Cubic twists of Legendre moments: cohomological descent and Frobenius trace formulas

Correction and status notice — 29 October 2026 A corrected revision of this preprint and its computational supplement is in preparation. Readers should take the following corrections and qualifications into account: • The cubic second-moment hypergeometric normalization requires a Jacobi-sum factor: M_(3,2)(p)+1 = -J(rho,phi*rho_bar)^2 H_p(A;1), with A={1/2,1/2,1/3,1/3;1,1,5/6,5/6} and compatible character conventions. This changes the stated normalization, not the underlying moment calculation. • The quartic finite-comparison argument must use primes above both 2 and 3 in its ramification set. The correction draft supplies a seven-vector test set. The quartic and sextic global conclusions still require their geometric hypotheses; finite agreement alone does not supply them. • The larger FFT-derived computations are rounded data, not independently certified exact point counts. The corrected supplement repairs the Weil validator and clarifies the limits of statistical and functional-equation diagnostics. Historical programs have not been rerun in preparing the present consolidated draft. • The conductor 1296 in the title remains a numerical candidate. Bad factors at 2 and 3, purity/Hodge identification, unrestricted irreducibility/Sato–Tate, analytic continuation and arithmetic special-value interpretations remain unproved in the present work. No elliptic-curve BSD theorem is claimed. The v1 files remain a historical version. The revision will document the changes and distinguish archived outputs from fresh verification. This notice does not announce a deposited v2, external peer review, or a completed novelty assessment. ORIGINAL V1 DESCRIPTION — 28 OCTOBER 2026 The text below preserves the original v1 description in plain-text notation. Its claims must be read subject to the correction and status notice above. Let a(λ) be the Frobenius trace of the Legendre curve y² = x(x−1)(x−λ) over F_p. We study twisted moments Σ ψ(λ) a(λ)^k for characters ψ of order 2, 3, 4, 6. (i) For the quadratic fourth moment we record a closed formula in terms of the newforms 8.4.a.a, 4.6.a.a and 8.6.a.a. It is a direct corollary of the formulas of Ahlgren and Ahlgren–Ono. (ii) For ψ of order n = 3, 4, 6 we show Re Σ ψ(λ) a(λ)^2 = −a_f(p) − p − 1, where f = 36.4.a.a, 16.4.a.a, 72.4.a.b are the forms attached by Long–Tu–Yui–Zudilin to rigid hypergeometric Calabi–Yau data. The proof uses Livné's criterion; some standard étale-cohomological inputs are sketched. The identities are verified by exact computation for p ≤ 1000, and for p ≤ 4999 when n = 3. (iii) For the cubic fourth moment we isolate a four-dimensional piece V of weight 5 and Hodge type {0,0,5,5}. Data for all p ≤ 4999 indicate that V is irreducible, not of GL₂-type, with Sato–Tate group USp(4). (iv) Descending V to Q gives a degree-four L-function. Numerically, it satisfies a functional equation with conductor 1296 = 2^4 3^4, sign +1 and gamma factor Γ_C(s)^2, to precision 1e−15. Its central value is L(W,3) ≈ 2.2413552. It is not among the L-functions currently listed in the LMFDB. We conjecture that it is the spin L-function of a non-cohomological paramodular Siegel form of level 1296 and weight Sym^4 ⊗ det^2. All statements are labelled as proved, sketched, or numerical. Code (Python/NumPy/SciPy) and data are included. AI assistance (ChatGPT, Claude) is disclosed in the preprint.