Related papers: Appendix: proof of the Uniformity Conjecture

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes…

We prove a compact embedding theorem in a class of spaces of piecewise H1 functions subordinated to a class of shape regular, but not necessarily quasi-uniform triangulations of a polygonal domain. This result generalizes the…

We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to…

The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobenius-semilinear endomorphism of a finite free module over the Robba ring (the ring of germs of rigid analytic…

We extend the Jacquet-Langlands'correspondence between the Hecke-modules of usual and quaternionic modular forms, to overconvergent p-adic forms of finite slope. We show that this correspondence respects p-adic families and is induced by an…

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and…

The notion of relative cuspidality for distinguished representations attached to $p$-adic symmetric spaces is introduced. A characterization of relative cuspidality in terms of Jacquet modules is given and a generalization of Jacquet's…

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's…

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(\varphi,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose…

We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian…

A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift (R^{d},+,0) and its Euclidean distance. This approach allows us to…

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the…

The classical Pfaff-Darboux theorem, which provides local 'normal forms' for $1$-forms on manifolds, has applications in the theory of certain economic models [Chiappori P.-A., Ekeland I., Found. Trends Microecon. 5 (2009), 1-151]. However,…

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does…

The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ann. of Math. 102 (1975), 223-290]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR…

This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully…

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms.…

The paper of S. Gulick [Sidney (Denny) L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math 18 No. 1, 1966] contains some good mathematics, but it also contains an error. It claims that for a Banach…

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…