Class field theory describes abelian extensions of a number field in terms of data intrinsic to the field. Hilbert’s 12th problem, or explicit class field theory, goes further and asks for explicit generators of abelian extensions of a number field as values of analytic functions. Stark’s conjectures provide a different perspective by postulating existence of special elements, now called Stark units, related to leading terms of L-functions at s=0 and generating abelian extensions of a number field. In this talk we will first review classical results on explicit class field theory for the field of rational numbers and imaginary quadratic fields. Next we review Stickelberger’s theorem on annihilation of class groups of abelian extensions of the field of rational numbers. Brumer-Stark conjecture is an analogue of Stickelberger’s theorem for arbitrary totally real fields. We will see the statement of this conjecture along with its refinements and our results towards them. We prove these conjectures away from p=2 Brumer-Stark conjecture gives existence of special elements called Brumer-Stark units. These generate abelian extensions of a totally real number field. A conjecture by one of us gives p-adic analytic formulae for Brumer-Stark units. We will discuss this conjecture. Our recent work proves this conjecture away from p=2 thus providing a p-adic analytic solution to Hilbert’s 12th problem for totally real number fields. We will end with a brief sketch of our proofs.
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This colloquium is part of the Distinguished Lecture Series (DLS) organized by The (Indian) Mathematics Consortium (TMC), and co-hosted by IIT-B and ICTS-TIFR. For more information about this lecture series, and to register, please visit the homepage of the TMC DLS.