Apparaît dans la collection : Mathematics is a long conversation: a celebration of Barry Mazur

In this talk we will give an adelic construction of an object that we call the Kato-Beilinson modular symbol for GL(2), extending constructions of Goncharov and Brunault. We obtain a modular symbol Ψ belonging to the compactly supported cohomology of arithmetic subgroups of GL(2) and taking values in a group of distributions valued in K2 of the tower of modular curves. We interpret Ψ as a “universal” L-value for modular forms and explain how it specializes to Kato's Euler systems, as well as its role in Fukaya and Kato's proof of Sharifi's conjecture. Our hope is that these ideas will also help us understand a conjecture of Darmon and Dasgupta about “elliptic units” associated to real quadratic fields.