Mathematics 205a

Number Theory

Hecke L-functions and their critical values

Fall Quarter 2019

Haruzo HIDA

**Meeting Time:** Mondays, Wednesdays and Fridays 2:00pm to
2:50pm in MS 7608.

**Office hours:** After class meetings, **from 3 PM (MW) until 3:50 PM**
in my office: MS6308.

## Lecture Starts on Monday September 30th at 2 PM in MS 7608

**Texts:** Lecture notes will be posted:

[Note No.0] (posted, a pdf file),

[Note No.1] (posted, a pdf file),

[Note No.2] (posted, a pdf file),

[Note No.3] (posted, a pdf file),

[Note No.4] (posted, a pdf file),

[Note No.5] (posted, a pdf file, last lecture notes).

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,
we suggest

[LFE] H. Hida, "Elementary Theory of L-functions
and Eisenstein Series",
LMSST **26**, Cambridge University Press, 1993,
[EDM] G. Shimura, "Elementary Dirichlet series and Modular forms", Springer Monographs in Mathematics, Springer 2007.
**Topics:** In this course, assuming basic knowledge
of complex analysis,
we describe the proof of Euler/Hurwitz/Shintani of rationality of Hecke L-values at non-positive integers.
We hope to cover the following four topics:

Analytic continuation/functional equation of Dirichlet L functions,
Analytic continuation/functional equation of Hecke L functions over totally real field (and possibly over general number fields),
Rationality and integrality of L-values,
If time allows, construction of Kubota-Leopoldt and Deligne-Ribet *p*-adic L functions.

**Prerequisite:**

Good understanding of complex analysis (for Riemann surfaces) and
basics of algebraic number theory (e.g. Dirichlet's unit theorem).