Let p be a fixed odd prime number. Throughout this paper Z, Q, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the ring p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p−vp(p) = p−1. If q ∈ Cp, we normally as...

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.

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 the algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(p) = 1 p . When one talks of q-extension, q is variously considered as an indeter...

Let L be a number field and let E/L be an elliptic curve with potential supersingular reduction at a prime ideal ℘ of L above a rational prime p. In this article we describe a formula for the slopes of the Newton polygon associated to the multiplication-by-p map in the formal group of E, that only depends on the congruence class of p mod 12, the ℘-adic valuation of the discriminant of a model f...

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.

For a prime number p > 2, we give a direct proof of Breuil’s classification of finite flat group schemes killed by p over the valuation ring of a p-adic field with perfect residue field. As application we establish a correspondence between finite flat group schemes and Faltings’s strict modules which respects associated Galois modules via the Fontaine-Wintenberger field-of-norms functor

We show that the p-adic valuation of the sequence of Fibonacci numbers is a p-regular sequence for every prime p. For p 6= 2, 5, we determine that the rank of this sequence is α(p) + 1, where α(m) is the restricted period length of the Fibonacci sequence modulo m.