Faltings heights and the sub-leading terms of adjoint L-functions (live stream)

The Kronecker limit formula may be interpreted as an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subheading terms of certain Artin L-functions. An averaged (over all abelian varieties with CM by the integer ring of a given CM field) version was proved by Andreatta—Goren—Howard—Madapusi and Yuan—Zhang.

In this talk we will formulate a “non-Artinian” generalization of Colmez conjecture, relating the following two quantities:
(1) the arithmetic intersection numbers from the Hodge bundle and certain cycles on unitary Shimura varieties, and 
(2) the sub-leading terms of the adjoint L-functions of (cohomological) automorphic representations of U(n). 

The case n=1 amounts to the averaged Colmez conjecture. We are able to prove our conjecture when n=2. Work in progress with Ryan Chen and Weixiao Lu.