The 1729 \(K3\) surface (Q2520546)

This paper studies the \(1729\) \(K3\) surface associated to Euler's cubic Diophantine equation \(a^3+b^3=c^3+d^3\) considered by Ramanujan. A geometric interpretation of Euler's equation yields the equation \((\star)\quad X^3+Y^3=k(T)\), where \(k(T)=63(3T^2-3T+1)(T^2+T+1)(T^2-3T+3)\). The main result of this short note is formulated as follows. {Theorem}: The smooth minimal surface associated to \((\star)\) is an elliptic \(K3\) surface with Picard number \(18\) over \(\overline{\mathbb{Q}}\). Regarding \((\star)\) as an elliptic curve \(E_{k(T)}\) over the function field \({\mathbb{Q}}(T)\), it is shown that it has rank \(2\), and that for infinitely many \(t\in{\mathbb{Q}}\), \(E_{k(t)}\) has rank \(\geq 2\) over \({\mathbb{Q}}\).