Spanning the isogeny class of a power of an elliptic curve
Kirschmer, Markus
Narbonne, Fabien
Ritzenthaler, Christophe
Robert, Damien
Applied Mathematics
Computational Mathematics
Algebra and Number Theory
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.
American Mathematical Society (AMS)
2021
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://ris.uni-paderborn.de/record/34912
Kirschmer M, Narbonne F, Ritzenthaler C, Robert D. Spanning the isogeny class of a power of an elliptic curve. <i>Mathematics of Computation</i>. 2021;91(333):401-449. doi:<a href="https://doi.org/10.1090/mcom/3672">10.1090/mcom/3672</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1090/mcom/3672
info:eu-repo/semantics/altIdentifier/issn/0025-5718
info:eu-repo/semantics/altIdentifier/issn/1088-6842
info:eu-repo/semantics/closedAccess