Rough syllabus for Math 254.

MATH 254 will be run in the form of a seminar accessible to beginning and second year graduate students (and advanced undergraduates), but more particularly, its level will be adjusted to be useful to the students participating in the seminar.

PREREQUISITES: We will assume some familiarity with the definitions and a few elementary properties of cohomology of groups, and basic algebraic number theory.

FORMAT FOR THE SEMINAR: I may give a few preliminary lectures, but once it gets going, the students will each give two lectures (Mondays and Wednesdays at 11 a.m.). On Fridays there will be no PUBLIC course meeting, but on Fridays the next student in line who is to give his or her lectures will give those lectures privately to me (and to Tom Weston who will be helping out) and we will help shape the lecture the student is to present, suggest things, and help the student write up a handout that covers the material to be presented (by him or her) the following Monday and Wednesday. Towards the end of the course, we will switch gears: Tom and I will give survey-lectures on some of the advanced material.

THE EMPHASIS: will be on examples. I want to think of this seminar as a basic introduction to some of Kolyvagin’s techniques, rather than as an elaborate exposition of the technical aspects of its famous consequences.

WEBSITE: http://www.math.harvard.edu/weston/mazur.html (which gives a lectureby-lecture write-up of a more advanced course course I gave on this topic two years ago, these being Tom Weston’s notes covering each lecture, plus my auxiliary handouts).

TENTATIVE SCHEDULE:

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Part I. First introduction to the Selmer group.

1. General survey of what we expect to cover in the seminar.

2. Review of some algebraic number theory, intro to Galois modules, intro to cohomology of groups and cup product, intro to Galois cohomology, and specifically for finite fields and beginning discussion of local fields.

References: Lecture 1 on the website.

3. Local fields, a discussion of the structure of the Galois groups of the algebraic closure of local fields, and higher ramification groups.

References: Lectures 2-4 on the website.

4. Back to group cohomology, Galois modules, interpretation of H^{1}, intro to Tate local duality.

References: Lecture 5 on the website.

5. Tate local duality.

References: Lecture 6 on the website.

6. Galois cohomology of global fields, local “conditions”, intro to the Selmer group, examples.

References: Lecture 7 on the website.

7. The Selmer group, finite generation of the the Selmer group; intro to elliptic curve and abelian variety examples.

References: Lecture 8 on the website.

8. Abelian varieties and their cohomology, examples. Intro to their Galois representations on torsion points.

References: Lectures 9-12 on the website.

Short Bibliography for part I.

[1]. Serre, J.-P.: Corps Locaux, 2-nd edition Hermann (1968) [2]. Cassels, J.W.S., Frohlich, A.: Algebraic Number Theory Thompson (1967) [3]. Milne, J. S.: Arithmetic Duality Theorems, Perspectives in Math, Academic

Press (1986)

[4]. Shatz, S.: Profinite groups, arithmetic and geometry Princeton Annals of Math studies 67 Princeton Univ. press (1972)

[5]. Tate, J.: Duality theorems in Galois cohomology over number fields, Proc. Intern. Cong. Math. Stockholm (1962) pp. 234-241

————————————————– Part II. Galois representations and their Selmer groups ————————————————– Part III. Intro to the method of Kolyvagin; applications to the theory of

circular units and the ideal class group.

————————————————– Part IV. The method of Kolyvagin; Euler systems, applications to the

arithmetic of elliptic curves.

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