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Isogeny graphs of ordinary abelian varieties
[article]
2016
arXiv
pre-print
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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. ...
arXiv:1609.09793v1
fatcat:qroutgyz2fd45gfayodyp3zqey
Isogeny graphs of ordinary abelian varieties
2017
Research in Number Theory
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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. ...
doi:10.1007/s40993-017-0087-5
fatcat:mehwdw2wpfco7fkhjnw4cpeaga
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
[article]
2017
arXiv
pre-print
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Fix an 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 ...
arXiv:1506.00522v2
fatcat:l6tthtqs7zdv7evkw2bpitso6i
Supersingular Non-Superspecial Abelian Surfaces in Cryptography
[article]
2022
IACR Cryptology ePrint Archive
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Instead, we propose to use supersingular non-superspecial 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. ...
dblp:journals/iacr/LeGrowTZ22
fatcat:ldojuicdxbafpgtazi23a7uya4
Isogeny graphs with maximal real multiplication
2019
Journal of Number Theory
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An 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. ...
doi:10.1016/j.jnt.2019.06.019
fatcat:iakoufei7zc2zgzltwdvgwzp6q
Isogeny graphs with maximal real multiplication
[article]
2019
arXiv
pre-print
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An 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. ...
arXiv:1407.6672v4
fatcat:syv4rplqdvhvfmw6o3vxwf7grq
Computing the endomorphism ring of an ordinary abelian surface over a finite field
[article]
2019
arXiv
pre-print
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We present a new algorithm for computing the endomorphism ring 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. ...
arXiv:1810.12270v2
fatcat:espnorn7bzgclbw2yt7m4nx7fy
Semisimple pointed isogeny graphs for abelian varieties
[article]
2018
arXiv
pre-print
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In this paper we show that if ϕ_i:A_i→A is a semisimple pointed K-rational ℓ-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. ...
arXiv:1803.05194v1
fatcat:2cqjh63mt5hvpb6z2eckljahe4
Principally polarized ordinary abelian varieties over finite fields
1995
Transactions of the American Mathematical Society
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Our result shows that every 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 ...
doi:10.1090/s0002-9947-1995-1297531-4
fatcat:coyc7klbcjh2viiuqfmtphujae
Principally Polarized Ordinary Abelian Varieties Over Finite Fields
1995
Transactions of the American Mathematical Society
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Our result shows that every 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 ...
doi:10.2307/2154828
fatcat:6vhqldvnhve5rfnwibf5s4yxvu
Deducing information about curves over finite fields from their Weil polynomials
[article]
2021
arXiv
pre-print
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We discuss methods for using the Weil polynomial 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 . ...
arXiv:2110.04221v1
fatcat:jwspwzghazenbf55tif6wera7y
Computing endomorphism rings of abelian varieties of dimension two
[article]
2013
arXiv
pre-print
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Generalizing a method 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. ...
arXiv:1209.1189v2
fatcat:dgyyc3jaancpbhov2wo2ntgfxy
How to not break SIDH
[article]
2019
IACR Cryptology ePrint Archive
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We include methods that fail to attack the pure 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). ...
dblp:journals/iacr/MartindaleP19
fatcat:4a6xfv463vcp7e6bdsqlcxrwvu
Computing endomorphism rings of abelian varieties of dimension two
2015
Mathematics of Computation
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Generalizing a method 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. ...
doi:10.1090/s0025-5718-2015-02938-x
fatcat:cqnh3tyjrbcuvcvpokastfaafy
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB
[article]
2020
arXiv
pre-print
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This collection consists 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 . ...
arXiv:2003.05380v2
fatcat:gpqdmbrvgbeupal2hk6z6on654
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