Student Number Theory Seminar

Derived Galois Deformation Rings

Spring 2020

This semester's edition of the Student Number Theory Seminar will focus on studying derived Galois deformation rings,
partially inspired by the 2017 Heidelberg Study Group/Forschungsseminar and the 2019 London Number Theory Study Group.

The goal is to follow the paper of Galatius—Venkatesh, define a pro-simplicial ring whose zeroth homotopy group is Mazur's Galois deformation ring, and
define a graded action of derived deformation rings on the homology of arithmetic groups in the sense of a hypothetical derived Langlands program.

Along the way, the seminar will provide a gentle introduction to the necessary homotopy theory and review the Taylor—Wiles method and its extensions.

The seminar meets from 4:30pm — 6:00pm, Wednesdays in Mathematics Room 507.
The talks will be followed by dinner.

The in-person seminar is indefinitely postponed due to the COVID-19 pandemic.
All talks after March 11 will be held remotely.

(Three "pre-seminar" talks from last semester are also listed here)

Date	Speaker	Title	References
November 8	Michael Harris	Organizational meeting	—
November 21	Ashwin Iyengar	Galois deformation theory	[St]
December 12	Sam Mundy	The Taylor—Wiles method	[Di]

February 5	Roy Magen	Homotopy background I	[LP], [GJ], [L1]§1.2.3, [R]
February 12	Inbar Klang	Model categories	—
February 19	Michael Harris	Simplicial rings	[Ma], [W]§8, [F], [GV]§2, [Si]
February 26	Stanislav Atanasov	Calegari—Geraghty method: General framework	[CG]§1, §6-10, [Gee], [GV]§13, [T]
March 4	Robin Zhang	Calegari—Geraghty method: Modularity lifting	[CG]§1-3, §5, [Ger]
March 11	Yogesh More (Remote)	Derived deformation theory (Slides)	[GV]§4.3-4.6, Appendix A, [Da], [L2]§6.2
March 18	Spring Break (no talk)	Spring Break (no talk)	Spring Break
March 25	Cancelled (no talk)	Cancelled (no talk)	Cancelled
April 2	Hung Chiang (Remote)	Simplicial Galois representations (Notes)	[GV]§4.6-4.7, §8-9, [B]
April 8	Haodong Yao (Remote)	The derived deformation ring and patching (Slides)	[GV]§13-14, [CG]§2, §6, [I]
April 22	Stanislav Atanasov (Remote)	Derived Hecke algebra I (Slides)	[V]
May 6	Chi Dung Nguyen (Remote)	Derived Hecke algebra II	[V]
May 13	Michael Harris (Remote)	Derived Hecke action at p and the ordinary Hida tower	[KR]

References

Articles

[CG] F. Calegari, D. Geraghty Modularity lifting beyond the Taylor—Wiles method. Invent. Math. 211(1) (2018), 297—433. (old draft)
[KR] C. Khare, N. Ronchetti Derived Hecke action at p and the ordinary Hida tower. Preprint 2020.
[GV] S. Galatius and A. Venkatesh, Derived Galois deformation rings. Adv. Math. 327 (2018), 470—623.
[PV] K. Prasanna and A. Venkatesh, Automorphic cohomology, motivic cohomology, and the adjoint L-function. arXiv pre-print (2017).
[V] A. Venkatesh. Derived Hecke algebra and cohomology of arithmetic groups. Forum of Mathematics, Pi (2019).

London Number Theory Study Group (Winter 2019) notes

[B] R. Bellovin (notes by A. Iyengar), Derived Galois deformation rings. 2019.
[Da] D. Davis (notes by A. Iyengar), Derived deformation theory in general. 2019.
[Di] F. Diamond (notes by P. van Hoften), The Taylor—Wiles method. 2019.
[Gee] T. Gee, The Calegari—Geraghty method. 2019.
[I] A. Iyengar, The derived deformation ring and patching. 2019.
[LP] L. La Porta, Notes on simplicial objects and things. 2019.
[Si] R. Singer (notes by P. van Hoften), Simplicial commutative rings. 2019.
[St] M. Steinmetz (notes by A. Iyengar), Deformations of representations of Galois groups. 2019.

Textbooks, surveys, and other notes

[F] G. Friedman, An elementary illustrated introduction to simplicial sets. Rocky Mountain Journal of Mathematics, 42(2) (2012), 353—423.
[Ger] D. Geraghty, Chapter 4: Notes on modularity lifting in the ordinary case in p-Adic Aspects of Modular Forms. World Scientific 2016.
[GJ] P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory. Birkhäuser 2012.
[L1] J. Lurie, Higher Algebra. 2017.
[L2] J. Lurie, Derived Algebraic Geometry.
[Ma] J. P. May, Simplicial objects in algebraic topology. University of Chicago Press 1993.
[Mo] S. Morel, Note sur les foncteurs de déformations dérivés et la conjecture de de Jong. 2018.
[R] E. Riehl, A leisurely introduction to simplicial sets. 2011.
[T] J. Thorne, Beyond the Taylor—Wiles method. 2016.
[W] C. A. Weibel, Introduction to homological algebra. Cambridge University Press 1994.

Organizers: Stanislav Atanasov, Michael Harris, and Robin Zhang

Student Number Theory SeminarDerived Galois Deformation Rings

Spring 2020

References

Student Number Theory Seminar

Derived Galois Deformation Rings