Lecture Course III:

The theory of p-adic numbers, developed at the end of the 19th Century, is fundamental in modern number theory. It is based on an alternative definition of absolute value, which derives from a hierarchical (as opposed to sequential) arrangement of the integers. This device opens the study of integer congruences to the methods of analysis, and results in a number system with rather unusual properties. Recently, p-adic numbers have shown up in other areas of mathematics; this course is an introduction to the lively research area of p-adic dynamical systems.

SYLLABUS:

• Gauss, chaos, and digits of rationals.
• p-adic numbers.
• Linear p-adic dynamics; recurrence.
• Hensel's lifting; linearisation and p-adic Siegel discs.
• Application: planar rotations with round-off errors.

Lecture Notes

BACKGROUND READING:

• Some knowledge of elementary number theory will be assumed, see, e.g., the Intute resources.
• F. Q. Gouvea, p-adic Numbers: an Introduction, third edition, Springer, Berlin (2020).
• A. Baker, An Introduction to p-adic Numbers