The 1729 K3 Surface

TL;DR

This paper explores Ramanujan's work on the number 1729, revealing his early discovery of a K3 surface with significant implications for arithmetic geometry and number theory.

Contribution

It uncovers Ramanujan's anticipation of deep structures in arithmetic geometry, specifically identifying a K3 surface with Picard number 18 linked to the taxicab number 1729.

Findings

Identifies a K3 surface associated with 1729.

Shows the surface can generate infinitely many cubic twists with rank ≥ 2.

Connects Ramanujan's work to modern arithmetic geometry phenomena.

Abstract

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 $\geq 2$.