Meeting Time: Mondays, Wednesdays and Fridays 12:00noon to
12:50pm via Zoom until June 5, 2020 .
Office hours: CCLE forum.
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).
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:
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).