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 $$\begin{aligned} a^3+b^3=c^3+d^3. \end{aligned}$$ 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 $$\mathbb {Q}$$ Q with rank $$\ge 2$$ ≥ 2 .
Highlights
We will show that Ek(T)/Q(T ) has rank two
1.1 1729 and Ramanujan Srinivasa Ramanujan is said to have possessed an uncanny memory for idiosyncratic properties of numbers
Parametrized solutions to X3 + Y 3 = Z3 + W 3 give us families of elliptic curves with rank at least two
Summary
We will show that Ek(T)/Q(T ) has rank two. To study this rank, we use a map, described in Proposition 1 of [12], from the Q(T ) points of Ek(T) to the vector space of holomorphic differentials on the auxiliary curve C/Q : S3 = k(T ). For each point P = (x(T ), y(T )) in Ek(T)(Q(T )), we define an element φP of MorQ(C, E), where E/Q is given by X3 + Y 3 = k(T ) as in the introduction, by φP(T, S) =. Is given by λ(P) = φP∗ωE, where φP∗ωE denotes the pullback via φP of the invariant differential ωE. Proposition 1 of [12] states that λ is a homomorphism with finite kernel.
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