Faltings heights and the sub-leading terms of adjoint L-functions

Based on work in progress with Ryan Chen and Weixiao Lu.
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 sub-leading terms of certain Artin L-functions. In this talk we will formulate a “non-Artinian” generalization of (averaged) Colmez conjecture, relating the following two quantities:

(1) the Faltings height of certain cycles on unitary Shimura varieties, and
(2) the sub-leading terms of the adjoint L-functions of (cohomological) automorphic representations of unitary groups U(n).

The case $n=1$ amounts to the averaged Colmez conjecture. We formulate a relative trace formula approach for the general $n$, and we are able to prove our conjecture when $n=2$.

Please note that this week there will not be a pre-talk.