Let K be a complete discrete valuation field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G=Gal(L/K) and suppose that the induced extension of residue fields kL/kK is separable. Let 𝕎n(⋅) denote the ring of p-typical Witt vectors of length n. Hesselholt [‘Galois cohomology of Witt vectors of algebraic integers’, Math. Proc. Cambridge Philos. Soc.137(3) (2004), 551–557] conjectured that the pro-abelian group {H1 (G,𝕎n (𝒪L))}n≥1 is isomorphic to zero. Hogadi and Pisolkar [‘On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt’, J. Number Theory131(10) (2011), 1797–1807] have recently provided a proof of this conjecture. In this paper, we provide a simplified version of the original proof which avoids many of the calculations present in that version.

Let K be a complete discrete valuation field of mixed characteristic (0,p), with possibly imperfect residue field. We prove a Hasse–Arf theorem for the arithmetic ramification filtrations on GK, except possibly in the absolutely unramified and non-logarithmic case, or the p=2 and logarithmic case. As an application, we obtain a Hasse–Arf theorem for filtrations on finite flat group schemes over 𝒪K.

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

Let K be a finite unramified extension of Qp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional -representations of GK and characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

Laumon introduced the local Fourier transform for ℓ-adic Galois representations of local fields, of equal characteristic p different from ℓ, as a powerful tool for studying the Fourier–Deligne transform of ℓ-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon’s formula relating the ε-factor to the determinant of the local Fourier transform under the same condition.

This paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.

Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the étale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces.

The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of for p≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of using B-pairs as defined by Berger.

Let K be a local field of equal characteristic p>2, let XK/K be a smooth proper relative curve, and let ℱ be a rank 1 smooth l-adic sheaf (l≠p) on a dense open subset UK⊂XK. In this paper, under some assumptions on the wild ramification of ℱ, we prove a conductor formula that computes the Swan conductor of the etale cohomology of the vanishing cycles of ℱ. Our conductor formula is a generalization of the conductor formula of Bloch, but for non-constant coefficients.

Let K be a real quadratic number field and let p be a prime number which is inert in K. We denote the completion of K at the place p by Kp. We propose a p-adic construction of special elements in Kp× and formulate the conjecture that they should be p-units lying in narrow ray class fields of K. The truth of this conjecture would entail an explicit class field theory for real quadratic number fields. This construction can be viewed as a natural generalization of a construction obtained by Darmon and Dasgupta who proposed a p-adic construction of p-units lying in narrow ring class fields of K.

We associate two almost Cp-representations to a (ϕ,Γ)-module, and we compute their dimensions and heights. As a corollary, we get a full faithfulness result for Be-representations.

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

Given a compact p-adic Lie group G over a finite unramified extension L/ℚp let GL/ℚp be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic GL/ℚp-representations that coincides with passage to analytic vectors in the case L=ℚp. On the other hand, we study the functor ‘passage to analytic vectors’ and its derived functors over general basefields. As an application we compute the higher analytic vectors in certain locally analytic induced representations.

We develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

We develop and study the epsilon factor of a ‘local system’ of p-adic coefficients over the spectrum of a complete discrete valuation field K with finite residue field of characteristic p>0. In the equal characteristic case, we define the epsilon factor of an overconvergent F-isocrystal over Spec(K), using the p-adic monodromy theorem. We conjecture a global formula, the p-adic product formula, analogous to Deligne’s formula for étale ℓ-adic sheaves proved by Laumon, which explains the importance of this local invariant. Namely, for an overconvergent F-isocrystal over an open subset of a projective smooth curve X, the constant of the functional equation of the L-series is expressed as a product of the local epsilon factors at the points of X. We prove the conjecture for rank-one overconvergent F-isocrystals and for finite unit-root overconvergent F-isocrystals. In the mixed characteristic case, we study the behavior of the epsilon factor by deformation to the field of norms.