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Spanning the isogeny class of a power of an elliptic curve.
| hal.structure.identifier | Mathematisches Institut der Universität Paderborn | |
| dc.contributor.author | KIRSCHMER, Markus | |
| hal.structure.identifier | Institut de Recherche Mathématique de Rennes [IRMAR] | |
| dc.contributor.author | NARBONNE, Fabien | |
| hal.structure.identifier | Institut de Recherche Mathématique de Rennes [IRMAR] | |
| dc.contributor.author | RITZENTHALER, Christophe | |
| hal.structure.identifier | Lithe and fast algorithmic number theory [LFANT] | |
| dc.contributor.author | ROBERT, Damien | |
| dc.date.accessioned | 2024-04-04T02:46:40Z | |
| dc.date.available | 2024-04-04T02:46:40Z | |
| dc.date.issued | 2021 | |
| dc.identifier.issn | 0025-5718 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191576 | |
| dc.description.abstractEn | Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre's obstruction for principally polarized abelian threefolds isogenous to $E^3$ and of the Igusa modular form in dimension $4$. We illustrate our algorithms with examples of curves with many rational points over finite fields. | |
| dc.description.sponsorship | Cryptographie, isogenies et variété abéliennes surpuissantes - ANR-19-CE48-0008 | |
| dc.description.sponsorship | Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020 | |
| dc.language.iso | en | |
| dc.publisher | American Mathematical Society | |
| dc.subject.en | Curves with many points overfinite fields | |
| dc.subject.en | Polarization | |
| dc.subject.en | Isogeny class | |
| dc.subject.en | Hermitian lattice | |
| dc.subject.en | Order in quadratic field | |
| dc.subject.en | Siegel modular form | |
| dc.subject.en | Theta constant | |
| dc.subject.en | Theta null point | |
| dc.subject.en | Algorithm | |
| dc.subject.en | Igusa modular form | |
| dc.subject.en | Serre’s obstruction | |
| dc.subject.en | Schottkylocus | |
| dc.title.en | Spanning the isogeny class of a power of an elliptic curve. | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1090/mcom/3672 | |
| dc.subject.hal | Mathématiques [math] | |
| dc.identifier.arxiv | 2004.08315 | |
| bordeaux.journal | Mathematics of Computation | |
| bordeaux.page | 401-449 | |
| bordeaux.volume | 91 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 333 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-02554714 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02554714v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematics%20of%20Computation&rft.date=2021&rft.volume=91&rft.issue=333&rft.spage=401-449&rft.epage=401-449&rft.eissn=0025-5718&rft.issn=0025-5718&rft.au=KIRSCHMER,%20Markus&NARBONNE,%20Fabien&RITZENTHALER,%20Christophe&ROBERT,%20Damien&rft.genre=article |
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