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Isogeny graphs of ordinary abelian varieties
[article]

2016
*
arXiv
*
pre-print

*Graphs*

*of*

*isogenies*

*of*degree a power

*of*ℓ are well-understood for elliptic curves, but not for higher-dimensional

*abelian*

*varieties*. ... We study the case

*of*absolutely simple

*ordinary*

*abelian*

*varieties*over a finite field. We analyse

*graphs*

*of*so-called l-

*isogenies*, resolving that they are (almost) volcanoes in any dimension. ...

*Graphs*

*of*l-

*isogenies*. Fix again a principally polarizable absolutely simple

*ordinary*

*abelian*

*variety*A

*of*dimension g over k, with endomorphism algebra K. ...

##
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Isogeny graphs of ordinary abelian varieties

2017
*
Research in Number Theory
*

*Graphs*

*of*

*isogenies*

*of*degree a power

*of*are well-understood for elliptic curves, but not for higher-dimensional

*abelian*

*varieties*. ... We study the case

*of*absolutely simple

*ordinary*

*abelian*

*varieties*over a finite field. We analyse

*graphs*

*of*so-called l-

*isogenies*, resolving that they are (almost) volcanoes in any dimension. ...

*Graphs*

*of*l-

*isogenies*. Fix again a principally polarizable absolutely simple

*ordinary*

*abelian*

*variety*A

*of*dimension g over k, with endomorphism algebra K. ...

##
###
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
[article]

2017
*
arXiv
*
pre-print

Fix an

arXiv:1506.00522v2
fatcat:l6tthtqs7zdv7evkw2bpitso6i
*ordinary**abelian**variety*defined over a finite field. ... Any subgroup*of*the class group, and generating set thereof, induces an*isogeny**graph*on the orbit*of*the*variety*for this subgroup. ...*Isogeny**graphs**of**ordinary**abelian**varieties*In this section, we describe the relation between our*graphs**of*interest -*graphs**of*horizontal*isogenies*between*ordinary**abelian**varieties*over finite fields ...##
###
Supersingular Non-Superspecial Abelian Surfaces in Cryptography
[article]

2022
*
IACR Cryptology ePrint Archive
*

Instead, we propose to use supersingular non-superspecial

dblp:journals/iacr/LeGrowTZ22
fatcat:ldojuicdxbafpgtazi23a7uya4
*isogeny**graphs*where such a product decomposition does not have a computable description via separable*isogenies*. ... We consider the use*of*supersingular*abelian*surfaces in cryptography. ... In dimension two,*isogeny**graphs**of**ordinary**abelian*surfaces assemble into a similar volcano structure. ...##
###
Isogeny graphs with maximal real multiplication

2019
*
Journal of Number Theory
*

An

doi:10.1016/j.jnt.2019.06.019
fatcat:iakoufei7zc2zgzltwdvgwzp6q
*isogeny**graph*is a*graph*whose vertices are principally polarizable*abelian**varieties*and whose edges are*isogenies*between these*varieties*. ... We describe the*isogeny**graphs*in that case, by considering cyclic*isogenies**of*degree , under the assumption that there is an ideal l*of*norm in K0 which is generated by a totally positive algebraic integer ... We are indebted to John Boxall for sharing his ideas regarding the computation*of**isogenies*preserving the real multiplication, and providing guidance for improving the writing*of*this paper. ...##
###
Isogeny graphs with maximal real multiplication
[article]

2019
*
arXiv
*
pre-print

An

arXiv:1407.6672v4
fatcat:syv4rplqdvhvfmw6o3vxwf7grq
*isogeny**graph*is a*graph*whose vertices are principally polarized*abelian**varieties*and whose edges are*isogenies*between these*varieties*. ... In his thesis, Kohel described the structure*of**isogeny**graphs*for elliptic curves and showed that one may compute the endomorphism ring*of*an elliptic curve defined over a finite field by using a depth ... We are indebted to John Boxall for sharing his ideas regarding the computation*of**isogenies*preserving the real multiplication, and providing guidance for improving the writing*of*this paper. ...##
###
Computing the endomorphism ring of an ordinary abelian surface over a finite field
[article]

2019
*
arXiv
*
pre-print

We present a new algorithm for computing the endomorphism ring

arXiv:1810.12270v2
fatcat:espnorn7bzgclbw2yt7m4nx7fy
*of*an*ordinary**abelian*surface over a finite field which is subexponential and generalizes an algorithm*of*Bisson and Sutherland for elliptic ... Thus we avoid the multiple heuristic assumptions on*isogeny**graphs*and polarized class groups which were previously required. ... Let V be any connected component*of*the l-*isogeny**graph*for the*isogeny*class*of*an*ordinary*, absolutely simple*abelian**variety*A with Frobenius π and maximal RM by F .1. ...##
###
Semisimple pointed isogeny graphs for abelian varieties
[article]

2018
*
arXiv
*
pre-print

In this paper we show that if ϕ_i:A_i→A is a semisimple pointed K-rational ℓ-

arXiv:1803.05194v1
fatcat:2cqjh63mt5hvpb6z2eckljahe4
*isogeny**graph**of*order n for a prime ℓ, then the group*of*ℓ-torsion points A[ℓ](K) contains a subspace*of*dimension n generated ... Furthermore, we give an explicit counterexample for*abelian**varieties**of*higher dimension to show that the semisimplicity condition is indeed necessary. ... Semisimple pointed*isogeny**graphs*We now turn to pointed K-rational ℓ-*isogeny**graphs*for*abelian**varieties**of*dimension g > 1. ...##
###
Principally polarized ordinary abelian varieties over finite fields

1995
*
Transactions of the American Mathematical Society
*

Our result shows that every

doi:10.1090/s0002-9947-1995-1297531-4
fatcat:coyc7klbcjh2viiuqfmtphujae
*isogeny*class*of*simple odd-dimensional*ordinary**abelian**varieties*over a finite field contains a principally polarized*variety*. ... We use our result to completely characterize the Weil numbers*of*the*isogeny*classes*of*two-dimensional*ordinary**abelian**varieties*over a finite field that do not contain principally polarized*varieties*... This paper is based on a portion*of*the author's doctoral ...##
###
Principally Polarized Ordinary Abelian Varieties Over Finite Fields

1995
*
Transactions of the American Mathematical Society
*

Our result shows that every

doi:10.2307/2154828
fatcat:6vhqldvnhve5rfnwibf5s4yxvu
*isogeny*class*of*simple odd-dimensional*ordinary**abelian**varieties*over a finite field contains a principally polarized*variety*. ... We use our result to completely characterize the Weil numbers*of*the*isogeny*classes*of*two-dimensional*ordinary**abelian**varieties*over a finite field that do not contain principally polarized*varieties*... This paper is based on a portion*of*the author's doctoral ...##
###
Deducing information about curves over finite fields from their Weil polynomials
[article]

2021
*
arXiv
*
pre-print

We discuss methods for using the Weil polynomial

arXiv:2110.04221v1
fatcat:jwspwzghazenbf55tif6wera7y
*of*an*isogeny*class*of**abelian**varieties*over a finite field to determine properties*of*the curves (if any) whose Jacobians lie in the*isogeny*class. ... Many*of*the techniques we discuss were inspired by methods that Serre used in his 1985 Harvard class on rational points on curves over finite fields. ... Suppose I is an*isogeny*class*of**ordinary**abelian**varieties*over F q , where q = q e 0 . ...##
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Computing endomorphism rings of abelian varieties of dimension two
[article]

2013
*
arXiv
*
pre-print

Generalizing a method

arXiv:1209.1189v2
fatcat:dgyyc3jaancpbhov2wo2ntgfxy
*of*Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings*of**ordinary**abelian**varieties**of*dimension two over finite ... Acknowledgments This work would never have seen the light*of*day without the author's prior collaborations with Andrew V. ...*of*the*graph**of*horizontal*isogenies*with that*of*polarized class groups*of*candidate rings. ...##
###
How to not break SIDH
[article]

2019
*
IACR Cryptology ePrint Archive
*

We include methods that fail to attack the pure

dblp:journals/iacr/MartindaleP19
fatcat:4a6xfv463vcp7e6bdsqlcxrwvu
*isogeny*problem, namely: looking at the Fpsubgraph, lifting to characteristic zero, and using Weil restrictions. ... and parameters*of*SIKE. ...*of*solutions), Tanja Lange, and Christophe Petit (alphabetical order). ...##
###
Computing endomorphism rings of abelian varieties of dimension two

2015
*
Mathematics of Computation
*

Generalizing a method

doi:10.1090/s0025-5718-2015-02938-x
fatcat:cqnh3tyjrbcuvcvpokastfaafy
*of*Sutherland and the author for elliptic curves [5, 1], we design a subexponential algorithm for computing the endomorphism ring structure*of**ordinary**abelian**varieties**of*dimension ... Certain results*of*this paper previously appeared in the author's thesis [2] . ... Acknowledgments This work would never have seen the light*of*day without the author's prior collaborations with Andrew V. ...##
###
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB
[article]

2020
*
arXiv
*
pre-print

This collection consists

arXiv:2003.05380v2
fatcat:gpqdmbrvgbeupal2hk6z6on654
*of*tables*of*Weil q-polynomials, which by the Honda-Tate theorem are in bijection with*isogeny*classes*of**abelian**varieties*over finite fields. ... Acknowledgements The authors began this project during the semester program "Computational aspects*of*the Lang- ... Table 2 : 2 Predicted versus actual values for the number*of**isogeny*classes*of**ordinary*/arbitrary*abelian**varieties**of*dimension g over F q . ...
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