JA25 Samit Dasgupta

Описание: Plenary lecture by Samit Dasgupta (Duke University) at the 33rd Journées Arithmétiques at the University of Luxembourg:
https://www.uni.lu/fstm-en/conference...

Title: On Special Values of Abelian L-functions: Stark’s conjectures, Explicit Class Field Theory, and Transcendence Theory

Abstract: In this talk I will try to tell a story spanning three central topics in number theory: explicit class field theory, special values of L-functions, and the transcendence theory of logarithms of algebraic numbers. We motivate our discussion with Hilbert’s 12th Problem, which asks for the explicit generation of abelian extensions of number fields in a fashion generalizing the theorem of Kronecker- Weber. We introduce Stark’s conjectures, which aim to address this question by studying the special values of abelian L-functions. Finally, we discuss the non-vanishing of regulators, which in our context are determinants of matrices of logarithms of algebraic numbers. Along the way, I will describe my joint work with Mahesh Kakde on these topics: an analytic description of the maximal abelian extension of a totally real field; a proof of the strongest form of the Brumer-Stark conjecture; and a conjecture on the vanishing of regulators along with an approach to attack the conjecture.