Mathematics 205a

Number Theory


Modular forms, congruence and L-values

Fall Quarter 2016

Haruzo HIDA

Meeting Time: Mondays and Wednesdays 2:00pm to 2:50pm in MS 6118 (note room change), Fridays 4:00pm-5:50pm in MS 5117 to be announced (sometimes, we meet at 10am on Fridays (in MS 6118) depending on schedule).
Office hours: Before class meetings, from 1:00pm (MW) in my office: MS6308.

Texts: Lecture notes are now posted:
Notes (pdf file, a tentative version).

Grading will be based on student presentation at the end of the Fall quarter (9th-10th week) 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:

  • Spaces of modular forms and its rational structure,
  • Modular L-functions,
  • Rationality and integrality of L-values,
  • Congruence among cusp forms.

    • Related topics may be touched upon in Math 290B student seminar on Wednesdays at 4:00pm-5:50pm in MS 5138 and possibly Mondays at 4:00pm-5:50pm in MS 5117 when number theory research seminar is not scheduled on the day (students are encouraged to attend).

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