We recall some basic constructions from p-adic Hodge theory, then describe some recent results in the subject. We chiefly discuss the notion of B-pairs, introduced recently by Berger, which provides a natural enlargement of the category of p-adic Galois representations. (This enlargement, in a different form, figures in recent work of Colmez, Bellaïche, and Chenevier on trianguline representati...

, for m in the range 0 ≤ m ≤ n, that are not divisible by p. We give a matrix product that generalizes Fine’s formula, simultaneously counting binomial coefficients with p-adic valuation α for each α ≥ 0. For each n this information is naturally encoded in a polynomial generating function, and the sequence of these polynomials is p-regular in the sense of Allouche and Shallit. We also give a fu...

We explore a conjecture posed by Eswarathasan and Levine on the distribution of p-adic valuations harmonic numbers $$H(n)=1+1/2+\cdots +1/n$$ that states set $$J_p$$ positive integers n such p divides numerator H(n) is finite. proved two results, using modular-arithmetic approach, one for non-Wolstenholme primes other Wolstenholme primes, an anomalous asymptotic behaviour valuation $$H(p^mn)$$ ...

In this article, we prove the integrality of $v$-adic multiple zeta values (MZVs). For any index $\mathfrak{s}\in\mathbb{N}^r$ and finite place $v\in A := \mathbb{F}\_q\[\theta]$, Chang Mishiba introduced notion MZVs $\zeta\_A(\mathfrak{s})\_v$, which is a function field analogue Furusho's $p$-adic MZVs. By estimating valuation show that $\zeta\_A(\mathfrak{s})\_v$ integer for almost all $v$. T...

Many mathematicians have studied Euler numbers and Euler polynomials( see [1-11]). Euler polynomials posses many interesting properties and arising in many areas of mathematics and physics. In this paper we introduce the generalized q-Euler numbers and polynomials with weak weight α. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q de...

This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M , for odd primes p, is bounded below by a linear function of the mod p first betti number of M . Sharper bounds using more delicate topological invariants are given as well.

Let p be a fixed prime. Throughout this paper Z, Zp, Qp, and Cp will, respectively, denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp, cf.[1, 2, 3]. Let vp be the normalized exponential valuation of Cp with |p| = p −vp(p) = p and let a+ pZp = {x ∈ Zp|x ≡ a( mod p N )}, where a ∈ Z lies i...

Let p be an odd prime number. 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. The normalized valuation in Cp is denoted by | · |p with |p|p = 1 p . We say that f is a uniformly differentiable function at a point a ∈ Zp and denote...