In the fall of 2019, I organized a seminar on the cohomology of arithmetic groups. Topics included: the Eichler-Shimura isomorphism, Matsushima's formula, Eisenstein classes, coherent cohomology, and Venkatesh's conjectures.

Ashwin Iyengar took some notes for the seminar and they are available at his website. Links to these notes can also be found for some of the lectures in the schedule below.

**Time and place:** Tuesdays 4:30-6:00 in Mathematics 507.

[Bel] Bellaiche, J. Eigenvarieties, Families of Galois Representations, p-adic L-functions.

[BW] Borel, A. and Wallach, N. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Second Edition. AMS, 2000.

[Franc] Franc, C. Notes on Hida Theory.

[Fra] Franke, J. Harmonic Analysis in Weighted L_2 Spaces. Annales Scientifiques de l'E.N.S., 1998.

[HT] Hansen, D. and Thorne, J. On the GL_n Eigenvariety and a Conjecture of Venkatesh. Selecta Math. 2017.

[Har1] Harris, M. Automorphic Forms and the Cohomology of Vector Bundles on Shimura Varieties. In the Ann Arbor Proceedings, Volume II, pages 41-91. 1989.

[Har2] Harris, M. Automorphic Forms of d-bar-Cohomology Type as Coherent Cohomology Classes. J. Diff. Geom., 1990.

[HV] Harris, M. and Venkatesh, A. Derived Hecke Algebra for Weight One Forms. Experimental Mathematics, 2018.

[Jac] Jacinto, J. Cohomology of Arithmetic Groups. Notes on Matsushima's formula from Ashwin Iyengar's website.

[Lan] Lan, K.-W. An Example-Based Introduction to Shimura Varieties.

[LS] Li, J.-S. and Schwermer, J. On the Eisenstein Cohomology of Arithmetic Groups. Duke, 2004.

[PV] Prasanna, K. and Venkatesh, A. Automorphic Cohomology, Motivic Cohomology, and the Adjoint L-function. From arXiv, last revised 2017.

[Ven1] Venkatesh, A. Cohomology of Arithmetic Groups and Periods of Automorphic Forms. Notes based on his Takagi Lectures in 2014.

[Ven2] Venkatesh, A. Derived Hecke Algebra and Cohomology of Arithmetic Groups. From arXiv, last revised 2019.

[VZ] Vogan, D. and Zuckerman, G. Unitary Representations with Non-zero Cohomology. Compositio Mathematica, 1984.

[Wil] Williams, C. Automorphic Forms in Cohomology via Examples. Notes from Ashwin Iyengar's website.

[Yo] Yoshida, T. Notes on Matsushima's Formula.

[You] Youcis, A. Shimura Varieties: Motivation (blog post).

Date | Title | Speaker | References |
---|---|---|---|

September 17 | Overview | Sam Mundy | |

September 24 | Eichler-Shimura Isomorphism | Ashwin Iyengar | [Bel], [Wil] |

October 1 | Eichler-Shimura Isomorphism, Continued | Ashwin Iyengar | [Bel], [Wil] |

Local Systems | Cailan Li | [BW] Ch. 7 | |

October 8 | Local Systems, Continued | Cailan Li | [BW] Ch. 7 |

Adelization, Hecke operators, (g,K)-cohomology | Yu-Sheng Lee | [BW] | |

October 15 | Cancelled | ||

October 22 | Adelization, Hecke operators, (g,K)-cohomology | Yu-Sheng Lee | [BW] |

Matsushima's Formula | Stan Atanasov | [Yo], [Jac] | |

October 29 | Matsushima's Formula, Continued | Stan Atanasov | [Yo], [Jac] |

Cohomology and Automorphic Representations | Sam Mundy | ||

November 5 | No Talk: Election Day | ||

November 12 | Introduction to the Prasanna-Venkatesh Conjecture: Motivation | Robin Zhang | [PV], [Ven1], [Ven2] |

November 19 | The Crystalline Realization of Venkatesh's Conjecture | Ashwin Iyengar | [Franc], [HT] |

November 26 | Coherent Cohomology of Shimura Varieties: Setup | Stan Atanasov | [Lan], [You] |

December 3 | Coherent Cohomology of Shimura Varieties | Stan Atanasov | [Har1], [Har2] |

December 10 | Numerical Approach to a Conjecture of Harris and Venkatesh | David Marcil | [HV], [Ven2] |

December 17 | The Eisenstein Cohomology of Arithmetic Groups | Sam Mundy | [Fra], [LS] |