189-726A: Topics in Number Theory: p-adic numbers, p-adic analysis, and applications

This is an introductory graduate course on p-adic numbers, given at McGill University in Fall 2001. We are not following any single textbook. Instead, we are merging portions of the treatments of several excellent texts:

• J.W.S. Cassels, "Local fields"
• F. Gouvea, "p-adic numbers"
• N. Koblitz, "p-adic numbers, p-adic analysis, and zeta functions"
• A. Robert, "A course in p-adic analysis"

Notes

Here are scans of the handwritten notes from which I have been lecturing (at the rate of approximately 1.5 pages per class).

Index of notes

• pages 1-5: introduction to ultrametric spaces
• pages 6-9: topological algebra
• pages 9-13: completions, valued fields, definition of p-adic numbers
• pages 14-19: basic properties of p-adics, inverse limits
• pages 19-23: Hensel's lemma and applications
• pages 23-25 : equivalence of absolute values, Ostrowski's theorem
• pages 26-27 : equivalence of norms on non-archimedian vector spaces
• pages 27-32: basic properties of finite extensions of Q_p
• pages 32-34: construction of C_p
• pages 35-40 : basics of p-adic analysis (interchanging double-sums, Strassman's theorem)
• pages 40-45: p-adic Weierstrass preparation theorem and applications
• pages 46-48: Newton polygons
• pages 48-50: interpolation, Mahler's theorem, binomial expansions
• pages 50-52: p-adic exponential and logarithm functions
• pages 52-56: p-adic Riemann zeta function
• pages 56-63: rationality of the zeta function of varities over finite fields

Homework