A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers

The main purpose of this paper is to construct a family of modified p-adic twisted functions, which interpolate the modified twisted q-Bernoulli polynomials and the generalized twisted q-Bernoulli numbers at negative integers. We also give some applications and examples related to these functions and numbers.     

Further congruences involving Bernoulli numbers

Congruences of Voronoi's type for Bernoulli numbers are proved via Bernoulli distributions and connections to some already known congruences of a similar type are briefly discussed.     

On p-adic differentiability

A p-adic-valued function on the p-adic integers has a continuous derivative, Mahler showed, whenever its interpolation coefficients decay at a certain rate. It is shown here that Mahler's decay condition is equivalent to the strict differentiability of the function. There is a discussion of the Banach-space structure of the space of strictly differentiable functions. It is shown, moreover, that there is no rate of decay common to all functions with continuous derivative. Specifically, given any decay condition, there exists a function with derivative identically zero, whose interpolation coefficients decay more slowly.     

On p-adic interpolating function for q-Euler numbers and its derivatives

In this paper we study a two-variable p-adic q-l-function lp,q(s,t|χ) for Dirchlet's character χ, with the property that

     

On Euler numbers, polynomials and related p-adic integrals

In this note we give a new proof of Witt's formula for Euler numbers, which are related to some known or new identities involving the Euler numbers. We also obtain a brief proof of a classical result on Euler numbers modulo of two due to M.A. Stern using the approach of p-adic integration, which was recently proved by G. Liu, and Z.-W. Sun. Finally some explicit formulas for Genocchi numbers are proved and applications are given.     

p-adic Arakelov theory

We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. We also prove analogues of the Adjunction formula and the Riemann-Roch formula.     

Zeros of p-adic forms

We show that all p-adic quintic forms in at least n>4562911 variables have a non-trivial zero. We also derive a new result concerning systems of cubic and quadratic forms.     

p-adic matrix valued inequalities

This paper explores the p-adic analogues of the matrix valued triangle inequality and the matrix valued norm inequality.     

Cyclotomic p-adic multi-zeta values

The cyclotomic p-adic multi-zeta values are the p-adic periods of ${\pi }_1^{uni}({\mathbb{G}}_m?{\mu }_M,?)$, the unipotent fundamental group of the multiplicative group minus the M-th roots of unity. In this paper, we compute the cyclotomic p-adic multi-zeta values at all depths. This paper generalizes the results in [9] and [10]. Since the main result gives quite explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these p-adic periods and also, through the use of p-adic Hodge theory, in proving non-triviality results for the corresponding p-adic Galois representations.     

Bounds of divided universal Bernoulli numbers and universal Kummer congruences

Let be a prime. We obtain good bounds for the -adic sizes of the coefficients of the divided universal Bernoulli number when is divisible by . As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo for the divided universal Bernoulli numbers for the case , which is a new result.

     

A p-adic Simpson correspondence

For curves over a p-adic field we construct an equivalence between the category of Higgs-bundles and that of “generalised representation” of the etale fundamental group. The definition of “generalised representations” uses p-adic Hodge theory and almost etale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group.     

Semisimple strata for p-adic classical groups

Let F₀ be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F₀. In this paper, we prove some fundamental results towards the classification of the representations of G via types [8]. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].     

On p-adic zeros of forms

Using the procedure of G. I. Arkhipov and A. A. Karatsuba (Math. USSR-Izv.19 (1982), 321–340), their exponential lower bound on the number of variables possible for a form of degree d having only the trivial p-adic zero is sharpened.     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

On the behaviour of p-adic L-functions

Text

Let Lp(s,χ) denote a Leopoldt-Kubota p-adic L-function, where p>2 and χ is a nonprincipal even character of the first kind. The aim of this article is to study how the values assumed by this function depend on the Iwasawa λ-invariant associated to χ. Assuming that λ?p−1, it turns out that Lp(s,χ) behaves, in some sense, like a polynomial of degree λ. The results lead to congruences of a new type for (generalized) Bernoulli numbers.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=5aaB1d6fZDs.     

Sheaves of bounded p-adic logarithmic differential forms

Let K be a local field, X the Drinfel'd symmetric space of dimension d over K and X the natural formal OK-scheme underlying X; thus G=GLd+1(K) acts on X and X. Given a K-rational G-representation M we construct a G-equivariant subsheaf of OK-lattices in the constant sheaf M on X. We study the cohomology of sheaves of logarithmic differential forms on X (or X) with coefficients in . In the second part we give general criteria for two conjectures of P. Schneider on p-adic Hodge decompositions of the cohomology of p-adic local systems on projective varieties uniformized by X. Applying the results of the first part we prove the conjectures in certain cases.     

The solubility of certain p-adic equations

Estimates are given for the number of variables required to solve p-adic equations. In particular, systems of homogeneous and of inhomogeneous additive equations, as well as single homogeneous equations in general, are studied.