents. Celle-ci étend celle d’Étesse et Le Stum...

Let $a$ be a $0-1$ sequence with a finite number of terms equal to 1. The distance sequence ${\delta }^{\left(a\right)}$ of $a$ is defined as a sequence of the numbers of $(1-1)$-couples of given distances. The paper investigates such pairs of $0-1$ sequences $a,b$ that a is different from $b$ and ${\delta }^{\left(a\right)}={\delta }^{\left(b\right)}$.

In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑n≥0 (an / bf(n)), where an ∈ Z and 1 ≤ |an| ≤ K for all n ≥ 0, is neither rational nor quadratic.

The alphabet ${𝐅}_3+v{𝐅}_3$ where $v^2=1$ is viewed here as a quotient of the ring of integers of $𝐐\left(\sqrt{-2}\right)$ by the ideal (3). Self-dual ${𝐅}_3+v{𝐅}_3$ codes for the hermitian scalar product give $2$-modular lattices by construction $A_K$. There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual...

Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality:$$\text{Br}\left(X\right)\times \text{Pic}\left(X\right)\to \mathbb{Q}/\mathbb{Z}$$between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\text{Br}\left(X\right)$ in ${\displaystyle \prod _{P\in X}\text{Br}\left(k_P\right)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\tilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every $X_{et}$-gerb which...

We explicitly perform some steps of a 3-descent algorithm for the curves $y^2=x^3+a$, $a$ a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.