Elliptic Reduction and Meromorphic Periods of the Effective Isosceles Pedal Correspondence

Description This paper investigates the algebraic and differential-geometric structure of the effective isosceles pedal correspondence through a birational reduction to an elliptic curve. Starting from the corrected second-iterate resultant model, we establish an explicit birational transformation between the effective correspondence and a quartic curve of the form [ y^2=p_t(s). ] The resulting quartic model has geometric genus one for every parameter (t∈(1/2,1)). Its discriminant is computed exactly, yielding [ Disc_s(p_t) =-16t^2(3t^4-10t^3+59t^2-28t+4). ] A positivity argument proves that the quartic factor does not vanish on the parameter interval under consideration. This shows that the distinguished dynamical parameters (t=1/2), (t=(3+√3)/6), and (t=1) cannot simply be identified with singular fibers of this elliptic model. The paper also studies the meromorphic differential inherited from the original correspondence. Its four simple poles are identified, together with their residues. This leads to an important distinction between the rank-two cohomology of the compact elliptic curve and the rank-five cohomology associated with the corresponding curve punctured at the four poles. The Gauss–Manin framework is then discussed in both settings. The existence of the relevant differential systems is distinguished carefully from the explicit computation of their matrices and scalar Picard–Fuchs equations. The work provides exact algebraic identities and a reproducibility script. It also identifies the remaining steps required for a complete Picard–Fuchs and monodromy analysis. Main contributions: Birational reduction of the effective correspondence to a quartic elliptic model. Exact computation and factorization of the discriminant. Proof of geometric genus one on the open parameter interval. Analysis of the meromorphic differential, its poles, and residues. Comparison between compact elliptic and punctured-curve cohomology. Precise formulation of the remaining Gauss–Manin and Picard–Fuchs problems. This work builds on the preceding studies of the second-iterate resultant and the monotonicity and hyperbolicity of the Jonah multipliers.