### ${\mathbb{F}}_{1}$-schemes and toric varieties.

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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 $\mathcal{L}$-invariant ${\mathcal{L}}_{f}^{FM}$. 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 $\mathcal{L}$-invariant ${\mathcal{L}}_{f}$, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $\mathcal{L}$-invariants are equal....

Nous étudions d’abord le foncteur cohomologique local. Ensuite, nous introduisons la notion de $\mathcal{D}$-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 $X$ 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 $\prod _{P\in X}\text{Br}\left({k}_{P}\right)$ 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.

Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$. The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

In this note, we consider a one-parameter family of Abelian varieties $A/\mathbb{Q}\left(T\right)$, and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.

The aim of this paper is to compare two modules of elliptic units, which arise in the study of elliptic curves E over quadratic imaginary fields K with complex multiplication by ${}_{K}$, good ordinary reduction above a split prime p and prime power conductor (over K). One of the modules is a special case of those modules of elliptic units studied by K. Rubin in his paper [Invent. Math. 103 (1991)] on the two-variable main conjecture (without p-adic L-functions), and the other module is a smaller one,...

Let $f$ be a weight $k$ holomorphic automorphic form with respect to ${\mathrm{\Gamma}}_{0}\left(N\right)$. We prove a sufficient condition for the integrality of $f$ over primes dividing $N$. This condition is expressed in terms of the values at particular $CM$ curves of the forms obtained by iterated application of the weight $k$ Maaß operator to $f$ and extends previous results of the Author.

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.

We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.