Math 252 (Euler Systems) Home Page

This is the home page for Harvard University Math 252, Spring 1998, taught by Professor Barry Mazur. The course meets Tuesday/Thursday 10-11:30 in Science Center Room 507.

There is also a seminar for the course, covering Galois cohomology, class formations, local and global Tate duality, and possibly some other topics, run (at the moment) by Tom Weston and Mark Dickinson. This meets Tuesday/Thursday 1-2 in Science Center Room 411.

These are my notes from the course, which are being posted as they are written. I make no guarantees that the notes are correct; any comments or corrections would be appreciated.

  • Lecture 1: Galois modules, quasi-finite fields, local fields.
  • Lecture 2: Discrete valuation rings, local fields, Galois theory of local fields
  • Lecture 3: Ramification groups, Witt vectors, projective limits of units of finite fields
  • Lecture 4: The absolute Galois group of a local field, Galois representations
  • Lecture 5: Group cohomology, Galois cohomology, Tate local duality
  • Lecture 6: Duality preliminaries, Tate local duality
  • Lecture 7: Finite/singular structures, generalized Selmer groups, Brauer groups
  • Lecture 8: Notations for generalized Selmer groups, a global pairing, finiteness of Selmer groups
  • Lecture 9: Abelian varieties, Selmer groups of abelian varieties
  • Lecture 10: Kummer theory, cohomology of abelian varieties
  • Lecture 11: L/K forms, cohomological interpretations
  • Lecture 12: Torsors for algebraic groups
  • Lecture 13: The Picard group of a curve, direct limits of Selmer groups, conditions on Galois representations
  • Lecture 14: Structures on Galois representations, depth and Kolyvagin-Flach systems
  • Lecture 15: The main theorem
  • Lecture 16: Other versions of the main theorem
  • Lecture 17: Modular curves, operators on modular curves
  • References
    These are the notes which Professor Mazur has been handing out.

  • Lectures 2-3
  • Lecture 4
  • Lecture 5
  • Lecture 6
  • Lectures 7-8
  • Lectures 9-10
  • Lectures 11-12
    These are some write-ups which are referred to occasionally in my course notes.

  • Cohomology of Finite and Profinite Groups
  • The Inflation-Restriction Sequence: An Introduction to Spectral Sequences
  • Local Fields: Algebraic and Topological Description
    Karl Rubin gave a course on Euler systems at Stanford University in the Fall of 1997 and is writing up the course into a book. Click here to get to the notes.
    Mail any comments to [email protected].