x∂F⊂a⊂OF (Na)k−1. Here the superscript ‘+′ stands for totally positive elements. Siegel further derived from this a simpler proof of his famous theorem that ζF (1− k) is rational for all k ≥ 1. Siegel’s construction is based on the simple observation that a Hilbert modular form becomes an elliptic modular form when restricting diagonally to the upper half plane. Indeed, Hecke constructed and pr...

We survey the results of [Fun02] and of our joint work with Bruinier [BF06] on using the theta correspondence for the dual pair SL(2)×O(1, 2) to realize generating series of values of modular functions on a modular curve as (non)-holomorphic modular forms of weight 3/2.

In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

In this paper, we reinterpret the Colmez conjecture on Faltings’ height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving Faltings’ height of a CM abelian surface and arithmetic intersection numbers, and prove that Colmez’s conjecture for CM abelian surfaces is equivalent to the cuspitality of this modular form.