Pk-torsion of Genus Two Curves over 𝔽pm

We determine the isogeny classes of abelian surfaces over Fq whose group of Fq-rational points has order divisible by q . We also solve the same problem for Jacobians of genus-2 curves. In a recent paper [4], Ravnshøj proved: if C is a genus-2 curve over a prime field Fp, and if one assumes that the endomorphism ring of the Jacobian J of C is the ring of integers in a primitive quartic CM-field, and that the Frobenius endomorphism of J has a certain special form, then p ∤ #J(Fp). Our purpose here is to deduce this conclusion under less restrictive hypotheses. We write q = pm where p is prime, and for any abelian variety J over Fq we let PJ denote the Weil polynomial of J , namely the characteristic polynomial of the Frobenius endomorphism πJ of J . As shown by Tate [6, Thm. 1], two abelian varieties over Fq are isogenous if and only if their Weil polynomials are identical. Thus, the following result describes the isogeny classes of abelian surfaces J over Fq for which q | #J(Fq). Theorem 1. The Weil polynomials of abelian surfaces J over Fq satisfying q | #J(Fq) are as follows: (1.1) X +X − (q + 2)X + qX + q (if q is odd and q > 8); (1.2) X −X2 + q; (1.3) X −X3 + qX − qX + q (if m is odd or p 6≡ 1 mod 4); (1.4) X − 2X + (2q + 1)X − 2qX + q; (1.5) X + aX + bX + aqX + q, where (a, b) occurs in the same row as q in the following table: