David R. Hayes

Visting Scholar, Fall Term

Office: #101, 65 Mt. Auburn

email: dhayes@math.umass.edu  
Lectures on Stark's Conjectures -- Final lecture on Dec. 2 at 2:30pm.

I will offer up to eight lectures/discussions on Stark's conjectures, depending on student interest. The class will meet TTh, 2:30-3:45pm, in SC 411, beginning October 19. Parts of these lectures will eventually be incorporated into a monograph I am writing on Stark's conjectures for the AMS/IP series Studies in Advanced Mathematics. I will hand out lecture notes from time to time.

What are Stark's conjectures?

The conjectures provide one solution to Hilbert's Twelth Problem, which challenged mathematicians of the 20th century to show how class fields may be constructed over any number field by the methods of complex analysis. For example, values of the exponential function at rational multiples of 2Pi i generate abelian extensions of the rational numbers.

The conjectures are precise enough for computation. In fact, the first order zero conjectures are used in the new version of Pari to compute Hilbert Class Fields of totally real number fields.

Stark introduced his conjectures in a series of papers in the 1970's. Except when the base field is the rational numbers or an imaginary quadratic field, they are still unproved in number fields. More progress has been made in function fields.

Proposed Lectures/Discussions

  1. Stark's conjectures as a solution to Hilbert's Twelfth Problem: An overview.
  2. Statement of the first order zero conjecture.
  3. The first order zero conjectures over the rationals as base field.
  4. The first order zero conjectures in function fields.
  5. The conjecture of Brumer-Stark and the refined p-adic conjecture.
  6. Algorithms for computing Brumer elements over real quadratic fields, I.
  7. Algorithms for computing Brumer elements over real quadratic fields, II.
  8. The higher order zero conjectures of Tate and Gross.
  9. The higher order zero conjectures of Rubin and Popescu.

Lectures Notes in PostScript Format

  1. Lecture 1
  2. Lecture 2 (updated 10/25)
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 9

Supplimentary Notes in PostScript Format

  1. Notes on Class Field Theory by M. Baker

 

Last updated: November 13, 1999
David R. Hayes
email: dhayes@math.umass.edu