p-adic Hodge theory (Summer 2020)

Instructor: Daniel Gulotta
Location and time: Tuesday and Thursday 3:30pm-4:30pm, online. The password is the number of roots of unity in Q₃₄₇₅₉.
Prerequisites: I will assume some familiarity with p-adic numbers and with algebraic geometry.

Notes will be posted here. I do not plan to give homework, but you may find it helpful to do some of the exercises in references [BC] and [D] listed below.

References

[B] L. Berger, An introduction to the theory of p-adic representations
[BC] O. Brinon and B. Conrad, CMI summer school notes on p-adic Hodge theory
[BMS] B. Bhat, M. Morrow, P. Scholze, Integral p-adic Hodge theory
[BS] B. Bhatt and P. Scholze, Prisms and prismatic cohomology
[C] P. Colmez, Une construction de B_dR⁺
[D] H. Diao, Period rings and period sheaves
[F] J.-M. Fontaine, Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux
[FO] J.-M. Fontaine and L. Ouyang, Theory of p-adic Galois representations
[K] K. Kedlaya, New methods for (φ,Γ)-modules
[LB] A.-C. Le Bras, Overconvergent de Rham cohomology over the Fargues-Fontaine curve
[SW] P. Scholze and J. Weinstein, Berkeley lectures on p-adic geometry
[T] J. Tate, p-divisible groups
[W] J. Weinstein, Arizona Winter School 2017: Adic spaces

Schedule of lectures

Date	Topic	References
28 Apr	Cohomology of algebraic varieties, Galois representations
30 Apr	φ-modules	[BC] 3, [FO] 2.2-2.3
5 May	(φ,Γ)-modules, Perfectoid fields, tilting, Witt vectors	[W] 2.0-2.2, [K] 1.1-1.3, 2.3, [B] III, [FO] 4.4
7 May	Untilting, Ax-Sen-Tate theorem	[W] 2.3, [K] 1.4-1.6, [T] 3
12 May	Ax-Sen-Tate theorem, B_dR	[T] 3, [BC] 4.4, [FO] 5.2
14 May	B_dR and differentials	[C] 3
19 May	B_dR and differentials, Hodge-Tate decomposition for abelian varieties, p-divisible groups	[C] 3, [F], [T], [BC] 7.1-7.2
21 May	p-divisible groups	[T], [BC] 7.1-7.2, [SW] 12.1
26 May	Classifications of p-divisible groups, Isocrystals	[BC] 7, [SW] 12.1
28 May	Filtered ϕ-modules	[BC] 7-8
2 Jun	B_cris, B_st	[BC] 9, [FO] 6, [B] II.3
4 Jun	B_st, adic spaces	[BC] 8-9, [FO] 6, [SW] 2-3, [W] 1
9 Jun	Adic spaces, Fargues-Fontaine curve	[SW] 2-3, 11.2-11.3, 13.5, [W] 1, 2.5, 4.2
11 Jun	Fargues-Fontaine curve	[SW] 11.2-11.3, 13.5, [W] 2.5, 4.2
16 Jun	Fargues-Fontaine curve, Prismatic cohomology	[SW] 11.2-11.3, 13.5, [W] 2.5, 4.2, [BMS], [BS]
18 Jun	Prismatic cohomology, de Rham cohomology over the Fargues-Fontaine curve	[BMS], [BS], [LB]