math.NT — Number Theory
BRUDNO’S FAMILIES OF EQUAL SUMS OF SIXTH POWERS AND BREMNER’S K3 SURFACE
Brudno (1970, 1976) and Brudno–Kaplansky (1974) found an elliptic family and a degree-four polynomial family of solutions of x^6 + y^6 + z^6 = u^6 + v^6 + w^6. We explain both in terms of the K3 surface V ⊂P5, the complete intersection of three quadrics studied by Bremner (1981) and Kuwata (2007). The Brudno–Kaplansky curve is Cremona’s curve 480f4 and is the fibre at λ=∞ of the elliptic pencil on V that Bremner calls the Brudno pencil; the curve z^2 = x^4 + 38x^2y^2 + y^4 of 1970 is 480f1, and the classical recursions of Lebesgue and Desboves that Brudno iterated are the multiplication maps [2] and [−3] on it. Brudno’s 1976 quartic is the unique rational curve of degree four on V (Bremner’s Theorem 4.4) and a section of the Brudno pencil; its parameter is the fibre coordinate λ, and the involution Brudno noticed is a linear symmetry of V. Using the Mordell–Weil lattice of the pencil we prove a closed formula for the degree of a section, deg C_S = 2⟨S−γ/4, S−γ/4⟩, enumerate all sections of degree ≤12 (four of them new, of degrees 7,10,11,11), and enumerate, by a lattice computation, all smooth rational curves of degree ≤12 on V over Q(i) and over Q, reproducing Bremner’s theorems and testing his conjecture. A census of the 74 primitive solutions with all terms ≤ 1343 identifies which of them lie on V, on known curves, on sections, and on fibres of jumping rank, and records some phenomena off V. In the second part we compute equations and parametrisations of the 48 new curves that are not sections and show that seven of the eight census solutions on no known curve lie on them, leaving a single unexplained point; we determine the geometry over C of the sextic fourfold M6, of the threefold W= M6 ∩M2 (which has exactly 710 nodes and defect 95) and of V (transcendental lattice U(2) ⊕⟨12⟩, zeta function through the newform 320f2), and discuss in what sense the geometry drives the arithmetic; and we show that Bremner’s conjecture follows, with non-explicit finite sets, from the theorems of Sterk and Bright–Logan–van Luijk, describe how to make it explicit (Borcherds–Shimada), and run the first step of Bremner’s own cone procedure, which identifies the degree-8 pencil involutions that the proof requires.