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.     

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.     

Bounds on degrees of p-adic separating polynomials

We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Štefankovi? (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl-Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances.     

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.     

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.     

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.     

p-adic valuations and k-regular sequences

A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x)∈Q_p[x], we consider the sequence , where v_p is the p-adic valuation. We show that this sequence is p-regular if and only if f(x) factors into a product of polynomials, one of which has no roots in Z_p, the other which factors into linear polynomials over Q. This answers a question of Allouche and Shallit.     

On the behaviour of p-adic L-functions

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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.     

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

On p-adic multiple zeta and log gamma functions

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We define p-adic multiple zeta and log gamma functions using multiple Volkenborn integrals, and develop some of their properties. Although our functions are close analogues of classical Barnes multiple zeta and log gamma functions and have many properties similar to them, we find that our p-adic analogues also satisfy reflection functional equations which have no analogues to the complex case. We conclude with a Laurent series expansion of the p-adic multiple log gamma function for (p-adically) large x which agrees exactly with Barnes?s asymptotic expansion for the (complex) multiple log gamma function, with the fortunate exception that the error term vanishes. Indeed, it was the possibility of such an expansion which served as the motivation for our functions, since we can use these expansions computationally to p-adically investigate conjectures of Gross, Kashio, and Yoshida over totally real number fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=I9Bv_CycEd8.     

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 a class of rational p-adic dynamical systems

In this paper we investigate the behavior of trajectories of one class of rational p-adic dynamical systems in complex p-adic field Cp. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.     

A p-adic algorithm for computing the inverse of integer matrices

A method for computing the inverse of an (n×n)$(n\times n)$ integer matrix A$A$ using p$p$-adic approximation is given. The method is similar to Dixon’s algorithm, but ours has a quadratic convergence rate. The complexity of this algorithm (without using FFT or fast matrix multiplication) is O(n⁴(logn)²)$O\left(n^4{(logn)}^2\right)$, the same as that of Dixon’s algorithm. However, experiments show that our method is faster. This is because our methods decrease the number of matrix multiplications but increase the digits of the components of the matrix, which suits modern CPUs with fast integer multiplication instructions.     

Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of . Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.     

On the behavior of some two-variable p-adic L-functions

We give some p-adic integral representations for the two-variable p-adic L-functions introduced recently by G. Fox. For powers of the Teichmüller character, we use the integral representation to extend the L-function to a larger domain, in which it is a meromorphic function in the first variable and an analytic element in the second. These integral representations imply systems of congruences for the generalized Bernoulli polynomials, improving previous results of Fox, Gunaratne, and the author; they also lead to generalizations of some formulas of Diamond and of Ferrero and Greenberg for p-adic L-functions in terms of the p-adic gamma and log gamma functions.     

On p-adic q-l-functions and sums of powers

In this paper, we give an explicit p-adic expansion of

     

Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along étale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.