Script de vérification numérique : Annulation structurelle de la matrice de Cartier–Manin de y² = xᵈ + αx + t

Let (d = 2g + 1 \geq 5) be an odd integer and (P^+(d) = d^2 - 4d + 2). Zhu proved that the family (y^2 = x^d + \alpha x + t) is generically ordinary for any prime (p > P^+(d)) not dividing the norm of (\alpha ), and observed, based on data from Katz, that this threshold cannot be lowered to (p \geq P^+(d)) in three cases. We show that when (p = P^+(d)) is prime, the (g)-th row of the Cartier–Manin matrix is identically zero: the coefficient of (x^{gp-j}) in ((x^d + Xx + Y)^{(p-1)/2}) is zero in (\mathbb{Z}[X, Y]) for (1 \leq j \leq g). It follows that no smooth curve (y^2 = x^d + \alpha x + t) is ordinary at this prime, regardless of (\alpha ) and (t). Zhu's threshold is therefore optimal for the 61 odd values of (d \leq 301) such that (P^+(d)) is prime, including the three cases from his Remark 1.5. We also show that his criterion (E < 0) is a sufficient, but not necessary, condition for the vanishing of the Hasse–Witt polynomial: at (p = P^+(d)), (E = g^2(g - 2) \geq 0) even though this polynomial is zero. The proofs are elementary; the numerical verifications are reproducible using the Python code provided separately.[ Teeoreme Annulation structurelle de la matrice de Cartier–Manin de.pdf ](url)