p-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 [1], 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 = p where p is prime, and for any abelian variety J over Fq we let PJ denote the characteristic polynomial of the Frobenius endomorphism πJ of J (as shown by Tate [4], PJ determines the isogeny class of J). Theorem. The following is a list of all polynomials PJ , where J is an abelian surface over Fq such that q | #J(Fq): (1) X +X − (q + 2)X + qX + q (if q odd and q > 8); (2) X −X + q; (3) X −X + qX − qX + q (if m odd or p 6≡ 1 (mod 4)); (4) X − 2X + (2q + 1)X − 2qX + q; (5) X + aX + bX + aqX + q, where either q = 13, a = 9, b = 42; or q = 9, a = 6, b = 20; or q = 7, a = 4, b = 16; or q = 5, (a, b) ∈ {(3, 6), (8, 26)}; or q = 4, (a, b) ∈ {(2, 5), (4, 11), (6, 17)}; or q = 3, (a, b) ∈ {(1, 4), (3, 5), (4, 10)}; or q = 2, (a, b) ∈ {(0, 3), (1, 0), (1, 4), (2, 5), (3, 6)}. The special form required of the Frobenius endomorphism in [1] 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 [1] follows from the above result. Date: June 12, 2007. 1991 Mathematics Subject Classification. 14H40.