Abstract:
We generalize the work of DeBacker and Reeder to the case of unitary groups split by a tame extension. The approach is broadly similar and the restrictions on the parameter the same, but many of the details of the arguments differ. Let $G$ be a unitary group defined over a local field $K$ and splitting over a tame extension $E/K$. Given a Langlands parameter $\varphi : \mathcal{W}_K \rightarrow {^L G}$ that is tame, discrete and regular, we give a natural construction of an $L$-packet $\Pi_\varphi$ associated to $\varphi$, consisting of representations of pure inner forms of $G(K)$ and parametrized by the characters of the finite abelian group $A_\varphi = \operatorname{Z}_{\hat{G}}(\varphi)$.

Abstract:
This paper generalizes work of Buzzard and Kilford to the case $p=3$, giving an explicit bound for the overconvergence of the quotient $E_\kappa / V(E_\kappa)$ and using this bound to prove that the eigencurve is a union of countably many annuli over the boundary of weight space.

Abstract:
We generalize the rectifier of Bushnell and Henniart, which occurs in the local Langlands correspondence for $GL_n(K)$, to certain Langlands parameters for unramified connected reductive groups.

Abstract:
Let $G$ be a connected reductive group over a non-Archimedean local field. We prove that its parahoric subgroups are definable in the Denef-Pas language, which is a first-order language of logic used in the theory of motivic integration developed by Cluckers and Loeser. The main technical result is the definability of the connected component of the N\'eron model of a tamely ramified algebraic torus. As a corollary, we prove that the canonical Haar measure on $G$, which assigns volume $1$ to the particular \emph{canonical} maximal parahoric defined by Gross, is motivic. This result resolves a technical difficulty that arose in Cluckers-Gordon-Halupczok and Shin-Templier and permits a simplification of some of the proofs in those articles. It also allows us to show that formal degree of a compactly induced representation is a motivic function of the parameters defining the representation.

Abstract:
We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type N\'eron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

Abstract:
We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 sequence.

Abstract:
Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.

Abstract:
Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F_2 and F_3 with arithmetic Picard number 1, and a smooth quintic over F_2 with geometric Picard number 1. We also produce some examples of smooth quartics with geometric Picard number 2, which by a construction of van Luijk also have trivial geometric automorphism group.

Abstract:
The computation of the dimension of linear systems of curves with imposed base multiple points on surfaces is a difficult problem, with open conjectures that are being approached only with partial success. Among others, blowup-based techniques and degenerations show some promise of leading to satisfactory answers. We present an overview of such blowup based techniques at an introductory level, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's blowup and twist.