Fall 2024   Topics in Number Theory: Special Values of L-functions

Class information

  • Instructor: Liang Xiao (肖撁)
  • Lectures: Tuesday 5-6 (odd weeks) and Thursday 7-8
  • Lecture room: Lecture Building #1, Room 204 (δΈ€ζ•™ 204)
  • Office Hours: by appointment
  • Email: lxiao at bicmr.pku.edu.cn
  • Syllabus: For a pdf version, click here.

    The course will focus on the arithmetic properties of special values of L-functions. More precisely, we hope to cover the following aspects in this course:
    • Kubota--Leopoldt p-adic L-function.
    • Cyclotomic units and Euler system argument for Iwasawa Main Conjecture.
    • General definition of L-functions, and Tate thesis.
    • Deligne's conjecture on special values of L-functions and periods, and periods of modular forms.
    • A quick introduction to Waldspurger formula and Gross--Zagier formula.
    • Introduction to Beilinson Conjecture, and examples.
    • I do not know of a good reference that covers all material. I will be hopefully keep updating the latexed notes while posting hand-written notes (below). Hopefully, I can provide a list of references that I used for preparing this lecture.

    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 9/10 Introduction and special values of Dirichlet L-functions (algebraicity). (notes, video)
    2 9/12 Kummer congruences and p-adic analysis over Zp. (notes, video)
    3 9/19 p-adic Dirichlet L-functions, and L-functions for Galois representations (notes)
    4 9/24 Analytic class number formula. (notes)
    5 9/26 Cyclotomic units, regulators. (notes)
    Happy National's Day!
    6 10/8 (φ, Γ)-modules and Galois cohomology (notes)
    7 10/10 Coleman power series. (notes)
    8 10/17 Iwasawa Main Conjecture. (notes)
    9 10/22 Iwasawa Main Conjecture II. (notes)
    10 10/24 A light introduction to motives. (notes)
    11 10/31 Hodge structures and Deligne's conjecture. (notes)
    12 11/5 Periods of Hecke characters, and $p$-adic analogue. (notes)
    13 11/7 L-functions attached to modular forms, periods. (notes)
    14 11/14 Introduction to Waldspurger's formula. (LX away, sub by Yuan) (notes)
    15 11/19 Weak Mordell--Weil theorem for elliptic curves. (notes)
    16 11/21 Height of points on an elliptic curve. (notes)
    17 11/28 Introduction to Gross--Zagier's formula (LX away, sub by Yuan) (notes)
    18 12/3 Overflow. (notes)
    19 12/5 Introduction to Beilinson's conjecture. (notes)
    20 12/12 Examples of Beilinson's conjecture: K2 of modular curves. (notes)
    21 12/17 Examples of Beilinson's conjecture: Rankin--Selberg case. (notes)
    22 12/19 Borel's regulator theorem? (notes)
    23 12/26 An overview of development on the study of special values of L-functions. (notes)
    Take-home Final Exam