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John Coates: (1/4) Classical algebraic Iwasawa theory [AWS 2018]

slides for this lecture: http://swc-alpha.math.arizona.edu/vid... lecture notes: http://swc.math.arizona.edu/aws/2018/... CLASSICAL ALGEBRAIC IWASAWA THEORY. JOHN COATES If one wants to learn Iwasawa theory, the starting point has to be the basic material covered in §1 - §8 of Iwasawa’s paper [1]. The aim of these lectures will be to give a concise account of this work, concentrating on proving one of the main results of the paper, which is the so called weak Leopoldt conjecture for the cyclotomic Zp-extension of any number field, i.e. that the p-adic defect of Leopoldt is always bounded as one goes up the cyclotomic Zp-extension. The course will also include a brief introduction to the notion of the Iwasawa algebra of Zp, and the structure theory of modules for this Iwasawa algebra. The background required for the lectures will be basic algebraic number theory, including a knowledge of the main facts of abelian global class field theory. References [1] K. Iwasawa On Zl-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246- 326. [2] J. Coates Infinite descent on elliptic curves with complex multiplication, in Arithmetic and Geometry, Progress in Mathematics 35 (1983), Birkhauser, 107-137. http://swc.math.arizona.edu/index.html