We prove that torsion codimension $2$ algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension $2$ on supersingular abelian varieties over the algebraic closure of finite fields.

The Griffiths group $\text{G}{{\text{r}}^{r}}\left( X \right)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)$ of a supersingular abelian variety ${{A}_{{\bar{k}}}}$ over the algebraic closure of a finite field of characteristic $p$ is at most a $p$-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of $\text{C}$. Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field $k$ of characteristic $p\,>\,2$, then the Griffiths group of any ordinary abelian threefold ${{A}_{{\bar{k}}}}$ over the algebraic closure of $k$ is non-trivial; in fact, for all but a finite number of primes $\ell \,\ne \,p$ it is the case that $\text{G}{{\text{r}}^{2}}\left( {{A}_{{\bar{k}}}} \right)\,\otimes \,{{\mathbb{Z}}_{\ell }}\,\ne \,0$.