Elliptic curves: From Fermat to Weil (Q2713750)

The present paper deals with the development of the investigations on the group structure of elliptic curves. The author starts with Fermat's observations on rational points lying on curves defined in 2- and 3-dimensional spaces. Then he turns to works of Newton and Euler. He states ``\dots{} that Newton originated the chord and tangent method''. Concerning Euler's work his method for computing twice and thrice a point lying on an elliptic curve is presented. The last chapter is devoted to the discovery of the group structure. Here Poincaré's conjecture is mentioned, i.e. elliptic curves over \({\mathbb Q}\) form a finitely generated group. Remarks on the proof of this conjecture by Mordell and its generalization by Weil close the paper.