$$ \DeclareMathAlphabet{\matheuler}{U}{eus}{m}{n} \newcommand\F{\mathbb F} \newcommand\P{\mathbb P} \newcommand\A{\mathbb A} \newcommand\T{\mathbb T} \newcommand\msO{\mathscr O} \newcommand\msL{\mathscr L} \newcommand\ms{\mathscr} \newcommand\GL{\opname{GL}} \newcommand\eps{\varepsilon} \newcommand\opname{\operatorname} \newcommand\Proj{\opname{Proj}} \newcommand\Pic{\opname{Pic}} \newcommand\hom{\opname H} \newcommand\Char{\opname{char}} \newcommand\Hom{\opname{Hom}} \newcommand\Sel{\opname{Sel}} \newcommand\inv[2][1]{#2^{-#1}} \newcommand\pinv[2][1]{\inv[#1]{\p{#2}}} \newcommand\p[1]{\left(#1\right)} $$

Modularity/Fermat Seminar

A learning seminar on Fermat's Last Theorem, modularity of (semistable) elliptic curves, and whatever other related topics come up along the way.

This is a learning seminar on the proof of Fermat’s Last Theorem and on modularity of elliptic curves. The main goal is to hopefully address the question, “How on Earth does one prove that some representation is modular?” Our main source will be the big book “Modular Forms and Fermat’s Last Theorem’’ edited by Cornell, Silverman, and Stevens, but we will use other sources as well (see below).

Schedule

Each meeting will have a specified topic and will start with someone giving a short (say, 20–40 minutes) talk/summary.\(^\dagger\) Afterwards, we will proceed with an informal discussion to try to work out each others confusions. Speakers are encouraged to write notes, and participants are encoraged to look at the reference(s) ahead of each meeting.

Meeting time: Fridays 5:30 - 7:00PM in 2-361
(Because of scheduling difficulties, this is subject to change in any given week)

My goal is to keep the schedule flexible (only the next 2-3 talks fixed at any time), so we can switch up topics as audience interests dictate.

  • If you want to talk about anything in particular, do let me know. If you see a topic below without a speaker, then this is a suggestion from me; ask me for more info if you might want to give that talk.
  • If you see a ??? in the Date, then the placement of this topic is still extra flexible, so it does not have to come up in the indicated position.
  • If you see a ? after the Date, then that means the indicated date is probably when the talk will happen, but possibly this could get changed (e.g. if the speaker can’t make the usual time).
Date Topic Speaker Notes References
Sep 1\(^*\) An Overview of the proof of Fermat Niven Achenjang here [FLT, Chap 1]
Sep 15 Galois representations associated to modular forms Vijay Srinivasan here [BCdS+, Chaps 4,5,10]
Sep 22 Langlands-Tunnell, part I Kenta Suzuki here [FLT, Chap 6, lecture 2],
[BH, Section 33]
Sep 29 Langlands-Tunnell, part II Daniel Hu here [FLT, Chap 6, lecture 3]
Oct 6 Serre’s Conjecture Dylan Pentland here [FLT, Chap 7]
Oct 13 Galois Deformation Rings &
Stating \(R=\T\) theorems		 Niven Achenjang here [Gee, Sect 3 up to 3.23],
[DDT, Sect 2.6,2.7,3.3]
Oct 20 \(R=\mathbb T\) for Galois characters Grant Barkley here [Cal, Sect 2]
Oct 27 Tangent spaces of deformation problems Daniel Hu here [Gee, Sect 3.23+]
Nov 3 Taylor-Wiles Deformations Kenta Suzuki here [Gee, Sect 3.32+]
Nov 10 Local deformation rings at \(p\) N/A N/A [Gee, 3.27-3.31],
[Stan, Lect 21]
Nov 17 Automorphic forms for quaternion algebras I Niven Achenjang here [Gee, Sect 4]
???\(^\ddagger\) Automorphic forms for quaternion algebras II N/A N/A [Gee, Sect 4]
Dec 1 Proof of [Gee, Theorem 5.2] I John Sim   [Gee, Sect 5]
Dec 8 Proof of [Gee, Theorem 5.2] II Eunsu Hur   [Gee, Sect 5]


\(^*\) at 4:30PM in 2-361
\(^\dagger\) If you want to speak for longer, that’s fine too; just let me know ahead of time. The main motivations for the default format are to leave room for plenty of audience participation and to not stress out the speaker if they end up having a lot going on around the time their talk is coming up.
\(^\ddagger\) I imagine one talk won’t be enough to adequately cover what all we need from section 4, but also I don’t want to have a talk during Thanksgiving week. So maybe what I’ll do is just ask everyone to read section 4 on their own.

Sources

[FLT] Too many actual authors to list, “Modular Forms and Fermat’s Last Theorem”
[DDT] H. Darmon, F. Diamond, and R. Taylor, “Fermat’s Last Theorem”
[Gee] T. Gee, “Modularity Lifting Theorems” \(^*\)
[BCdS+] D. Bump, J. W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, S. S. Kudla, “An Introduction to the Langlands Program”
[BH] C. J. Bushnell, G. Henniart, “The local Langlands conjecture for \(\GL(2)\)”
[Kis] M. Kisin, “Moduli of finite flat group schemes, and modularity”
[CDT] B. Conrad, F. Diamond, R. Taylor, “Modularity of certain potentially Barsotti-Tate Galois representations”
[Stan] Too many actual authors to list, “Stanford modularity lifting seminar”
[Hida] Hida, “Taylor-Wiles Patching Lemmas a la Kisin”
[BBD+] L. Berger, G. Böckle, L. Dembélé, M. Dimitrov, T. Dokchitser, J. Voight, “Elliptic Curves, Hilbert Modular Forms and Galois Deformations”
[Cal] F. Calegari, “Motives and L-functions”

\(^*\)There are videos of Gee’s lectures on the AWS website.

Additional Helpful Notes

I haven’t read most of these, but they could be useful if you want further sources.

Recordings of the Conference on which [FLT] is based
Lectures on Modular Forms and Hecke Operator
Lectures on Serre’s conjectures
UChicago Galois def + modularity lifting seminar
Notes associating Artin representations to weight one modular forms
Elliptic Curves and the Weil-Deligne Group
A tameness criterion for Galois reps associated to modular forms (mod \(p\))
Notes from an MIT course on modularity lifting/Gee’s notes.\(^*\)
Notes from a Stanford course on automorphy lifting
Lectures on Deformations of Galois Representations

(I’ll keep adding more as I discover more)

\(^*\) The website for this course has problem sets (though no lecture notes of its own)