Superelliptic curves: hidden vanishings of the Cartier–Manin matrix

Hidden vanishings for y^n = x^d + σx^r + 1 over F_p Research code and notes on the Cartier–Manin matrix of the superelliptic family y^n = x^d + σx^r + 1 over F_p (p ∤ n, gcd(d,n) = 1). The case n = 2 (hyperelliptic) is the control case. ## What is here - Block decomposition of the Cartier matrix, indexed by k ↦ k·p⁻¹ mod n, and the p-rank as the stabilised (Fitting) rank of the block cycles. - "Hidden vanishings": cases where the support of the return map Φ admits a perfect matching, yet rk Φ < g₀ for every smooth σ ∈ F_p. - Hand-checkable proofs: kernel of a block, corank formula, duality k ↔ n−k. - Scripts reproducing every numerical claim, with saved outputs. ## Status of results Statements are labelled [D] proved, [V] numerically verified, [O] open. No claim of novelty is made: the literature has not been exhaustively checked. ## Documents - `vanissements_yn_theory_en.pdf`: main article (v4.3) - `proofs_blocks.pdf`: complete proofs note ## Reproduce `bash run_checks.sh`: pure Python, no dependencies for the main checks (Sage optional for two scripts). ## Author Mohamed Sofiene, independent researcher: [email protected] Technical Head, Office National de Télédiffusion de Tunisie (ONT); B.Sc. IoT and Communication Networks; B.Sc. Electrical Engineering, University of Tunis. ## License MIT