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,...

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.