(A preprint and an official version are available online.)
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. Here is our main theorem. (Note that we assume we are given the Weil polynomial of an isogeny class of abelian surfaces, so to apply the theorem you have to know how to use the Honda-Tate theorem to determine what those polynomials are. We explain how to do this in an appendix to the paper.)
Theorem. Let f = x^{4} + ax³ + bx² + aqx + q² be the Weil polynomial of an isogeny class A of abelian surfaces over F_{q}, where q is a power of a prime p.
p-rank of A | Condition on p and q | Conditions on s and t |
---|---|---|
— | — | |s − t| = 1 |
2 | — | s = t and t² − 4q∈{−3,−4,−7} |
2 | q = 2 | |s| = |t| = 1 and s≠ t |
1 | q square | s² = 4q and s−t squarefree |
1 | p > 3 | s² ≠ t² |
1 | p = 3 and q nonsquare | s² = t² = 3q |
0 | p = 3 and q square | s − t is not divisible by 3√q |
0 | p = 2 | s² − t² is not divisible by 2q |
0 | q = 2 or q = 3 | s = t |
0 | q = 4 or q = 9 | s² = t² = 4q |
p-rank of A | Condition on p and q | Conditions on a and b |
---|---|---|
— | — | a² − b = q and b < 0 and all prime divisors of b are 1 mod 3 |
2 | — | a = 0 and b = 1 − 2q |
2 | p > 2 | a = 0 and b = 2 − 2q |
2 | p ≡ 11 mod 12 and q square | a = 0 and b = −q |
0 | p = 3 and q square | a = 0 and b = −q |
0 | p = 2 and q nonsquare | a = 0 and b = −q |
0 | q = 2 or q = 3 | a = 0 and b = −2q |