Hilbert modular polynomials

Chloe Martindale

doi:10.1016/j.jnt.2019.11.019

Hilbert modular polynomials

Chloe Martindale^*

^*Corresponding author for this work

Department of Computer Science

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

74 Downloads (Pure)

Abstract

We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the ‘set of Hilbert modular polynomials’, concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field . In the 2-dimensional case with we also provide an implementation together with some optimisations specific to this case. We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.

Original language	English
Pages (from-to)	464-498
Number of pages	35
Journal	Journal of Number Theory
Volume	213
Early online date	18 Mar 2020
DOIs	https://doi.org/10.1016/j.jnt.2019.11.019
Publication status	Published - 1 Aug 2020

Keywords

Abelian varieties
Cyclic isogenies
Genus two
Hilbert modular polynomials
Maximal real multiplication

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10.1016/j.jnt.2019.11.019

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Accepted author manuscript, 409 KBLicence: CC BY-NC-ND

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title = "Hilbert modular polynomials",

abstract = "We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the {\textquoteleft}set of Hilbert modular polynomials{\textquoteright}, concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field . In the 2-dimensional case with we also provide an implementation together with some optimisations specific to this case. We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.",

keywords = "Abelian varieties, Cyclic isogenies, Genus two, Hilbert modular polynomials, Maximal real multiplication",

author = "Chloe Martindale",

year = "2020",

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doi = "10.1016/j.jnt.2019.11.019",

language = "English",

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T1 - Hilbert modular polynomials

AU - Martindale, Chloe

PY - 2020/8/1

Y1 - 2020/8/1

N2 - We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the ‘set of Hilbert modular polynomials’, concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field . In the 2-dimensional case with we also provide an implementation together with some optimisations specific to this case. We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.

AB - We present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the ‘set of Hilbert modular polynomials’, concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field . In the 2-dimensional case with we also provide an implementation together with some optimisations specific to this case. We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials.

KW - Abelian varieties

KW - Cyclic isogenies

KW - Genus two

KW - Hilbert modular polynomials

KW - Maximal real multiplication

UR - http://www.scopus.com/inward/record.url?scp=85081966959&partnerID=8YFLogxK

UR - http://www.martindale.info/research/hilbert.pdf

U2 - 10.1016/j.jnt.2019.11.019

DO - 10.1016/j.jnt.2019.11.019

M3 - Article (Academic Journal)

SN - 0022-314X

VL - 213

SP - 464

EP - 498

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Hilbert modular polynomials

Abstract

Keywords

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