Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\le 4$

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 in corpus (12)

• Computing the endomorphism ring of an ordinary elliptic curve over a finite field
• The Torelli locus and special subvarieties
• Computing Igusa class polynomials
• Computing isogenies between abelian varieties
• A CRT algorithm for constructing genus 2 curves over finite fields
• Constructing genus 3 hyperelliptic Jacobians with CM
• Abelian varieties isogenous to a power of an elliptic curve
• Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves
• Schottky Algorithms: Classical meets Tropical
• The characteristic polynomials of abelian varieties of higher dimension over finite fields
• The transformation laws of algebraic theta functions
• Incremental Algorithms for Lattice Problems