A Ramanujan Pell Elliptic K3 Surface and Primitive Five-Cube Near Misses

Abstract

We construct a trace-16 Ramanujan--Pell family of primitive positive five-cube near misses \[ a_n^3+b_n^3+c_n^3+d_n^3+e_n^3=t_n^3+(-1)^{n+1}, \] obtained from a six-cube identity of quadratic forms and the negative Pell orbit generated by \(8+\sqrt{65}\). The six coefficient sequences have rational recurrence generating functions with common reciprocal denominator \[ R(q)=1-257q-257q^2+q^3=(1+q)(1-258q+q^2). \] The quadratic identity is derived from a conic source, explaining the constants, while the Pell mechanism accounts for both the alternating error term and the denominator. The same construction yields the Mordell curve \[ E_K:\ y^2=x^3-432K(u)^2,\qquad K(u)=(1-13u+26u^2)^3+(6+182u^2)^3. \] We prove that its minimal smooth projective model is an elliptic K3 surface with geometric fibre configuration \(6IV\); over \(\mathbb Q\), the singular-fibre divisor is supported at one degree-two and one degree-four closed point. The section induced by the displayed decomposition of \(K\) has canonical height \(4/3\). The torsion groups over \(\mathbb Q(u)\), \(\mathbb Q(\sqrt{-3})(u)\), and \(\overline{\mathbb Q}(u)\) are respectively \(0\), \(\mathbb Z/3\mathbb Z\), and \(\mathbb Z/3\mathbb Z\). Over \(\mathbb Q(\sqrt{-3})(u)\), the complex multiplication orbit of the section gives an Eisenstein Mordell--Weil sublattice and a visible generated rank-16 sublattice of the geometric Neron--Severi group of discriminant \(-108\); no fullness or primitivity is claimed. We identify the remaining free Mordell--Weil problem with an explicit equivariant Hom module, exhibit an anti-symplectic reciprocal involution, and show that \(w^3=K(u)\) has a Fermat-cubic quotient. No modularity assertion is made for the recurrence functions.