Mathematics 207c

Topics in Number Theory

Universal Galois deformation ring

Spring Quarter 2020

Haruzo HIDA

**Meeting Time:** Mondays, Wednesdays and Fridays 12:00noon to
12:50pm via Zoom until June 5, 2020 .

**Office hours:** CCLE forum.

## Zoom Lecture ended

**Comments**

I originally planned to give a course on analytic properties of L-functions continuing my Fall 19 course 205a.1.19f.
However, the analytic material is technical with many formulas, and therefore, I am afraid that it cannot be very well described without having in-person class meetings.
Because of this, I have chosen to give survey lectures fitting well with Zoom on deformation theory of ordinary 2-dimensional representations
without much technical details but full of open questions (perhaps useful for your Ph D study).
Staring with 1-dimensional case, and admitting the existence of the universal deformation ring and certain property of the corresponding Hecke algebra,
we explore structure theorems of the universal ring and *p*-local indecomposability of modular *p*-adic Galois representations.
Hopefully this works well although I never did lectures using Zoom.

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

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

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

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

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

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

Grading will be based on student report paper on topics close to the course material.
The report paper in tex pdf file (or dvi file) is due at the end of 9th week (as I am organizing a conference in France just after the 10th week).
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,
[MFG] H. Hida, "Modular Forms and Galois Cohomology", Cambridge studies in advanced mathematics, **69**, Cambridge University Press 2000,
[GME] H. Hida, "Geometric Modular Forms and Elliptic Curves", World Scientific, 2012.
**Topics:** In this course, assuming basic knowledge
of elliptic modular forms and Hecke operators acting on them,
we describe Galois deformation theory.
We hope to cover the following four topics:

Universal ring for Galois characters (GL(1) case),
2-dimensional case of universal rings,
Structure theorems of universal rings,
Adjoint Selmer groups and local indecomposability.

**Prerequisite:**

Good understanding of complex analysis (for Riemann surfaces), modular forms with Hecke operators and
basics of algebraic number theory (e.g. Dirichlet's unit theorem).