Hyperelliptic curves, Cartier - Manin matrices and Legendre polynomials

We investigate the hyperelliptic curves of the form C₁ : y² = x^2g+1 + ax^g+1 + bx and C₂ : y² = x^2g+2 + ax^g+1 + b over the finite field F_q, q = pⁿ, p > 2. We transform these curves to the form C_1)P : y² = x^2g+1 - 2px^g+1 + x and C₂,_p : y² = x^2g+² - 2px^g+1 + 1 and prove that the coefficients of corresponding Cartier - Manin matrices are Legendre polynomials. As a consequence, the matrices are centrosym-metric 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 gcd(p,g) = 1, the matrices are monomial, and we prove that characteristic polynomial of the Frobenius endomorphism x(^) (mod 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 x(A) (mod p) for the case of gcd(p,g) = 1, g E {1,..., 7} and the curve C₁ is over F_p or F_p2.

Keywords

Authors

References