Quadratic Rational Maps with a Five-Periodic Critical Point

The paper proves that Per5(0)cm is an elliptic curve C5 over Q, specifically 17a4, realized as the smooth plane cubic x^3 + y^2 z − 3xyz + xz^2 = 0 via a Hurwitz-space/COR compactification, and enumerates its boundary to show there are exactly 10 punctures, the four Q-rational points plus six algebraic points of infinite order; it also identifies exactly 20 PCF points where both critical points lie on the same 5-cycle, all of infinite order. These statements appear explicitly as Theorem 1.1, Theorem 1.2, and Corollary 1.3, and the cubic model with the linear change of projective coordinates to 17a4 is given in Section 5 (including the map [x:y:z] -> [-x+z : -x+y+2z : z]) . The paper also details the boundary analyses establishing the 10 punctures and the 20 interior PCF points (C5 ∩ RI,…,RIV) with transversality, clarifying that the latter are not punctures . The candidate solution follows the same route and matches these results, even using the same cubic and coordinate change, and correctly deduces Per5(0)cm(Q)=∅ (no quadratic Q-map with a 5-periodic critical point) .