On the Brumer-Stark Conjecture and Refinements

Explore a comprehensive lecture on the Brumer–Stark conjecture and its refinements, delivered by Mahesh Kakde and Samit Dasgupta for the International Mathematical Union. Delve into advanced topics in algebraic number theory, including group rings, Stickelberger elements, class groups, and Fitting ideals. Examine the Strong Brumer-Stark conjecture and various refinements, such as the Burns-Kurihara-Sano and Equivariant Tamagawa Number conjectures. Investigate Ribet's method, group ring-valued modular forms, and Galois representations. Discuss the historical context, including Hilbert's 12th Problem and the Kronecker-Weber Theorem. Study the Gross-Stark conjecture, Eisenstein measures, and computational examples. Conclude with a proof of the exact formula for units, providing a thorough understanding of this complex mathematical topic.

Intro
SETUP
SMOOTHING
THE BRUMER-STARK CONJECTURE
BRUMER-STARK AWAY FROM 2
GROUP RINGS AND STICKELBERGER ELEMENTS
CLASS GROUP
DEFINITION OF FITTING IDEAL
STRONG BRUMER-STARK
REFINEMENTS CONJECTURES BURNS-KURIHARA-SANO
REFINEMENT: EQUIVARIANT TAMAGAWA NUMBER CONJECTURE
RIBETS METHOD
GROUP RING VALUED MODULAR FORMS
GROUP RING CUSP FORM
GALOIS REPRESENTATION
KRONECKER-WEBER THEOREM
HILBERT'S 12TH PROBLEM (1900)
THE BRUMER-STARK AND GROSS-STARK CONJECTURES
EXACT FORMULA FOR THE UNITS
EISENSTEIN MEASURE
COMPUTATIONAL EXAMPLE
A LARGER EXAMPLE
PROOF OF EXACT FORMULA