Class invariants for quartic CM fields

One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S-units in certain abelian extensions of K, where S is effectively determined by K. It also yields class polynomials for primitive quartic CM fields whose coefficients are S-integers.Comment: 14 page

The canonical subgroup: a "subgroup-free" approach

Beyond the crucial role they play in the foundations of the theory of overconvergent modular forms, canonical subgroups have found new applications to analytic continuation of overconvergent modular forms. For such applications, it is essential to understand various ``numerical'' aspects of the canonical subgroup, and in particular, the precise extent of its overconvergence. We develop a theory of canonical subgroups for a general class of curves (including the unitary and quaternionic Shimura curves), using formal and rigid geometry. In our approach, we use the common geometric features of these curves rather than their (possible) specific moduli-theoretic description.Comment: 16 pages, 1 figur

Faltings heights of abelian varieties with complex multiplication

Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over 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.Comment: Final version. To appear in Annals of Mat

Canonical Subgroups over Hilbert Modular Varieties

We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.Comment: 56 page

A theta operator on Picard modular forms modulo an inert prime

We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions