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2013 A Gross–Zagier formula for quaternion algebras over totally real fields
Eyal Goren, Kristin Lauter
Algebra Number Theory 7(6): 1405-1450 (2013). DOI: 10.2140/ant.2013.7.1405


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 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 Bp,, the definite quaternion algebra over 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 pp 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 that 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.


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Eyal Goren. Kristin Lauter. "A Gross–Zagier formula for quaternion algebras over totally real fields." Algebra Number Theory 7 (6) 1405 - 1450, 2013.


Received: 23 February 2012; Revised: 5 October 2012; Accepted: 3 November 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1300.11068
MathSciNet: MR3107568
Digital Object Identifier: 10.2140/ant.2013.7.1405

Primary: 11G15 , 11G16
Secondary: 11G18 , 11R27

Keywords: CM abelian varieties , quaternion algebras , Singular moduli , superspecial orders

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 6 • 2013
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