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

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

A Gross-Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \infty}$, the definite quaternion algebra over \QQ ramified only at $p$ and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo $\frak{p} | p$ between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields, and to give a bound on those supersingular primes which can appear. In the special case of Jacobians of genus 2 curves, this provides information about the factorizations of numerators of Igusa invariants, and so is also relevant to the problem of constructing genus 2 curves for use in cryptography.Comment: 32 page