The 1729 K3 Surface

We revisit the mathematics that Ramanujan developed in connection with the famous "taxi-cab" number $1729$. A study of his writings reveals that he had been studying Euler's diophantine equation $$a^3+b^3=c^3+d^3.$$ It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a $K3$ surface with Picard number $18$, one which can be used to obtain infinitely many cubic twists over $\mathbb{Q}$ with rank $\geq 2$.