Algebraic Number Theory, Part III Lent 2014

Lectures: 11:00-12:00, Tuesdays, Thursdays and Saturdays

Replacement lecture: Tuesday 11 March, 1:00-2:00, MR5 (I'll give people time to grab a snadwich before starting.)

First examples class: Wednesday 19 February, 2:00-3:30, MR3

Example sheet #1 (corrected) - Solutions

Second examples class: Wednesday 5 March, 2:00-3:30, MR3

Example sheet #2 - Solutions

Third examples class: (Thursday 24 April, 11:00-1:00)

Example sheet #3 - Solutions

Optional Easter term lectures on class field theory: handwritten notes

Course description (from the Part III handbook):

In recent years one of the most growing areas of research in number theory has been Arithmetic Algebraic Geometry, in which the techniques of algebraic number theory and abstract algebraic geometry are applied to solve a wide range of deep number-theoretic problems. These include the celebrated proof of Fermat's Last Theorem, the Birch-Swinnerton-Dyer conjectures, the Langlands Programme and the study of special values of L-functions. In this course we will study one half of the picture: Algebraic Number Theory. I will assume some familiarity with the basic ideas of number fields, although these will be reviewed briefly at the beginning of the course. (The relevant algebra will also be found in the Commutative Algebra course.)

Topics likely to be covered (not in order):

• Decomposition of primes in extensions, decomposition and inertia groups. Discriminant and different.
• Completion, adeles and ideles, the idele class group. Application to class group and units.
• Dedekind zeta function, analytic class number formula.
• Class field theory (statements and applications). L-functions.

Prerequisite Mathematics

A first course in number fields (or equivalent reading). Basic algebra up to and including Galois theory is essential.

Literature

• 1. J.W.S. Cassels and A. Frohlich, Algebraic Number Theory. London Mathematical Society 2010 (2nd ed.)
• 2. A. Frohlich, M.J. Taylor, Algebraic Number Theory. Cambridge, 1993.
• 3. J. Neukirch, Algebraic number theory. Springer, 1999.

a.j.scholl@dpmms.cam.ac.uk