Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Spanning the isogeny class of a power of an elliptic curve
HTML articles powered by AMS MathViewer

by Markus Kirschmer, Fabien Narbonne, Christophe Ritzenthaler and Damien Robert HTML | PDF
Math. Comp. 91 (2022), 401-449

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.
References
Similar Articles
Additional Information
  • Markus Kirschmer
  • Affiliation: Universität Paderborn, Fakultät EIM, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany
  • MR Author ID: 892539
  • Email: markus.kirschmer@math.upb.de
  • Fabien Narbonne
  • Affiliation: Univ. Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
  • Email: fabien.narbonne@univ-rennes1.fr
  • Christophe Ritzenthaler
  • Affiliation: Univ. Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
  • MR Author ID: 702917
  • Email: christophe.ritzenthaler@univ-rennes1.fr
  • Damien Robert
  • Affiliation: INRIA Bordeaux–Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence Cedex, France; and Institut de Mathématiques de Bordeaux, 351 cours de la liberation, 33405 Talence cedex, France
  • MR Author ID: 919800
  • Email: damien.robert@inria.fr
  • Received by editor(s): April 15, 2020
  • Received by editor(s) in revised form: December 18, 2020, May 9, 2021, and May 25, 2021
  • Published electronically: September 16, 2021
  • © Copyright 2021 by the authors
  • Journal: Math. Comp. 91 (2022), 401-449
  • MSC (2020): Primary 14H42, 14G15, 14H45, 16H20
  • DOI: https://doi.org/10.1090/mcom/3672
  • MathSciNet review: 4350544