Math 849, A Proof of Fermat's Last Theorem - Spring 2018

Nigel Boston

Contact Information

Telephone: 263-4753.

E-mail: boston@math.wisc.edu

Homepage

Course Syllabus

Office Hours: WF 11-12, or by appointment.

Sections

• Lecture: MWF 9:55-10:45, B235 VV.

Useful Web Materials

Videos from the 1995 Conference on Fermat's Last Theorem
A Taylor-Made Proof of Fermat's Last Theorem by N.Boston
Fermat's Last Theorem by H.Darmon, F.Diamond, and R.Taylor
Review of Marilyn vos Savant's Book by N.Boston and A.Granville
Wiles' Paper from the Annals of Math
Taylor-Wiles Paper from the Annals of Math
Serre's paper on Galois properties of torsion points of elliptic curves
Ribet's summary of and background to FLT proof
Ribet shows Shimura-Taniyama implies FLT
The Fontaine-Mazur conjectures
Diamond-Im on modular forms and modular curves
BBC Horizon programme on Wiles and Fermat's Last Theorem
TimeMagazine article

To Get a Grade

(1) Take notes for two classes and produce a careful clear TeX version
(2) Write a class project (10 to 20 pages) on one of the topics listed below or something of your own choice
(3) Give a 10 to 15 minute presentation to the class and to me on the same project

Suggested Topics

1. The abc conjecture.
2. Fermat-like equations.
3. Examples of Serre's conjecture and applications.
4. Families of elliptic curves with given mod p Galois representation.
5. More on Ribet's raising and lowering the level.
6. Group schemes and work of Khare et al.
7. More cases of the Fontaine-Mazur conjecture.
8. Kummer's work on regular primes.
9. Framed deformations.
10. Further properties of modular curves.
11. Profinite group theory.
12. Pseudo-representations.
13. The presentation of the absolute Galois group of the p-adics.
14. Explicit examples of modular curves.
15. Comparing the different constructions of universal deformation rings.
16. Fermat's last theorem over other rings (number rings, function fields, etc.).
17. Congruences of modular forms and Galois representations.
18. The modularity of mod 3 Galois representations.
19. Even Galois representations.
20. Brumer-Kramer proof of non-existence of elliptic curves of certain conductors.
21. Faltings-Serre method for finding all elliptic curves of small conductor.
22. Modular symbols and calculating with elliptic curves.

TeX Write-Ups of Sections

Current Version of Class Notes
Jan 29 & 31 Daniel Hast
Feb 2 & 5 Yuan Liu
Feb 7 & 9 Solly Parenti
Feb 12 & 14 Zinan Wang
Feb 16 & 19 Thomas Hameister
Feb 21 & 23 Soumya Sankar
Feb 26 & 28 Sun Woo Park
Mar 2 & 9 Xiaocheng Li
Mar 12 & 14 Brandon Boggess
Mar 16 & 19 Jiuya Wang
Mar 21 & 23 Niudun Wang
Apr 2 & 4 Ewan Dalby
Apr 6 & 9 Peter Ruan
Apr 11 Zinan Wang
Apr 13 Jiuya Wang
Apr 16 Xiaocheng Li
Apr 18 Thomas Hameister
Apr 20 Soumya Sankar
Apr 23 Sun Woo Park
Apr 25 Yuan Liu
Apr 27 Brandon Boggess
Apr 30 Solly Parenti
May 2 Niudun Wang

Schedule of Final Presentations

Friday, May 4 (B235 VV).
9:55 Zinan Wang, "Kummer's Lemma and Fermat's Last Theorem for Regular Primes".
10:10 Thomas Hameister, "The Fontaine-Mazur Conjecture".
10:25 Daniel Hast, "The Fontaine-Mazur Conjecture".

Monday, May 7 (B235 VV).
1:00 Soumya Sankar, "Congruences of Modular Forms".
1:15 Brandon Boggess, "Families of Elliptic Curves with Prescribed mod p Representation".
1:30 Solly Parenti, "Ribet's Converse to Herbrand-Ribet".
1:45 Yuan Liu, "The Faltings-Serre Method".
2:00 Sun Woo Park, "Recent Progress on Generalizations of Fermat's Last Theorem".

Tuesday, May 8 (B235 VV).
10:30 Niudun Wang, "The Modularity of mod 3 Representations".
10:45 Xiaocheng Li, "Certain Analytic Aspects of Modular Forms".
11:00 Jiuya Wang, "The Conductor of Elliptic Curves and the Ideal Class Group".
11:15 Ewan Dalby, "Framed Deformations".
11:30 Peter Ruan, "Profinite Groups: Virtues and Applications".

Some Final Projects