On continued fractions over the field of p-adic numbers

Title

On continued fractions over the field of p-adic numbers

Speaker

Giuliano Romeo (Politecnico di Torino)

Time and Location

Thursday, April 3, 18:30, 2025, E2-102 SOUTH CAMPUS

Abstract

Continued fractions over the field of real numbers are a classical and powerful tool in Diophantine approximation. Therefore, it has been natural to introduce them also in the field of p-adic numbers, where they behave quite differently from the classical case. In fact, unlike continued fractions over the real numbers, there is not a unique standard algorithm, due to the fact that there is not a unique canonical way to define the integral part of a p-adic number. One of the main open problems in this framework is the research of an algorithm that produces an ultimately periodic continued fraction for every p-adic quadratic irrational, i.e. the analogue of the famous Lagrange’s Theorem.

In this talk we provide an overview on the theory of p-adic continued fractions and we discuss the most important algorithms that have been defined throughout the years. We focus in particular on the properties of convergence, finiteness and periodicity, together with the most recent developments towards the proof a p-adic version of Lagrange’s Theorem. Moreover, we highlight a strong connection between the periodicity of some p-adic continued fractions and the convergence to a quadratic irrational in the field of real numbers.

Free pizza will be provided.