Prismatic Polylogarithms

Polylogarithms are analytic functions that appear naturally, at least conjecturally, in the formulas expressing special values of zeta functions. By the work of Borel, Beilinson, and Deligne, it was understood that these analytic functions actually come from a more fundamental "motivic" incarnation in algebraic geometry, which should in particular explain why these functions appear both in complex and in p-adic geometry. In this talk, after explaning some of the history on polylogarithms, I want to explain a new approach, based on prismatic and q-de Rham cohomologies, to express p-adic polylogarithms as syntomic Chern classes, leading in particular to integral refinements of the existing results. This is a joint work in progress with Quentin Gazda.