Denominators of Igusa class polynomials

Kristin Lauter[1]; Bianca Viray[2]

  • [1] Microsoft Research, 1 Microsoft Way, Redmond, WA 98062, USA
  • [2] Department of Mathematics, Box 1917, Brown University, Providence, RI 02912, USA

Publications mathématiques de Besançon (2014)

  • Volume: 137, Issue: 2, page 5-29
  • ISSN: 1958-7236

Abstract

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In [22], the authors proved an explicit formula for the arithmetic intersection number CM ( K ) . G 1 on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K . These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross and Zagier. The current paper combines the arguments of [21, 22] and presents a direct proof of the main arithmetic intersection formula. We focus on providing a stream-lined account of the proof such that the algorithm for implementation is clear, and we give applications and examples of the formula in corner cases.

How to cite

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Lauter, Kristin, and Viray, Bianca. "Denominators of Igusa class polynomials." Publications mathématiques de Besançon 137.2 (2014): 5-29. <http://eudml.org/doc/275721>.

@article{Lauter2014,
abstract = {In [22], the authors proved an explicit formula for the arithmetic intersection number $\left(\operatorname\{CM\}(K). G_1\right)$ on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field $K$. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross and Zagier. The current paper combines the arguments of [21, 22] and presents a direct proof of the main arithmetic intersection formula. We focus on providing a stream-lined account of the proof such that the algorithm for implementation is clear, and we give applications and examples of the formula in corner cases.},
affiliation = {Microsoft Research, 1 Microsoft Way, Redmond, WA 98062, USA; Department of Mathematics, Box 1917, Brown University, Providence, RI 02912, USA},
author = {Lauter, Kristin, Viray, Bianca},
journal = {Publications mathématiques de Besançon},
keywords = {Gross-Zagier’s formula; intersection number; complex multiplication; Igusa class polynomials},
language = {eng},
number = {2},
pages = {5-29},
publisher = {Presses universitaires de Franche-Comté},
title = {Denominators of Igusa class polynomials},
url = {http://eudml.org/doc/275721},
volume = {137},
year = {2014},
}

TY - JOUR
AU - Lauter, Kristin
AU - Viray, Bianca
TI - Denominators of Igusa class polynomials
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
VL - 137
IS - 2
SP - 5
EP - 29
AB - In [22], the authors proved an explicit formula for the arithmetic intersection number $\left(\operatorname{CM}(K). G_1\right)$ on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field $K$. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross and Zagier. The current paper combines the arguments of [21, 22] and presents a direct proof of the main arithmetic intersection formula. We focus on providing a stream-lined account of the proof such that the algorithm for implementation is clear, and we give applications and examples of the formula in corner cases.
LA - eng
KW - Gross-Zagier’s formula; intersection number; complex multiplication; Igusa class polynomials
UR - http://eudml.org/doc/275721
ER -

References

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