TY - JOUR
AB - 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.
AU - Kirschmer, Markus
AU - Narbonne, Fabien
AU - Ritzenthaler, Christophe
AU - Robert, Damien
ID - 34912
IS - 333
JF - Mathematics of Computation
KW - Applied Mathematics
KW - Computational Mathematics
KW - Algebra and Number Theory
SN - 0025-5718
TI - Spanning the isogeny class of a power of an elliptic curve
VL - 91
ER -