Description:

Dirichlet’s class number formula expresses residue at s=1 of Dedekind zeta function in terms of arithmetic invariants such as class number and regulator. We will start this course with Dirichlet’s class number formula. Stark formulated a series of conjectures in the 1970s in an attempt to generalise the class number formula to Artin L-functions. We will mostly follow Tate’s treatment of Stark’s conjectures. We will then consider integral refinements of the Stark’s conjectures due to Rubin. This naturally lead to connections with Iwasawa theory and Equivariant Tamagawa Number Conjecture (ETNC). We will explore refinements of integral Stark’s conjectures as well as connections with the ETNC.

Prerequisite: Algebraic Number Theory

Reference:

1. Les Conjectures de Stark sur les Fonctions L d’Artin en s=0. John Tate

2. On the values of abelian L-functions at s=0. Benedict Gross

3. A Stark conjecture over Z for abelian L-functions with multiple zeros. Karl Rubin

4. Integral and p-adic Refinements of the Abelian Stark Conjectures. Christian Popescu

Target Audience: Graduate students

Teaching Language: English

Personal Website: https://math.iisc.ac.in/~maheshkakde/

Registration: https://www.wjx.top/vm/wwoc7ZJ.aspx#

Recording: https://archive.ymsc.tsinghua.edu.cn/pacm_course?html=Stark_s_conjectures.html