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.