Mathematics 205c

Number Theory


Arithmetic of adjoint L-values

Spring Quarter 2015

Haruzo HIDA

Meeting Time: Mondays and Wednesdays 10:00am to 10:50am in MS 5148, Fridays 4:00pm-5:50pm in MS 5117 (sometimes, we meet at 10am on Fridays (in MS 5148) depending on schedule).

This is a continuation of Course 205a in Fall 2014.
Here is a link to Fall 205a course webpage [html]
Office hours: After the class meeong on Monday 11am-11:50am and before class meeting on Friday 2pm-3pm (F) at my office: MS6308.

Lecture Starts on Monday March 30 in MS 5148


Lecture notes: Tentative lecture notes:
Notes (pdf file, a tentative version posted on 3/11/2015).

Grading will be based on student presentation at the end of courses on topics close to the course material. No final exam is planned.

As reference books, we suggest

  • T. Miyake, "Modular Forms", Springer Monograph of Mathematics, New York-Tokyo, 2006,
  • H. Hida, "Elementary Theory of L-functions and Eisenstein Series", LMSST 26, Cambridge University Press, Cambridge, 1993,
  • H. Hida, "Geometric Modular Forms and Elliptic Curves", Second Expanded Edition, World Scientific Publishing Co., Singapore, 2011,

    • Topics: In this course, assuming basic knowledge of complex analysis, we describe basics of elliptic modular forms. We hope to cover the following four topics:

  • Galois deformation theory (sketch),
  • Hecke algebra as deformation rings,
  • Integrality of adjoint L-values,
  • Congruences among cusp forms and a non-abelian class number formula.

    • Prerequisite:
    Good understanding of commutative and non-commutative algebra, complex analysis (for Riemann surfaces) and algebraic number theory.