Fall 2025   Topics in Number Theory: Introduction to p-adic Hodge theory

Class information

  • Instructor: Liang Xiao (肖撁)
  • Lectures: Tuesday 5-6 (odd weeks) and Thursday 7-8
  • Lecture room: Lecture Building #2, Room 416 (δΊŒζ•™ 416)
  • Office Hours: by appointment
  • Email: lxiao at bicmr.pku.edu.cn
  • Syllabus: For a pdf version, click here.

    This course gives an introduction to $p$-adic Hodge theory, including the following topics:
    • Galois theory for completed valued fields.
    • Lubin-Tate formal group construction for local class field.
    • Scholze's tilting equivalence for perfectoid fields.
    • Tate's normalized trace and Sen theory for Galois representations.
    • Fontaine's (ϕ, Γ)-modules.
    • Period rings in p-adic Hodge theory: BdR, Bcrys, and etc.
    • Overconvergent (ϕ, Γ)-modules and beyond.
    • We do not know of a perfect reference for this topic. We will follow relatively closely Berger's IHP notes: Galois representations and (ϕ, Γ)-modules. I can also keep available my hand-written lecture notes.

    Grade Distribution

    • Homeworks: 60%, due on Thursdays of Week 3, 6, 8, 10, 12, 14, 16, in total 7 times, with lowest grade dropped.
    • Take-home final exam: 40%, to be announced (probably one or two French-style long problems).

    Homework policy

    • Homework problems will be posted on this course webpage. You are welcome and encouraged to work with other students on the problems, but you should write up your homework independently.

    List of Homeworks

    Course Schedule

    The course schedule is subject to change.
    Tentative schedule
    Lecture Date Content
    1 9/9 Introduction and motivation. (notes)
    2 9/11 Grothendieck's monodromy theorem, Toolbox of Newton polygon. (notes)
    3 9/18 Analytic functions on p-adic annuli, Ax-Sen-Tate Theorem. (notes)
    4 9/23 Lubin-Tate formal groups. (notes)
    5 9/25 Lubin-Tate's construction of local class field theory. (notes)
    Happy National's Day!
    6 10/9 Perfectoid fields and tilting process (notes)
    7 10/16 Tilting equivalence. (notes)
    8 10/21 Tate's normalized trace. (notes)
    9 10/23 Tate-Sen theory for Cp-representations. (notes)
    10 10/30 Colmez--Tate--Sen theory, imperfect period rings, and (ϕ, Γ)-modules. (notes)
    11 11/4 Galois cohomology in terms of (ϕ, Γ)-modules. (notes)
    12 11/6 ψ-operators and Tate duality in terms of (ϕ, Γ)-modules. (notes)
    13 11/13 Crystalline and de Rham period rings. (LX away, taught by Yiwen Ding) (notes)
    14 11/18 p-adic Hodge theory in terms of period rings. (notes)
    15 11/20 Ddif. (notes)
    16 11/27 Overconvergent period rings, Colmez-Tate-Sen theory for overconvergent elements. (notes)
    17 12/2 Overconvergent (ϕ, Γ)-modules. (notes)
    18 12/4 (ϕ, Γ)-modules over Robba rings. (notes)
    19 12/11 Berger's functor I. (notes)
    20 12/16 Berger's functor II. (notes)
    21 12/18 Cohomology of (ϕ, Γ)-modules over Robba rings. (notes)
    22 12/25 Triangulline (ϕ, Γ)-modules and introduction to global triangulation. (notes)
    Take-home Final Exam