We prove a higher dimensional generalization of Gross and Zagier\u2019s theorem on the factorization of di\ufb00erences 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 di\ufb00erent imaginary quadratic \ufb01elds K and K0, 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 \ufb01eld into particular maximal orders in Bp,\u221e, the de\ufb01nite quaternion algebra over Q rami\ufb01ed only at p and in\ufb01nity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM \ufb01elds into superspecial orders in de\ufb01nite quaternion algebras over totally real \ufb01elds of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo p|p between abelian varieties with CM by di\ufb00erent \ufb01elds. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of di\ufb00erences of values of Siegel modular functions at CM points associated to two di\ufb00erent CM \ufb01elds, 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.<\/p>\n","protected":false},"excerpt":{"rendered":"
We prove a higher dimensional generalization of Gross and Zagier\u2019s theorem on the factorization of di\ufb00erences of singular moduli. 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