Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

A motivic approach to the Gross-Zagier conjecture

The Gross-Zagier conjecture, now a theorem of Li and Bruinier-Li-Yang for Hecke congruence subgroups, concerns the log-algebraicity of special values of higher Green's functions on modular curves at CM points. In this talk, I'll explain an approach to this conjecture via the theory of periods and motives, connecting it to standard problems in algebraic geometry. As a proof of concept, we show this framework yields a purely geometric proof in level one and weight four by leveraging properties of the moduli space M_{1,3}. I will also introduce several new objects arising in the proof, including 'meromorphic modular forms of the third kind', mixed modular motives, and matrix-valued higher Green's functions. This is joint work with Francis Brown.