The Kronecker limit formula is 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 subleading terms of certain Artin L-functions.  In this talk we will formulate a “non-Artinian” generalization of Colmez conjecture, relating the following two quantities:

(1) the Faltings height of certain arithmetic Chow 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 $n=1$ case of our conjecture recovers the averaged Colmez conjecture.  We are able to prove our conjecture when $n=2$, using a relative trace formula approach that is formulated for the general $n$. 

The “arithmetic relative Langlands” morally suggests that there should be a lot of other similar (conjectural) phenomena involving subleading terms of L-functions and Faltings-like heights of algebraic cycles on Shimura varieties, and I will give a few more examples.

Joint work with Ryan Chen and Weixiao Lu.