We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

Let $G$ be the $n$-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $G$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions.

In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated with the scaling function for multiresolution analysis of ${{L}^{2}}\left( K \right)$. We use probability and martingale methods to provide such a characterization.

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$ , motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$ , and we complete its proof by reducing the positive characteristic case to characteristic $0$ . For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$ : we find an elliptic curve $E^{\prime }$ defined over a $p$ -adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$ .

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.

We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the K-theory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This correspondence arises from theta correspondences in E6 and E7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G2 to a single conjecture about the parameterization for PGSp 6.

We derive a power series formula for the p-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of p.

Dans ce papier nous étudions une correspondance de Jacquet–Langlands locale pour toutes les représentations lisses irréductibles. La correspondance est caractérisée par le fait qu’elle respecte la correspondance de Jacquet–Langlands classique et commute avec le foncteur d’induction parabolique. Elle est compatible dans un sens à préciser au foncteur de Jacquet et à l’involution d’Aubert–Schneider–Stuhler. Nous utilisons cette correspondance pour montrer qu’une certaine classe de représentations d’une forme intérieure de $\mathrm{GL}_n$ sur un corps $p$-adique sont unitarisables. C’est le premier pas dans la preuve de la conjecture U1 de Tadić.

We study a local Jacquet–Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet–Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet’s functor and the involution of Aubert–Schneider–Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet–Langlands correspondence and we deduce that a certain class of representations of an inner form of $\mathrm{GL}_n$ over a $p$-adic field are unitarizable. This is the first step towards the proof of Conjecture U1 of Tadić.

If G is a (split) Kac–Moody group over a field K endowed with a real valuation $\omega$, we build an action of G on a geometric object $\mathcal I$. This object is called a building, as it is an union of apartments, with the classical properties of systems of apartments. However, these apartments are more exotic: that associated to a torus T may be seen as the gluing of all Satake compactifications of affine apartments of T with respect to spherical parabolic subgroups of G containing T. Another geometric realization of these apartments makes them look more like the apartments of $\Lambda$-buildings; then the translations of the Weyl group act only on infinitely small elements of the apartment, so we call these buildings microaffine.

Let $F$ be a $p$-adic field and $K/F$ a finite, cyclic, tamely ramified field extension. We show, without any restriction, that base change from $F$ to K, in the sense of Arthur and Clozel, is compatible with the operation of $K/F$-lifting for simple characters defined by the authors in an earlier paper. This extends and completes the previous work, and represents a substantial step towards the goal of describing tamely ramified base change in terms of the structure theory of representations of Bushnell and Kutzko. We also give a consequence for the Langlands correspondence. The main step in the proof is to show that lifting of simple characters is compatible with automorphic induction, in the sense of Henniart and Herb, through an unramified extension. This is achieved via a new and independently interesting result concerning the conductor of a pair of supercuspidal representations. It gives a complete description of the minima of (a suitably normalized version of) the conductor when one of the representations is fixed. All results concerning the conductor are also valid for non-Archimedean local fields of positive characteristic.

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.