Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an overconvergent isocrystal on a variety over a perfect field of positive characteristic along a boundary divisor; this leads to an analogous construction for certain p-adic representations of the étale fundamental group of a variety. We then demonstrate some variational pro...

The divisor function σ(n) denotes the sum of divisors positive integer n. For a prime p and m∈N, p-adic valuation m is highest power which divides m. Formulas for νp(σ(n)) are established. p=2, these involve only odd primes dividing These expressions used to establish bound ν2(σ(n))≤⌈log2⁡(n)⌉, with equality if n product distinct Mersenne primes, an p, νp(σ(n))≤⌈logp⁡(n)⌉, related solutions Lju...

In the recent, many mathematicians studied the multiple zeta function in the complex number field. In this paper we construct the p-adic analogue of multiple zeta function which interpolates the generalized multiple Bernoulli numbers attached to χ at negative integers. §1. Introduction Let p be a fixed prime. Throughout this paper Z p , Q p , C and C p will, respectively, denote the ring of p-a...

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the q-Genocchi numbers and polynomials see 1–13 . In this paper, we construct a new type of q-Genocchi numbers G α,β n,q and polynomials G α,β n,q x with weight α and weak weight β. Throughout this paper, we ...

We complete our proof that given an overconvergent F -isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F -isocrystals overconvergent inside a partial compactification....

Let f1 (resp. f2) denote two (elliptic) newforms of prime level N , trivial character and weight 2 (resp. k + 2, where k ∈ {8, 12}). We provide evidence for the Bloch-Kato conjecture for the motive M = ρf1⊗ρf2 (−k/2−1) by proving that under some assumptions the p-valuation of the order of the Bloch-Kato Selmer group of M is bounded from below by the p-valuation of the relevant L-value (a specia...

Recently, many mathematicians have studied in the area of the Euler numbers and polynomials see 1–15 . The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In 14 , we introduced that Euler equation En x 0 has symmetrical roots for x 1/2 see 14 . It is the aim of this paper to observe an interesting phenomenon of “scattering”...

Using computer, a realistic study for new analogs of weighted q-Euler numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the weighted q-Euler polynomials E (α) n,q (x). The outline of this paper is as follows. In Section 2, we study weighted q-Euler polynomials E (α) n,q (x). In Section 3, we describe the...

We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are p-adic in nature.