Isogeny graphs with maximal real multiplication
Identifieur interne : 000665 ( Main/Exploration ); précédent : 000664; suivant : 000666Isogeny graphs with maximal real multiplication
Auteurs : Sorina Ionica [France] ; Emmanuel Thomé [France]Source :
English descriptors
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Abstract
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 first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2.
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Affiliations:
- France
- Aquitaine, Grand Est, Lorraine (région), Nouvelle-Aquitaine
- Bordeaux, Metz, Nancy
- Université de Bordeaux, Université de Lorraine
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CS 25233 - 54052 Nancy cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.univ-lorraine.fr/</ref></desc></org></tutelle><tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno><idno type="IdRef">02636817X</idno><orgName>Centre National de la Recherche Scientifique</orgName><orgName type="acronym">CNRS</orgName><date type="start">1939-10-19</date><desc><address><country key="FR"></country></address><ref type="url">http://www.cnrs.fr/</ref></desc></org></tutelle></tutelles></hal:affiliation><country>France</country><placeName><settlement type="city">Nancy</settlement><settlement type="city">Metz</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Université de Lorraine</orgName></affiliation></author></analytic></biblStruct></sourceDesc></fileDesc><profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>computation</term><term>endomorphism ring</term><term>genus 2</term><term>isogeny graph</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">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 first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2.</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Aquitaine</li><li>Grand Est</li><li>Lorraine (région)</li><li>Nouvelle-Aquitaine</li></region><settlement><li>Bordeaux</li><li>Metz</li><li>Nancy</li></settlement><orgName><li>Université de Bordeaux</li><li>Université de Lorraine</li></orgName></list><tree><country name="France"><region name="Nouvelle-Aquitaine"><name sortKey="Ionica, Sorina" sort="Ionica, Sorina" uniqKey="Ionica S" first="Sorina" last="Ionica">Sorina Ionica</name></region><name sortKey="Thome, Emmanuel" sort="Thome, Emmanuel" uniqKey="Thome E" first="Emmanuel" last="Thomé">Emmanuel Thomé</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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