Arizona Winter School 1999
Wetherell's Project Description


Start with Poonen's "Computational aspects of curves of genus at least 2." This is a great survey article with which to get everyone warmed up.

Once you are primed you can go to the source, Coleman's "Effective Chabauty". It's a fairly readable paper if you assume that the p-adic integration results work as quoted. Note: think about what a residue class is -- how does it relate to choosing a model for the curve?

The next two are optional, but you might want to read the article by Grant and Chabauty's original note at this point. Grant because it is easy and Chabauty for the historical interest. Also, Chabauty's note is a bit vague on the p-adic analytic foundations, which serves as a motivation to look at these foundations more closely. Did Chabauty actually prove anything? (Chabauty's note is a bit hard to find, email me if you want me to send you a copy.)

There are at least four approaches to the p-adic analysis.

  • Coleman, "Torsion points..."
  • Flynn, "A flexible method ..."
  • McCallum, "The arithmetic of Fermat curves"
  • Wetherell, Ph.D. Dissertation, Chapter 1 (see attached)

I will take my dissertation as the required text and the other three as optional, but I recommend you at least skim through all four of these papers. It is a good idea to get a sense of the various different approaches. We could have a good discussion of how these approaches relate to each other.

That maybe does it for an introduction. Then we divide into two groups.

Computational Coleman-Chabauty

This group will go back and read my dissertation and Flynn's "Flexible method" more thoroughly. Flynn, Poonen, Schaefer is also very good. Then we extract the necessary bits from Flynn and Wetherell (not yet in press, I will send it in email). I talk you through putting the pieces together for a specific, challenging problem.

(1-zeta_n)-descents on cyclic n-covers of P^1

The definitive paper here will be McCallum's Inventiones article. Other useful references are Poonen $amp; Schaefer and Schaefer's "Computing ... using functions on the curve". Bill will be putting together the basic description for this project.

General Citations

Poonen B 
Computational aspects of curves of genus at least 2 
LECTURE NOTES IN COMPUTER SCIENCE 1996, Vol 1122, pp 283-306 

Coleman RF 
Effective Chabauty 
DUKE MATHEMATICAL JOURNAL 1985, Vol 52, Iss 3, pp 765-770 

Grant D (optional)
A curve for which Coleman effective Chabauty bound is sharp 

Coleman RF (optional)
Torsion points on curves and p-adic abelian-integrals 
ANNALS OF MATHEMATICS 1985, Vol 121, Iss 1, pp 111-168 

Flynn EV (optional)
A flexible method for applying Chabauty's Theorem 
COMPOSITIO MATHEMATICA 1997, Vol 105, Iss 1, pp 79-94 

McCallum WG (recommended but optional)
The arithmetic of Fermat-curves 
MATHEMATISCHE ANNALEN 1992, Vol 294, Iss 3, pp 503-511 

Citations for Computational Coleman-Chabauty

Flynn EV    (again, required here)
A flexible method for applying Chabauty's Theorem 
COMPOSITIO MATHEMATICA 1997, Vol 105, Iss 1, pp 79-94 

Flynn EV; Poonen B; Schaefer EF 
Cycles of quadratic polynomials and rational points on a genus-2 curve 
DUKE MATHEMATICAL JOURNAL 1997, Vol 90, Iss 3, pp 435-463 

Citations for (1-zeta_n)-descents on cyclic n-covers of P^1

McCallum WG
On the Shafarevich-Tate group of the jacobian of a quotient of the Fermat
INVENTIONES MATHEMATICAE 1988, Vol 93, pp 637-666

Poonen B; Schaefer EF (optional)
Explicit descent for Jacobians of cyclic covers of the projective line 

Schaefer EF (optional)
Computing a Selmer group of a Jacobian using functions on the curve 
MATHEMATISCHE ANNALEN 1998, Vol 310, Iss 3, pp 447-471