We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.

We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ --- the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven ``averaged" version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.

Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on $M$ to the central derivatives of certain $L$-functions.

As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.

The Colmez conjecture is a formula expressing the Faltings height of an abelian variety with complex multiplication in terms of some linear combination of logarithmic derivatives of Artin $L$-functions. The aim of this paper to prove an averaged version of the conjecture, which was also proposed by Colmez.