Related papers: On approximation of p-adic numbers by p-adic algeb…

The Lie algebras over the algebra of dual numbers are introduced and investigated.

Let A be an abelian fourfold. We prove the Standard Conjecture of Hodge type for A. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

We give a dynamical construction of an infinite sequence of distinct totally $p$-adic algebraic numbers whose Weil heights tend to the limit $\frac{\log p}{p-1}$, thus giving a new proof of a result of Bombieri-Zannier. The proof is…

We derive "numerical" criteria for the existence of embeddings of representations of finite dimensional algebras.

We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness…

Motivated by the study of algebraic classes in mixed characteristic we define a countable subalgebra of $\bar{\mathbb{Q}}_p$ which we call the algebra of Andr\'e's $p$-adic periods. We construct a tannakian framework to study these periods.…

$p$-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review $p$-adic Hodge Theory for algebraic varieties, present current developments in $p$-adic Hodge Theory for analytic…

We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric…

We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic…

We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.

We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial…

By using p-adic q-integrals, we study the q-Bernoulli numbers and polynomials of higher order.

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…

This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.