---
_id: '34912'
abstract:
- lang: eng
text: '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 E⁹
. 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 E³ and of the Igusa modular form in dimension 4. We illustrate our
algorithms with examples of curves with many rational points over finite fields. '
author:
- first_name: Markus
full_name: Kirschmer, Markus
id: '82258'
last_name: Kirschmer
- first_name: Fabien
full_name: Narbonne, Fabien
last_name: Narbonne
- first_name: Christophe
full_name: Ritzenthaler, Christophe
last_name: Ritzenthaler
- first_name: Damien
full_name: Robert, Damien
last_name: Robert
citation:
ama: Kirschmer M, Narbonne F, Ritzenthaler C, Robert D. Spanning the isogeny class
of a power of an elliptic curve. *Mathematics of Computation*. 2021;91(333):401-449.
doi:10.1090/mcom/3672
apa: Kirschmer, M., Narbonne, F., Ritzenthaler, C., & Robert, D. (2021). Spanning
the isogeny class of a power of an elliptic curve. *Mathematics of Computation*,
*91*(333), 401–449. https://doi.org/10.1090/mcom/3672
bibtex: '@article{Kirschmer_Narbonne_Ritzenthaler_Robert_2021, title={Spanning the
isogeny class of a power of an elliptic curve}, volume={91}, DOI={10.1090/mcom/3672},
number={333}, journal={Mathematics of Computation}, publisher={American Mathematical
Society (AMS)}, author={Kirschmer, Markus and Narbonne, Fabien and Ritzenthaler,
Christophe and Robert, Damien}, year={2021}, pages={401–449} }'
chicago: 'Kirschmer, Markus, Fabien Narbonne, Christophe Ritzenthaler, and Damien
Robert. “Spanning the Isogeny Class of a Power of an Elliptic Curve.” *Mathematics
of Computation* 91, no. 333 (2021): 401–49. https://doi.org/10.1090/mcom/3672.'
ieee: 'M. Kirschmer, F. Narbonne, C. Ritzenthaler, and D. Robert, “Spanning the
isogeny class of a power of an elliptic curve,” *Mathematics of Computation*,
vol. 91, no. 333, pp. 401–449, 2021, doi: 10.1090/mcom/3672.'
mla: Kirschmer, Markus, et al. “Spanning the Isogeny Class of a Power of an Elliptic
Curve.” *Mathematics of Computation*, vol. 91, no. 333, American Mathematical
Society (AMS), 2021, pp. 401–49, doi:10.1090/mcom/3672.
short: M. Kirschmer, F. Narbonne, C. Ritzenthaler, D. Robert, Mathematics of Computation
91 (2021) 401–449.
date_created: 2022-12-23T11:02:02Z
date_updated: 2023-04-04T07:52:43Z
department:
- _id: '102'
doi: 10.1090/mcom/3672
intvolume: ' 91'
issue: '333'
keyword:
- Applied Mathematics
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 401-449
publication: Mathematics of Computation
publication_identifier:
issn:
- 0025-5718
- 1088-6842
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: Spanning the isogeny class of a power of an elliptic curve
type: journal_article
user_id: '93826'
volume: 91
year: '2021'
...