Jacobians in isogeny classes of abelian surfaces over finite fields

Abstract

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.

On the Number of Rational Points on Prym Varieties Over Finite Fields

“…If q = 3, then by [6], E × E is isogenous to a Jacobian, so suppose that q ≥ 5. The elliptic curve E has 2 rational branch points, so applying to P 1 some suitable rational automorphism, we can assume that z 1 = 0 and z 2 = 1.…”

“…For the third step, we use Proposition 13 combined with results from [6] (which gives a description of the set of isogeny classes of abelian surfaces containing a Jacobian), Proposition 14 and Lemma 15.…”

$p^k$-torsion of genus two curves over $\mathbb {F}_{p^m}$

“…Condition ( (1,4), (2,5), or (3,6) Proof. Let J be an abelian surface over F q whose Weil polynomial P J = X 4 + aX 3 + bX 2 + aqX + q 2 satisfies one of (1.1)-(1.5).…”

“…13 (9, 42) 9 (6, 20) 7 (4,16) 5 (3,6) or (8,26) 4 (2,5), (4, 11), or (6, 17) 3 (1,4), (3,5), or (4, 10) 2 (0, 3), (1,0), (1,4), (2,5), or (3,6) The special form required of the Frobenius endomorphism in [4] has an immediate consequence for the shape of its characteristic polynomial, and by inspection the above polynomials do not have the required shape. Thus the main result of [4] follows from the above result.…”

Zeta Function and Cryptographic Exponent of Supersingular Curves of Genus 2

“…Thus, it is possible to write down all isogeny classes of supersingular abelian surfaces by adding to the simple classes given in Table 1 the split isogeny classes. By [HNR06] we know exactly what isogeny classes of abelian surfaces do not contain Jacobians and they can be dropped from the list. By the results of Xing and Zhu we can distribute the remaining isogeny classes according to the possible values of rk 2 .…”