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Isogeny graphs of ordinary abelian varieties [article]

Ernest Hunter Brooks and Dimitar Jetchev and Benjamin Wesolowski
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.  ... 
arXiv:1609.09793v1 fatcat:qroutgyz2fd45gfayodyp3zqey

Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks, Dimitar Jetchev, Benjamin Wesolowski
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.  ... 
doi:10.1007/s40993-017-0087-5 fatcat:mehwdw2wpfco7fkhjnw4cpeaga

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

Dimitar Jetchev, Benjamin Wesolowski
2017 arXiv   pre-print
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]

Jason T. LeGrow, Yan Bo Ti, Lukas Zobernig
2022 IACR Cryptology ePrint Archive  
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

Sorina Ionica, Emmanuel Thomé
2019 Journal of Number Theory  
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]

Sorina Ionica
2019 arXiv   pre-print
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]

Caleb Springer
2019 arXiv   pre-print
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]

Paul Alexander Helminck
2018 arXiv   pre-print
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

Everett W. Howe
1995 Transactions of the American Mathematical Society  
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

Everett W. Howe
1995 Transactions of the American Mathematical Society  
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]

Everett W. Howe
2021 arXiv   pre-print
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]

Gaetan Bisson
2013 arXiv   pre-print
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]

Chloe Martindale, Lorenz Panny
2019 IACR Cryptology ePrint Archive  
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

Gaetan Bisson
2015 Mathematics of Computation  
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]

Taylor Dupuy, Kiran Kedlaya, David Roe, Christelle Vincent
2020 arXiv   pre-print
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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