The purpose of this paper is to derive formulae for the sums of products of the q-Euler polynomials and numbers, since many identities can be obtained from our sums of products of the q-Euler polynomials and numbers. In 1 , Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Ja...

The theory of valuations was started in 1912 by the Hungar-ian mathematician Josef K urschh ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic elds as deened by Kurt Hensel. In the following decades we can observe a quick development of valuation theory , triggered mainly by the discovery that much of algeb...

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...

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...

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non-vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of S...

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...

, 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 prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

In the present paper we provide a formula that allows to compute number of stable digits any integer tetration base \in {\mathbb N}_0. The digits, at given height power tower, indicates how many last (generic) are frozen. Our is exact for every which not coprime 10, although maximum gap equal V(a)+1 (where V(a) denotes constant congruence speed a) can occur, in worst-case scenario, between uppe...

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.