# Notes

## Expository talks I’ve given

The following are notes from expository talks that I’ve given. They likely contain errors: use with caution!

- Fermat’s Last Theorem, elliptic curves, and modular forms, a talk given to the Johns Hopkins University undergraduate math society.
- London number theory study group:
- $G$-bundles on the Fargues–Fontaine curve, from the study group on Caraiani–Scholze.
- The Décalage Functor, from the study group on Integral $p$-adic Hodge theory.
- Higher Coleman Theory, from the study group on Higher Coleman Theory.
- The spectral action, for study group on Xiao–Zhu.
- Stark’s Conjecture for Imaginary Quadratic Fields, from the study group on Stark’s Conjecture.
- Arithmetic level-raising in the even rank case, for the study group on Liu–Tian–Xiao–Zhang–Zhu.
- One Motives, for the London number theory study group on motives.

- Columbia student study groups:
- The Eichler–Shimura Isomorphism, from the study group on the cohomology of arithmetic groups.
- Hansen–Thorne, from the study group on the cohomology of arithmetic groups.
- Introduction to Deformation Theory, from the study group on derived deformation theory.

- Introduction to Schubert varieties and Deligne-Lusztig varieties, from a study group on Deligne–Lusztig theory (notes taken by Kamilla Rekvényi.
- Geometric equivalence of tori, from a study group on Deligne–Lusztig theory.
- Simplicial sets, for a study group on obstructions to rational points.

## Talks I’ve seen

The following are notes from talks that I’ve listened to. Any errors are mine: use with caution!

- Matthew Emerton’s Bonn Lectures, from the Hausdorff school on the Emerton–Gee stack.
- Emerton–Gee have since published this survey article, based on these notes.

- Breuil-Mézard and Automorphy, given by James Newton at the Hausdorff school on the Emerton–Gee stack.
- Cohomology of Arithmetic Groups: an Overview, by Sam Mundy for the Columbia student seminar on the cohomology of arithmetic groups.

## Derived structures in the Langlands Program

The following are notes from a study group on derived structures in the Langlands program from 2019, organized jointly with Pol van Hoften, Alice Pozzi, and Carl Wang–Erickson

## $p$-adic local Langlands for $\mathrm{GL}_2(\mathbb{Q}_p)$

Speaker | Date | Topic | Notes | References |
---|---|---|---|---|

Study group plan | See PDF | |||

Abi | May 1, 2020 | Mod $p$ and integral $p$-adic representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ | Notes | [Eme1], [Her1], [New] |

Ashwin | May 8, 2020 | Irreducible smooth admissible mod $p$ representations of $\mathrm{GL}_n(F)$ | Notes | [AHHV], [Bre], [GK], [Her2], [Her3] |

Andy | May 15, 2020 | Ext groups between irreducible representations | Typed notes and written notes | [Eme2], [Oll], [Paš1], [Vig] |

Ashvni | May 22, 2020 | Banach $L$-representations | Slides and handwritten proofs | [Paš2] |

Pol | May 28, 2020 | Locally finite abelian categories | [Gab] | |

Ashwin | June 5, 2020 | Paškūnas’s deformation theory | Notes | [Paš2] |

Sam | June 19, 2020 | Galois representations and $(\varphi,\Gamma)$-modules | Notes and Video (pw: 2g@0M%8#) | [Ber] |

Waqar | June 26, 2020 | Colmez’s Montréal Functor | [Col] | |

Ashwin | July 31, 2020 | Deformation Theory for Supersingular Representations | Notes | [Kis], [Paš1], [Paš2] |

### References

- [AHHV] Abe, Henniart, Herzig, Vignéras, A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups
- [Ber] Berger, Galois Representations and $(\varphi, \Gamma)$-modules (L14-L18)
- [Bre] Breuil, Sur quelques représentations modulaires et $p$-adiques de $\mathrm{GL}_2(\mathbb{Q}_p)$: I
- [Col] Colmez, Représentations de $\mathrm{GL}_2(\mathbb{Q}_p)$ et $(\varphi, \Gamma)$-modules
- [Eme1] Emerton, Ordinary Parts of Admissible Representations of $p$-adic Reductive Groups I: Definition and First Properties
- [Eme2] Emerton, Ordinary Parts of Admissible Representations of $p$-adic Reductive Groups II: Derived functors
- [Gab] Gabriel, Des catégories abéliennes
- [GK] Grosse-Klönne, On special representations of $p$-adic reductive groups
- [Her1] Herzig, $p$-modular and locally analytic representation theory of $p$-adic groups
- [Her2] Herzig, The classification of irreducible admissible mod $p$ representations of a $p$-adic $\mathrm{GL}_n$
- [Her3] Herzig, A Satake isomorphism in characteristic $p$
- [Kis] Kisin, Deformations of $G_{\mathbb{Q}_p}$ and $\mathrm{GL}_2(\mathbb{Q}_p)$ representations
- [New] Newton, LTCC Course: Representations of $p$-adic groups
- [Oll] Ollivier, Le foncteur des invariants sous l’action du pro-$p$-Iwahori de $\mathrm{GL}_2(F)$
- [Paš1] Paškūnas, Extensions for supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
- [Paš2] Paškūnas, The image of Colmez’s Montréal functor
- [Vig] Vignéras, Representations modulo $p$ of the $p$-adic group $\mathrm{GL}(2, F)$

## Notes from the Padova Summer School

In 2019 I attended the Padova school on Serre’s conjecture and the $p$-adic Langlands program. Here are some notes from the courses taught in the conference.

- Eugen Hellmann, $p$-adic Hodge theory and deformations of Galois representations
- Florian Herzig, $p$-modular and locally analytic representation theory of $p$-adic groups
- Benjamin Schraen, $p$-adic automorphic forms
- Sug Woo Shin, The local Langlands correspondence and local-global compatibility for $\mathrm{GL}_2$