Explore a comprehensive lecture on Brumer-Stark units and Hilbert's 12th Problem in this Distinguished Lecture Series presentation. Delve into complex mathematical concepts including Diophantine equations, Abelian extensions, Stickelberger's theorem, and the Brumer-Stark-Tate conjecture. Examine refinements of the Brumer-Stark conjecture, explicit class field theory, and new results in the field. Gain insights into computational examples, the Ritter-Weiss module, Ribet's method, and group ring valued modular forms. Conclude with an exploration of Galois representation and p-adic integral formulas for units. This in-depth talk offers a thorough examination of advanced number theory topics for mathematicians and researchers in related fields.
On Brumer-Stark Units and Hilbert’s 12th Problem by Mahesh Kakde and Samit Dasgupta
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
A live interactive session with the speaker will be hosted online on February 23, 2022, at 06:00 PM Indian Standard Time. Viewers can send in their questions for the speaker in advance of the live interactive session by filling out the Google form below before AM IST on February 21, 2022.
Start
Getting Started
Speakers
The Principle Aim & Format
Introduction
Preface
Diophantine Equations
Abelian extensions
Abelian extensions of imaginary quadratic fields
Hilbert's 12th problem
Stickelberger's theorem
Sickelberger's theorem
Analogue of Stickelberger's theorem for other number fields
General Stickelberger element
Brumer-Stark-Tate conjecture
Refinements of Brumer-Stark conjecture
Brumer-Stark-Tate from refinements
First reduction
MOTIVATION
EXPLICIT CLASS FIELD THEORY
THE BRUMER-STARK AND GROSS-STARK CONJECTURES
SOME OF MY PRIOR WORK IN THIS AREA
NEW RESULTS WITH MAHESH KAKDE
EXACT FORMULA FOR THE UNITS
COMPUTATIONAL EXAMPLE
A LARGER EXAMPLE
REFINEMENTS: CONJECTURES OF KURIHARA, BURNS, AND SANO
DEFINITION OF FITTING IDEAL
RITTER-WEISS MODULE
INCLUSION IMPLIES EQUALITY
RIBET'S METHOD
GROUP RING VALUED MODULAR FORMS
GROUP RING CUSP FORM
GALOIS REPRESENTATION
P-ADIC INTEGRAL FORMULA FOR UNITS
Thank you!
Taught by
International Centre for Theoretical Sciences
