MATH 6370 p-adic Hodge Theory (Spring 2025)

Instructor: Daniel Gulotta, dgulotta@math.utah.edu
Location and time: TR 2-3:20 in WEB 2470
Credits: 3
Homework: Homework exercises will be listed below. These are optional for PhD students.

References

We will cover chapters 1-9 of [BC] and chapters 1-8 of [FO] (these two references cover roughly the same material), chapters 2-13 of [SW], and some additional topics. Depending on interest, these may include [Sch], [BS], [BL].

Notes will be posted here after each class.

Main references

[BC] O. Brinon and B. Conrad, CMI summer school notes on p-adic Hodge theory
[BL] B. Bhatt and J. Lurie, Absolute prismatic cohomology
[BS1] B. Bhatt and P. Scholze, Prisms and prismatic cohomology
[FO] J.-M. Fontaine and Y. Ouyang, Theory of p-adic Galois representations
[Sch] P. Scholze, p-adic Hodge theory for rigid analytic varieties
[SW] P. Scholze and J. Weinstein, Berkeley lectures on p-adic geometry

Other references

[Bel] J. Bellaiche, An introduction to the conjecture of Bloch and Kato
[Ber] L. Berger, An introduction to the theory of p-adic representations
[Bos] S. Bosch, Lectures on formal and rigid geometry
[C] P. Colmez, Une construction de B_dR⁺
[F] J.-M. Fontaine, Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux
[H] R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces
[K1] K. Kedlaya, p-adic cohomology: from theory to practice
[K2] K. Kedlaya, New methods for (φ,Γ)-modules
[Mil] J. Milne, Field theory and Galois theory
[R] K. Rubin, Euler systems and Kolyvagin systems
[Ser] J.-P. Serre, Local fields
[Sil] J. Silverman, The arithmetic of elliptic curves
[Sta] Stacks Project
[T] J. Tate, p-divisible groups
[W] J. Weinstein, Arizona Winter School 2017: Adic spaces

Here is a link to a course on p-adic Hodge theory that I taught in 2020.

If you are looking for exercises, there are some in [BC] and in these two references:

[D] H. Diao, Period rings and period sheaves
[Mie] Y. Mieda, Adic spaces and perfectoid spaces

Tentative Schedule

Date	Topic	References
1/7	Introduction
1/9	Infinite Galois theory	[Mil] 3, 7
1/14	Elliptic curves	[Sil] 3
1/16	Formal groups	[Sil] 4
1/21	Elliptic curves	[Sil] 3
1/23	Elliptic curves, p-adic fields	[Sil] 3, [Bos] A
1/28	Elliptic curves over p-adic fields, Newton polygons	[Sil] 7, [Bos] A
1/30	φ-modules	[BC] 3, [FO] 3.2
2/4	φ-modules and (φ,Γ)-modules, perfectoid fields	[BC] 3, [FO] 3.2, 5.3, [Ber] III, [K2] 2.3, [W] 2.0
2/6	Perfectoid fields	[W] 2.0-2.2, [K2] 1.1-1.4
2/11	Perfectoid fields, group cohomology	[K2] 1.5, [Bos] A, [Ser] VII, [R] 1
2/13	Group cohomology	[Ser] VII, X, [R] 1, [FO] 1.5, [Bel] 2.1
2/18	Ax-Sen-Tate theorem	[T] 3
2/20	B_dR	[BC] 4.4, 6, [FO] 6.2
2/25	B_dR and differentials	[C]
2/27	B_dR and differentials	[C]
3/4	Sheaf and de Rham cohomology	[Sta] 00UZ, 0FK4, 03N1, [K1] 1
3/6	Sites, étale and pro'etale cohomology	[Sta] 01FQ, 03N1, 0965
3/18	Hodge-Tate decomposition for abelian varieties, Huber rings	[F], [SW] 2
3/20	Huber rings	[SW] 2
3/25	Adic spaces	[SW] 3-4
3/27	Adic spaces	[SW] 4, [Bos] 2, [H] 1.3, 1.6

References

Main references

Other references

Tentative Schedule

Homework