# Spanning the isogeny class of a power of an ordinary elliptic curve.

3 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-02554714
Contributor : Christophe Ritzenthaler <>
Submitted on : Monday, April 12, 2021 - 6:26:12 PM
Last modification on : Sunday, May 2, 2021 - 3:11:26 AM

### File

Article_obstruction.pdf
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### Identifiers

• HAL Id : hal-02554714, version 1
• ARXIV : 2004.08315

### Citation

Markus Kirschmer, Fabien Narbonne, Christophe Ritzenthaler, Damien Robert. Spanning the isogeny class of a power of an ordinary elliptic curve.. 2021. ⟨hal-02554714v1⟩

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