The subject of elliptic curves has played an important role in the development of the theory of numbers. It has been especially significant in connection with studies on the rational solutions of Diophantine equations and, as indicated in Section 8.4, it has led most famously to a proof of Fermat’s last theorem. A brief discussion of elliptic curves was given in Section 8.3 and we broaden this now into a deeper and more advanced exposition.
In a refined geometrical sense, an elliptic curve E over the rationals is a smooth curve of genus 1 with a specified rational point. By the Riemann–Roch theorem, E is then ...