Related papers: On generalized Fuchs theorem over relative $p$-adi…

In this paper, we study finite projective differential modules on $p$-adic polyannuli satisfying the Robba condition. Christol and Mebkhout proved the decomposition theorem (the $p$-adic Fuchs theorem) of such differential modules on one…

Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$\varphi$-modules and $G$-$(\varphi,\nabla)$-modules over the Robba ring, which are exact faithful $F$-linear tensor functors…

This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification…

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an…

The aim of this article is to generalize Kato's (commutative) p-adic local epsilon-conjecture [Ka93b] for families of (phi,Gamma)-modules over the Robba ring. In particular, we prove the generalized local epsilon-conjecture for rank one…

We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s=2-p. We use this generalization to give a statement on the…

Building on foundations introduced in a previous paper, we give several p-adic analytic descriptions of the categories of etale Zp-local systems and etale Qp-local systems on an affinoid algebra over a finite extension of Qp (or more…

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In…

This work is devoted to study of algebraicty modulo p of Siegel's G-functions. Our goal is to emphasize the relevance of the notion of strong Frobenius structure, clasically studied in the theory of the p-adic diffenrential equations, for…

(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the…

We give a simple proof of an isomorphism between the two $\mathbb{C}[t]$-modules: the module of relative cohomologies $\Lambda^2/dH\land \Lambda^1$ and the module of Abelian integrals corresponding to a regular at infinity polynomial $H$ in…

We investigate a question of Burns and Sano concerning the structure of the module of Euler systems for a general $p$-adic representation. Assuming the weak Leopoldt conjecture, and the vanishing of $\mu$-invariants of natural Iwasawa…

The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape…

We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$,…

In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient…

Farkas' lemma is an ubiquitous tool in optimisation, as it provides necessary and sufficient conditions to have $b \in A(P)$, where $P$ is a closed convex cone, $A$ is a (continuous) linear mapping and $b$ is a fixed vector. The standard…

In this paper, we answer a question due to Y. Andr\'e related to B. Dwork's conjecture on a specialization of the logarithmic growth of solutions of $p$-adic linear differential equations. Precisely speaking, we explicitly construct a…