Spanning the isogeny class of a power of an elliptic curve.
Language
en
Article de revue
This item was published in
Mathematics of Computation. 2021, vol. 91, n° 333, p. 401-449
American Mathematical Society
English 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 ...Read more >
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.Read less <
English Keywords
Curves with many points overfinite fields
Polarization
Isogeny class
Hermitian lattice
Order in quadratic field
Siegel modular form
Theta constant
Theta null point
Algorithm
Igusa modular form
Serre’s obstruction
Schottkylocus
ANR Project
Cryptographie, isogenies et variété abéliennes surpuissantes - ANR-19-CE48-0008
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
Origin
Hal imported