Class Invariants for Quartic CM Fields

Eyal Z. Goren[1]; Kristin E. Lauter[2]

  • [1] McGill University Department of Mathematics and Statistics 805 Sherbrooke St. W. Montreal H3A 2K6, QC (Canada)
  • [2] Microsoft Research One Microsoft Way Redmond, WA 98052 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 457-480
  • ISSN: 0373-0956

Abstract

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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 . Such constructions were given by de Shalit-Goren and Lauter. 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 a reflex field of K , where S is effectively determined by K , and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured by Lauter.

How to cite

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Goren, Eyal Z., and Lauter, Kristin E.. "Class Invariants for Quartic CM Fields." Annales de l’institut Fourier 57.2 (2007): 457-480. <http://eudml.org/doc/10228>.

@article{Goren2007,
abstract = {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 $K$. Such constructions were given by de Shalit-Goren and Lauter. 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 a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured by Lauter.},
affiliation = {McGill University Department of Mathematics and Statistics 805 Sherbrooke St. W. Montreal H3A 2K6, QC (Canada); Microsoft Research One Microsoft Way Redmond, WA 98052 (USA)},
author = {Goren, Eyal Z., Lauter, Kristin E.},
journal = {Annales de l’institut Fourier},
keywords = {Class invariant; modular form; complex multiplication; polarization; superspecial abelian variety; units; Igusa invariants; quaternion algebra; class invariant; genus 2},
language = {eng},
number = {2},
pages = {457-480},
publisher = {Association des Annales de l’institut Fourier},
title = {Class Invariants for Quartic CM Fields},
url = {http://eudml.org/doc/10228},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Goren, Eyal Z.
AU - Lauter, Kristin E.
TI - Class Invariants for Quartic CM Fields
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 457
EP - 480
AB - 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 $K$. Such constructions were given by de Shalit-Goren and Lauter. 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 a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured by Lauter.
LA - eng
KW - Class invariant; modular form; complex multiplication; polarization; superspecial abelian variety; units; Igusa invariants; quaternion algebra; class invariant; genus 2
UR - http://eudml.org/doc/10228
ER -

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