Fortuny–Grau–Oller-Marcén–Rúa conjecture on power sums of matrices over finite commutative rings

Let $d>1$ and let $R$ be a finite commutative ring. For each positive integer $k$, consider the power sum

$$\sum_{A\in M_{d\times d}(R)}A^k.$$

Fortuny–Grau–Oller-Marcén–Rúa conjecture. All such power sums vanish unless all of the following conditions hold: $d=2$; $|R|\equiv 2\pmod 4$ and $1<k\equiv 0,\pm1\pmod 6$; and the unique element $e\in R\setminus\{0\}$ satisfying $2e=0$ is idempotent, $e^2=e$. When these conditions hold, the power sum equals

$$e\cdot \operatorname{Id}_{2\times2}.$$

The conjecture concerns the precise exceptional cases in which matrix power sums over finite commutative rings can be nonzero; the surrounding theorem establishes the corresponding classification for the finite rings treated in the paper, and the conjecture is presented as a conjecture from earlier work that the paper settles.