Spanning the isogeny class of a power of an elliptic curve

M. Kirschmer, F. Narbonne, C. Ritzenthaler, D. Robert, Mathematics of Computation 91 (2021) 401–449.

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Journal Article | Published | English
Author
Kirschmer, MarkusLibreCat; Narbonne, Fabien; Ritzenthaler, Christophe; Robert, Damien
Abstract
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an ordinary elliptic curve over a finite field and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Superscript g"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>g</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">E^g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E cubed"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">E^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Igusa modular form in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We illustrate our algorithms with examples of curves with many rational points over finite fields.</p>
Publishing Year
Journal Title
Mathematics of Computation
Volume
91
Issue
333
Page
401-449
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Cite this

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
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
@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} }
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.
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.
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.

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