Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

In this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

Generic cuspidal representations of 𝑈(2, 1)

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .

Semistable models of elliptic curves over residue characteristic 2

Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$ , we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $ . We provide several examples along with an easy corollary concerning $2$ -torsion.

TOWARDS A NON-ARCHIMEDEAN ANALYTIC ANALOG OF THE BASS–QUILLEN CONJECTURE

We suggest an analog of the Bass–Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles $K_{0}$.

Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions {E/F} , where F is a finite unramified field extension of {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice {(L,H)} , when a base field is unramified over {\mathbb{Q}} at a prime {(2)} .

CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

COMPUTINGL-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES

We give an explicit description of the stable reduction of superelliptic curves of the formyn=f(x) at primes$\mathfrak{p}$whose residue characteristic is prime to the exponentn. We then use this description to compute the localL-factor and the exponent of conductor at$\mathfrak{p}$of the curve.

ON THE LOCAL LANGLANDS CORRESPONDENCE FOR SPLIT CLASSICAL GROUPS OVER LOCAL FUNCTION FIELDS

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

Ramification theory and perfectoid spaces

Let $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ whose ramifications are bounded by $j$ are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl’s theory of higher fields of norms with the ramification theory of Abbes–Saito, and the integrality of small Artin and Swan conductors of $p$-adic representations with finite local monodromy.