The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the  $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.

Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← N /p p. ...

We prove a conjecture of Denef on parameterized p-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalyt...

Let tn be a sequence that satisfies a first order homogeneous recurrence tn = Q(n)tn−1, where Q ∈ Z[n]. The asymptotic behavior of the p-adic valuation of tn is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.

We give an explicit construction of the antiequivalence of the category of finite flat commutative group schemes of period 2 defined over a valuation ring of a 2-adic field with algebraically closed residue field. This result extends the earlier author’s approach to group schemes of period p > 2 from Proceedings LMS, 101, 2010, 207-259.

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a nu...

Following Sun and Moll [4], we study vp(T (N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behaviour, by means of Fourier coefficients. The method is the Mellin-Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.

Let R be a complete discrete valuation ring, S = R[[u]] and d a positive integer. The aim of this paper is to explain how to compute efficiently usual operations such as sum and intersection of sub-S-modules of S. As S is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, ...

For a prime number p > 2, we give a direct proof of Breuil’s classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings’s strict modules can be identified via the Fontaine-Wintenb...

Let p be a fixed odd prime. Throughout this paper Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p and let q be regarded as either a complex number q ∈ C or a p-adic number q...