Algebra & Number Theory

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

Eyal Goren and Kristin Lauter

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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.

Article information

Algebra Number Theory, Volume 7, Number 6 (2013), 1405-1450.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G16: Elliptic and modular units [See also 11R27]
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11R27: Units and factorization

CM abelian varieties singular moduli quaternion algebras superspecial orders


Goren, Eyal; Lauter, Kristin. A Gross–Zagier formula for quaternion algebras over totally real fields. Algebra Number Theory 7 (2013), no. 6, 1405--1450. doi:10.2140/ant.2013.7.1405.

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  • F. Andreatta and E. Z. Goren, “Geometry of Hilbert modular varieties over totally ramified primes”, Int. Math. Res. Not. 2003:33 (2003), 1786–1835.
  • E. Bachmat and E. Z. Goren, “On the non-ordinary locus in Hilbert–Blumenthal surfaces”, Math. Ann. 313:3 (1999), 475–506.
  • C.-L. Chai, “Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli”, Invent. Math. 121:3 (1995), 439–479.
  • D. X. Charles, E. Z. Goren, and K. E. Lauter, “Families of Ramanujan graphs and quaternion algebras”, pp. 53–80 in Groups and symmetries (Montréal, 2007), edited by J. Harnad and P. Winternitz, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009.
  • D. X. Charles, K. E. Lauter, and E. Z. Goren, “Cryptographic hash functions from expander graphs”, J. Cryptology 22:1 (2009), 93–113.
  • B. Conrad, “Gross–Zagier revisited”, pp. 67–163 in Heegner points and Rankin $L$-series, edited by H. Darmon and S.-W. Zhang, Math. Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, 2004.
  • D. R. Dorman, “Special values of the elliptic modular function and factorization formulae”, J. Reine Angew. Math. 383 (1988), 207–220.
  • D. R. Dorman, “Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves”, pp. 108–116 in Théorie des nombres (Québec, 1987), edited by J.-M. De Koninck and C. Levesque, de Gruyter, Berlin, 1989.
  • D. R. Dorman, “Singular moduli, modular polynomials, and the index of the closure of ${\bf Z}[j(\tau)]$ in ${\bf Q}(j(\tau))$”, Math. Ann. 283:2 (1989), 177–191.
  • E. Z. Goren, Lectures on Hilbert modular varieties and modular forms, CRM Monograph Series 14, Amer. Math. Soc., Providence, RI, 2002.
  • E. Z. Goren and K. E. Lauter, “Evil primes and superspecial moduli”, Int. Math. Res. Not. 2006 (2006), Art. ID 53864, 19.
  • E. Z. Goren and K. E. Lauter, “Class invariants for quartic CM fields”, Ann. Inst. Fourier $($Grenoble$)$ 57:2 (2007), 457–480.
  • E. Z. Goren and K. E. Lauter, “The distance between superspecial abelian varieties with real multiplication”, J. Number Theory 129:6 (2009), 1562–1578.
  • E. Z. Goren and K. E. Lauter, “Genus 2 curves with complex multiplication”, Int. Math. Res. Not. 2012:5 (2012), 1068–1142.
  • B. H. Gross, “On canonical and quasicanonical liftings”, Invent. Math. 84:2 (1986), 321–326.
  • B. H. Gross and D. B. Zagier, “On singular moduli”, J. Reine Angew. Math. 355 (1985), 191–220.
  • S. Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics 110, Springer, New York, 1986.
  • M.-H. Nicole, Superspecial abelian varieties, theta series and the Jacquet–Langlands correspondence, Ph.D. thesis, McGill University, 2005,
  • M.-H. Nicole, “Superspecial abelian varieties and the Eichler basis problem for Hilbert modular forms”, J. Number Theory 128:11 (2008), 2874–2889.
  • M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics 800, Springer, Berlin, 1980. In French.
  • P. van Wamelen, “Examples of genus two CM curves defined over the rationals”, Math. Comp. 68:225 (1999), 307–320.
  • W. C. Waterhouse and J. S. Milne, “Abelian varieties over finite fields”, pp. 53–64 in 1969 Number Theory Institute (Stony Brook, NY, 1969), Proc. Sympos. Pure Math. 20, Amer. Math. Soc., Providence, R.I., 1971.
  • C.-F. Yu, “The isomorphism classes of abelian varieties of CM-type”, J. Pure Appl. Algebra 187:1-3 (2004), 305–319.