Math 847, A Proof of Fermat's Last Theorem - Spring 2013

Nigel Boston

Contact Information

Telephone: 263-4753.

E-mail: boston@math.wisc.edu

Homepage

Office Hours: Tuesdays 10:45-12:15 (in 303 VV), Wednesdays 1:30-3:00 (in 3619 EH).

Sections

• Lecture: TR 9:30-10:45, B129 Van Vleck.

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

To Get a Grade

(1) Take notes one class and produce a 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

Jan 22 by Rachel Davis Tex version
Jan 24 by Sara Jensen Tex version
Jan 29 by Carolyn Abbott Tex version
Feb 5 by Vladimir Sotirov Tex version
Feb 7 by Eric Ramos Tex version
Feb 12 by Yihe Dong Tex version
Feb 14 by Vlad Matei Tex version
Feb 19 by Cathryn Holm Tex version
Feb 21 by Elaine Brow Tex version
Feb 26 by Megan Maguire Tex version
Feb 28 by Ed Dewey Tex version
Mar 5 by Yueke Hu Tex version
Mar 7 by Rohit Nagpal Tex version
Mar 12 by Nathan Clement Tex version
Mar 14 by Serkan Sakar Tex version
Mar 19 by Jon Lima Tex version
Mar 21 by Jeff Poskin Tex version
Apr 2 by Peng Yu Tex version
Apr 4 by Ahmet Kabakulak Tex version
Apr 9 by Lalit Jain Tex version

Schedule of Final Presentations

Tuesday, May 7 (B129 VV).
9:30 Rachel Davis, "Comparing Constructions (of Universal Deformation Rings)"
9:45 Cathryn Holm, "Not as Easy as ABC!"
10:00 Megan Maguire, "Cohomology of Profinite Groups"
10:15 Carolyn Abbott, "Kummer's Proof of Fermat's Last Theorem for Regular Primes"
10:30 Eric Ramos, "Modularity Lifting Theorems and Serre's Conjecture"

Thursday, May 9 (B129 VV).
9:30 Sara Jensen, "Finite Index Subgroups of Profinite Groups"
9:45 Peng Yu, "Genus and Defining Equations of Modular Curves"
10:00 Serkan Sakar, "Galois Cohomology and its Applications"
10:15 Ahmet Kabakulak, "Fermat's Last Theorem over Other Rings"
10:30 Jonathan Lima, "Modular Forms and Hecke Operators"

Tuesday, May 14 (B223 VV).
2:30 Nathan Clement, "Neron Models: They're Flat out Amazing"
2:45 Yihe Dong, "A Glimpse into Hodge Structures"
3:00 Vlad Matei, "The Faltings-Serre Method"
3:15 Lalit Jain, "Families of Curves with Constant mod N Representations"
3:30 Elaine Brow, "Keeping the Spirit Alive: Fermat-Like Equations"

Thursday, May 16 (B223 VV).
2:30 Vladimir Sotirov, "Galois Groups of p-closed Extensions of p-adic Fields"
2:45 Yueke Hu, "Ribet's Level-Lowering"
3:00 Rohit Nagpal, "Langlands-Tunnell Theorem"
3:15 Jeffrey Poskin,
3:30 Edward Dewey, "Following Your Nose to Schlessinger's Theorem"

Some Final Projects

Rachel Davis, Different constructions of universal deformation rings
Edward Dewey, Following your nose to Schlessinger's criteria for representability
Yueke Hu, On Ribet's level lowering theorem
Sara Jensen, Subgroups of finite index in profinite groups
Megan Maguire, Cohomology of profinite groups
Vlad Matei, Faltings-Serre method