This theorem was the start of the whole set of investigations on elliptic curves, and indeed on arithmetic geometry. Andre Weil in his Ph. D. thesis created the subject of arithmetic of algebraic varieties and Galois cohomology, to prove his strengthened version of this theorem and to understand Mordell's calculations. I also believe that for him the motivation to re-write the foundations of algebraic geometry was also motivated by the desire to give the Mordell-Weil theorem a cleaner form, thoughs the officially stated motivation is for putting his proof of Riemann hypothesis for function fields over finite fields on a firm ground. And, the subject grew, flowered, through greats like Grothendieck, and one must remark the work of Faltings on Mordell conjecture on the same direction proposed in the same paper, which could be proved only so many years later, after Weil failed in his Ph. D. time. Indeed, Fermat's last theorem proof also belongs to the same subject. Looking back, rejection of Mordell's groundbreaking paper is so unbelievable.

The details of Weil's work can be found in his autobiography, "Apprenticeship of a mathematician"..

Excerpt from source:

Mordell submitted his subsequent work on indeterminate equations of the third and fourth degree when he became a candidate for a Fellowship at St John's College, but he was not successful. His paper on this topic was rejected for publication by the London Mathematical Society but accepted by the Quarterly Journal. Mordell was bitterly disappointed at the way his paper had been received. He wrote at the time on an offprint of the paper:-

This paper was originally sent for publication to the L.M.S. in 1913. It was rejected ... Indeterminate equations have never been very popular in England (except perhaps in the 17th and 18th centuries); though they have been the subject of many papers by most of the greatest mathematicians in the world: and hosts of lesser ones ...

Such results as [those in the paper] ... marks the greatest advance in the theory of indeterminate equations of the 3rd and 4th degrees since the time of Fermat; and it is all the more remarkable that it can be proved by quite elementary methods. ... We trust that the author may be pardoned for speaking thus of his results. But the history of this paper has shown him that in his estimation, it has not been properly appreciated by English mathematicians.

The Mordell-Weil theorem, when submitted by Mordell to the London mathematical society's journal, was rejected.

The Mordell-Weil theorem, when submitted by Mordell to the London mathematical society's journal, was rejected.

The Mordell-Weil theorem, when submitted by Mordell to the London mathematical society's journal, was rejected.