MA 354: Topics in Number Theory

Credits: 3:0

Pre-requisites :

1. a good background in commutative algebra (inverse limits, $I$-adic completion, Galois theory, possibly some familiarity with Dedekind domains),
2. some previous knowledge of algebraic number theory should be useful.

The goal is to give an introduction to adeles and some of their uses in modern number theory, discussing also some topics which are not too common in textbooks.

Topics to be covered: absolute values and Ostrowski’s Theorem; classification of locally compact fields; definition of adeles and some applications (finiteness of class number and of the generators of the group of S-units; structure of modules over Dedekind domains; applications to the geometry of curves); an introduction to the Strong Approximation Theorem; adelic points of varieties and schemes; possibly other topics (depending on time left and interests of the audience; for example Tate’s thesis, quasi-characters of the idele class group and p-adic L-functions).

Suggested books and references:

1. J. W. S. Cassels and A. Fröhlich (editors), Algebraic Number Theory, Papers from the conference held at the University of Sussex, Brighton, September 1–17, 1965.
2. A. Weil, Basic Number Theory, Classics in Mathematics, Springer 1974.
3. B. Conrad, Weil and Grothendieck approaches to adelic points, Enseign. Math. (2) 58 (2012), no. 1-2, 61–97.

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