Hyperelliptic curves, Cartier–Manin matrices and Legendre polynomials

We investigate the hyperelliptic curves of the form $C_1\colon y^2=x^{2g+1}+ax^{g+1}+bx$ and $C_2\colon y^2=x^{2g+2}+ax^{g+1}+b$ over the finite field $\mathbb F_q$, $q=p^n$, $p>2$. We transform these curves to the form $C_{1,\rho}\colon y^2=x^{2g+1}-2\rho x^{g+1}+x$ and $C_{2,\rho}\colon y^2=x^{2g+2}-2\rho x^{g+1}+1$ and prove that the coefficients of corresponding Cartier–Manin matrices are Legendre polynomials. As a consequence, the matrices are centrosymmetric and, therefore, it's enough to compute a half of coefficients to compute the matrix. Moreover, they are equivalent to block-diagonal matrices under transformation of the form $S^{(p)}WS^{-1}$. In the case of $\operatorname{gcd}(p,g)=1$, the matrices are monomial, and we prove that characteristic polynomial of the Frobenius endomorphism $\chi(\lambda)\pmod p$ can be found in factored form in terms of Legendre polynomials by using permutation attached to the monomial matrix. As an application of our results, we list all the possible polynomials $\chi(\lambda)\pmod p$ for the case of $\operatorname{gcd}(p,g)=1$, $g\in\{1,\dots,7\}$ and the curve $C_1$ is over $\mathbb F_p$ or $\mathbb F_{p^2}$.