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 on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field . These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 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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