Editing Proof by infinite descent (section)

== Number theory==
In the [[number theory]] of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of [[algebraic number theory]] and the study of [[L-function]]s. The structural result of [[Mordell]], that the rational points on an elliptic curve ''E'' form a [[finitely-generated abelian group]], used an infinite descent argument based on ''E''/2''E'' in Fermat's style.

To extend this to the case of an [[abelian variety]] ''A'', [[André Weil]] had to make more explicit the way of quantifying the size of a solution, by means of a [[height function]] – a concept that became foundational. To show that ''A''(''Q'')/2''A''(''Q'') is finite, which is certainly a necessary condition for the finite generation of the group ''A''(''Q'') of rational points of ''A'', one must do calculations in what later was recognised as [[Galois cohomology]]. In this way, abstractly-defined cohomology groups in the theory become identified with ''descents'' in the tradition of Fermat. The [[Mordell–Weil theorem]] was at the start of what later became a very extensive theory.