the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922.
It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.
The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century.
The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group
It can be proved by direct analysis of the doubling of a point on E. Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation[1] published in 1928.
More abstract methods were required, to carry out a proof with the same basic structure.
The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of
Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates.
Both halves of the proof have been improved significantly by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms).