# Introduction to automorphic forms and Galois representations

This is a joint seminar of the Center for Advanced Studies at Skoltech and International laboratory for Mirror Symmetry and Automorphic Forms offered by myself in the 2019-2020 academic year. The lectures are scheduled on Wednesday 19:00, room 249, Skoltech (1 Nobel Street, red building; public transport).

## Description

Automorphic forms serve as a systematic common generalization of various diverse objects such as Hecke characters and modular (or Hilbert, Mass, Siegel, ...) forms. They appear in number theory, representation theory, topology, harmonic analysis and mathematical physics. Most prominently, automorphic forms conjecturally provide a representation-theoretic style description of Galois representations of local and global fields via Langlands correspondence. Although Langlands conjectures are only partially proved at the moment, there are plenty of unconditional results purely on the automorphic side, interesting in their own right. In this seminar we will study these results and some known cases of Langlands correspondence.

During the first part of the course we will concentrate on the basic structure theory of complex automorphic forms and representations. In the second part we will move on to a more advanced topics and arithmetic applications, such as the local Langlands correspondence for $\mathrm{GL}_n$ and $p$-adic automorphic forms (if we have enough time).

### Plan

1. Reminder of the class field theory.

#### Complex automorphic representations

2. Double adelic quotients, approximation theorems, Hecke algebra, Satake isomorphism, Bernstein decomposition.
3. Admissible representations, Whittaker models, multiplicity one theorem, Flath's theorem.
4. Eisenstein series.
5. Arthur-Selberg trace formula.
6. Jacquet-Langlands correspondence.
7. #### Arithmetic applications

8. Shimura varieties.
9. $p$-divisible groups and their deformations, Lubin-Tate spaces.
10. Etale cohomology of rigid analytic spaces.
11. Langlands-Kottwitz method.
12. Local Langlands for $\mathrm{GL}_n$.

## Talks

TalkSpeakerDate, place
Overview of the courseArtem PrikhodkoSeptember, 18
Local class field theoryGleb TerentiukSeptember, 25
Global class field theory IDmitrii KrekovOctober, 2
Global class field theory IIDmitrii KrekovOctober, 5
Structure theory of reductive groups, approximation theoremsPavel PopovOctober, 9
Smooth and admissible representations, Hecke algebraArtem PrikhodkoOctober, 16
Skipped due to my graduationOctober, 23
Cuspidal representations, Bernstein decompositionArtem PrikhodkoOctober, 30
Geometric Satake equivalenceIvan PerunovNovember, 6
Aside: higher traces and fixed point theoremsArtem PrikhodkoNovember, 13
Witt vector affine GrassmannianIvan PerunovNovember, 20
Satake equivalence in mixed chacteristicIvan PerunovNovember, 27
Aside: non-commutative Hodge-to-de Rham degenerationAndrei KonovalovDecember, 4
Traces and shtukas IRoman KositsinDecember, 11
Traces and shtukas IIRoman KositsinDecember, 18
New year's talk: Frobenius action on commutative algebras and refined Mandell's theoremNikolai KonovalovDecember, 25
Matsushima formula IVasya RogovJanuary, 22
Matsushima formula IIVasya RogovJanuary, 29
Flath theoremAlexander NovikovFebruary, 5
Whittaker models and Multiplicity One theoremArtem PrikhodkoFebruary, 12
Automorphic $L$-functionsArtem PrikhodkoFebruary, 19
SkippedFebruary, 26
Etale cohomology of rigid-analytic varieties IDmitrii KrekovMarch, 4
Etale cohomology of rigid-analytic varieties IIDmitrii KrekovMarch, 11
SkippedMarch, 18
Jacquet-Langlands correspondence IArtem PrikhodkoMarch, 25
Jacquet-Langlands correspondence IIArtem PrikhodkoApril, 1
Cartier-Dieudonne theory of $\pi$-divisible groupsArtem PrikhodkoApril, 15
Aside: Tannakian reconstruction in derived algebraic geometryArtem PrikhodkoMay, 6
Dieudonne-Manin classificationAlexander NovikovMay, 13
Deformation spaces of $\pi$-divisible groupsArtem PrikhodkoMay, 20
Jacquet-Langlands III: two towers IArtem PrikhodkoMay, 28
Jacquet-Langlands III: two towers IIArtem PrikhodkoJune, 4
Aside: Drinfeld's approach to prismsKirill MagidsonJune, 18

## References

### Basics

1. Sam Raskin, Lecture notes on cohomological class field theory (notes by Oron Propp).
2. Ivan Losev, Vasia Krylov, Lecture notes on reductive groups and category O.
3. ### Complex automorphic representations

4. Stephen Gelbart, Automorphic Forms on Adele Groups.
5. Jayce R. Getz, An introduction to automorphic representations.
6. James W. Cogdell, Lectures on $L$-functions, Converse Theorems, and Functoriality for $\mathrm{GL}_n$.
7. Joseph Bernstein, Representations of p-adic groups.
8. Michael Harris, An Introduction to the Stable Trace Formula.
9. Alexandu Ioan Badulescu, The Jacquet-Langlands correspondence.
10. ### Satake equivalence

11. Xinwen Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence.
12. Xinwen Zhu, Affine Grassmannians and the geometric Satake in mixed characteristic.
13. Bhargav Bhatt, Peter Scholze, Projectivity of the Witt vector affine Grassmannian.
14. ### Local arithmetic Langlands

15. Pierre Deligne, Travaux de Shimura.
16. Alain Genestier, Bao Chau Ngo, Lectures on Shimura varieties.
17. Torsten Wedhorn, The Local Langlands Correspondence for GL(n) over p-adic Fields.
18. Robert E. Kottwitz, Points on some Shimura varieties over finite fields.
19. Michael Harris, Richard Taylor, The geometry and cohomology of some simple Shimura varieties.
20. Peter Scholze, The Langlands-Kottwitz approach for the modular curve and The Langlands-Kottwitz approach for some simple shimura varieties.
21. Peter Scholze, The local langlands correspondence for $\mathrm{GL}_n$ over $p$-adic fields.
22. ### $p$-adic automorphic forms

23. Armand Borel, Nolan R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups.
24. Matthew Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms.
25. Matthew Emerton, A new proof of a theorem of Hida.
26. Eric Urban, Eigenvarieties for reductive groups.
27. Peter Scholze, On torsion in the cohomology of locally symmetric varieties.
28. A. Caraiani, D. R. Gulotta, C.-Y. Hsu, C. Johansson, L Mocz, E. Reinecke, S.-C. Shih, Shimura varieties at level $\Gamma_1(p^\infty)$ and Galois representations.