Mathematics 205c

Number Theory

Arithmetic of adjoint L-values

Spring Quarter 2017

Haruzo HIDA

**Meeting Time:** Mondays and Wednesdays 2:00pm to
2:50pm in MS 7608, Fridays
4:00pm-5:50pm in **MS 5117** (sometimes, we meet at 2pm on Fridays (in MS 7608) depending on schedule).

**This is a continuation of Course 205b in Winter 2017.**

Here is a link to Fall 205b course webpage [html]

**Office hours:** Before the class meeting on Monday/Wednesday 1pm-1:50pm
at my office: MS6308.

## Lecture Starts on Monday April 10 at 2pm in MS 7608

**Lecture notes:** Tentative lecture
notes:

Notes (pdf file, a tentative version).

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,
Hecke algebra as deformation rings,
Integrality of adjoint L-values,
Congruence 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.