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
Office hours: time TBA in JWB 106
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
[BS] 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

[M] J. Milne, Field theory and Galois theory
[Sil] J. Silverman, The arithmetic of elliptic curves

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:

H. Diao, Period rings and period sheaves
Y. Mieda, Adic spaces and perfectoid spaces

Tentative Schedule

Date	Topic	References
1/7	Introduction
1/9	Infinite Galois theory, examples of Galois representations	[M] 3, 7, [BC] 1, [FO] 2
1/14	Elliptic curves	[Sil] 3
1/16	Formal groups	[Sil] 4
1/21	Elliptic curves and p-adic fields	[Sil] 3, 7
1/23	φ-modules	[BC] 3, [FO] 3.2

References

Main references

Other references

Tentative Schedule

Homework