Mahler measures of algebraic varieties

Thời gian: 10:00 đến 11:00 Ngày 24/07/2018

Địa điểm:

Tóm tắt:

Introduced by Mahler in the early 1960s, the (logarithmic) Mahler measure of a nonzero $n$-variate Laurent polynomial $P$ is defined to be the arithmetic mean of$\log |P|$ over the $n$-dimensional torus. Despite its purely analytic definition, the Mahler measures of certain polynomials turn out to encode interesting arithmetic information. More precisely, it has been conjectured since the 1990s that they are expressible in terms of special values of $L$-functions associated to the algebraic varieties defined by zero loci of the underlying polynomials. Most cases of these conjectures are still wide open despite a huge amount of effort to tackle them. In this talk, we will give a brief introduction to Mahler measures and present some recent progress and open problems on Mahler measures and their connection with $L$-values.