Hilbert theta series and invariants of genus 2 curves

In this paper, we compute pull-backs of Siegel theta functions to the Hilbert moduli space and consider their application to generating genus 2 curves for cryptography. We express invariants of genus 2 curves such as the Gundlach invariants and Rosenhain invariants in terms of these Hilbert theta fu...

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Bibliographic Details
Published inJournal of number theory Vol. 161; pp. 146 - 174
Main Authors Lauter, Kristin, Naehrig, Michael, Yang, Tonghai
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2016
Subjects
Cryptography
Genus 2 curves
Hilbert moduli space
Invariants
Theta functions
11G18
Theta functions
Genus 2 curves
Hilbert moduli space
Cryptography
14G35
Invariants
11F55
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Summary:In this paper, we compute pull-backs of Siegel theta functions to the Hilbert moduli space and consider their application to generating genus 2 curves for cryptography. We express invariants of genus 2 curves such as the Gundlach invariants and Rosenhain invariants in terms of these Hilbert theta functions. A result of independent interest is a simple formula in terms of these functions for the Eisenstein series of weight 2, which is not the pull-back of any Siegel modular form of level 1. We present an algorithm to compute minimal polynomials for the invariants, including a description of CM points and how to compute them, along with numerical examples.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2015.02.020