401-3119-70L  p-Adic Numbers

SemesterAutumn Semester 2020
LecturersP. Bengoechea Duro
Periodicitynon-recurring course
Language of instructionEnglish


AbstractThis course is an introduction to the p-adic numbers. We will see how the field of p-adic numbers Q_p is build. We will explore the (strange) topology and the arithmetic of Q_p, as well as some elementary analytic concepts such as functions, continuity, integrals, etc. We will explain an algebraic and an analytic reasons of interest for the existence of p-adic numbers.
Objective
Content- Absolute values on Q and Completions
- Topology and Arithmetic of Q_p, p-adic Integers
- Equations over p-adic numbers and Hensel's Lemma
- Local-global principle
- Hasse-Minkowski's Theorem on binary quadratic forms
- Elementary Analysis in Q_p
- the p-adic Riemann zeta function
Literature"p-adic Numbers. An Introduction", Fernando Q. Gouvea (Springer)
"p-adic Numbers, p-adic Analysis, and Zeta-Functions", Neal Koblitz (Springer)
"p-adic numbers and Diophantine equations", Yuri Bilu (online notes 2013)
Prerequisites / NoticeThe courses Topology, Measure and Integration, Algebra I/II are required prerequisites.