Narrow Results

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

This is a continuation of the papers [5, 6] on saturated distinguished chains (SDCs) over a local field K. We consider a totally ramified Galois extension L of K whose Galois group is a p-group and has exactly two proper higher ramification groups, where p is the characteristic of the residual field. Take a prime element π of L, and set π_i = N_L/Mi(π), where M_i are the corresponding fields to higher ramification groups such that K ⊂ M₁ ⊂ M₂ ⊂ L and N_L/Mi denotes the norm map from L to M_i for each i. We consider a SDC for α = π₁ + π₁π₂ + π₁π₂π over K, and the result is given as Main Theorem. There are thirteen cases according to sizes of extension degrees [L : M₂], [M₂ : M₁] and [M₁ : K].

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to a measure on the Skorokhod space of paths, D [0, ∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman–Kac formula in the local field setting as a consequence of the Hille–Yosida theory of semigroups.

We consider the arithmetic background of integral representations of finite groups over $p$-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite $p$-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

Let $K/F$ be a quadratic tamely ramified extension of a non-Archimedean local field $F$ of characteristic zero. In this paper, we give an explicit formula for Langlands’ lambda function ${\lambda }_{K/F}$.

The Sobolev space over local fields $H^s(𝕂)$ is defined. A multiresolution analysis for the Sobolev space is developed. Orthonormal wavelets with respect to these space are constructed and an example is also presented.

In this paper, we defined Calder$\acute{\text{o}}$n–Zygmund operator $T$ as $(T\varphi )(x)={\int }_{𝕂}\mathcal{𝒦}(x,y)\varphi (y)dy$ on local fields ($𝕂$) under certain conditions on $\mathcal{𝒦}(x,y)$ and then find the boundedness of operator $T$. Using result on boundedness of $T$ and orthonormal wavelet basis of $L^2(𝕂)$, we provide unconditional wavelet bases for $L^p$-norm over a local fields of positive characteristic by the wavelet coefficients.

We show the existence of abelian surfaces over ℚ_p having good reduction with supersingular special fiber whose associated p-adic Galois module is not semisimple.

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr_F/E(g(x), h(x)) = δ_{g, h} for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist.

Now let K be a finite extension of ℚ_p, let L/K be a finite abelian Galois extension of odd degree and let be the valuation ring of L. We define A_L/K to be the unique fractional -ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A_L/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.

Let ϕ(x) be an Eisenstein polynomial of degree n over a local field and let α be a root of ϕ(x). The Newton polygon of ρ(x) = ϕ(αx+α)/αⁿ is called the ramification polygon of ϕ(x). We attach an additional invariant, the segmental inertia degree, to each segment of the ramification polygon and use the slopes of the segments and their segmental inertia degrees to describe the splitting field of ϕ(x). Furthermore we present a method for determining the Galois group of ϕ(x) when the ramification polygon consists of one segment.

We explicitly compute the largest subtree, in the local Bruhat–Tits tree for ${\mathrm{\text{PSL}}}_2(k)$, whose vertices correspond to maximal orders containing a fixed order generated by a pair of orthogonal pure quaternions. In other words, we compute the set of maximal integral valued lattices in a ternary quadratic space, whose discriminant is a unit, containing a pair of orthogonal vectors, extending thus previous computations by Schulze-Pilot. The maximal order setting makes these computations simpler. The method presented here can be applied to arbitrary sub-orders or sublattices. The shape of this subtree is described, when it is finite, by a set of two invariants. In a previous work, the first author showed that determining the shape of these local subtrees allows the computation of representation fields, a class field determining the set of isomorphism classes, in a genus of Eichler orders, containing an isomorphic copy of a given order. Some further applications are described here.

Let $k$ be a local field and let $𝔄$ be the two-by-two matrix algebra over $k$. In our previous work, we developed a theory that allows the computation of the set of maximal orders in $𝔄$ containing a given suborder. This set is given as a subgraph of the Bruhat–Tits (BT)-tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the BT-tree in terms of balls in $k$. This is easier for orders spanning a split commutative subalgebra, i.e. an algebra isomorphic to $k\times k$. In fact, we have successfully used this idea in the past to compute embedding numbers for the split algebra. In the present work, we develop a theory of branches over field extensions that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions, generalizing previous results due to the first author and Saavedra. We assume throughout that $\mathrm{\text{char}}(k)\ne 2$.

In this paper, we present a description of the $\ell$-adic Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and ${\mathrm{\text{SL}}}_2({𝔽}_3)$, and in each case, we need to distinguish whether the inertia degree of $K$ over ${\mathbb{Q}}_2$ is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.