On characteristic polynomials for some genus $2$ and $3$ curves with $p$-rank $1$

In the paper, we study characteristic polynomials for some families of $p$-rank $1$ genus $2$ and $3$ hyperelliptic curves over finite field. $p$-Rank is an important invariant of the curves. It imposes restrictions on the coefficients of the characteristic polynomials and, therefore, on the order of the Jacobian. In this work, we distinguish several classes of $p$-rank $1$ curves among curves with authomorphims and find characteristic polynomials for these curves modulo $p$.