According to a theorem of Mordell, the set of rational points on an elliptic curve form a finitely generated commutative group. The rank of an elliptic curve over a number field is defined to be the rank of the subgroup of elements of infinite order.

Already Pierre de Fermat succeeded in proving that the rank of one particular elliptic curve is 0, which enabled him to prove that x⁴+y⁴=z⁴ does not have positive integral solutions.

Although a great deal of research has been done on the subject, the only unconditional method of getting upper bounds on the rank of an elliptic curve is essentially Fermat's method of infinite descent.

In this talk, I will discuss how one can perform this construction in a more modern language. I will be following an approach taken by Cassels, Flynn and Schaefer rather than the style used by Birch, Cremona and Swynnerton-Dyer. This approach takes better advantage of the interplay of several groups involved and is sometimes referred to as the "number field method" rather than the "homogeneous spaces method".

It has the advantage of being less susceptible to combinatorial explosion and of being readily generalisable to number fields other than the rationals.

If time permits, I will show how one can use descent techniques over quadratic extensions to sharpen the rank bound obtained from a descent over the base field. This relates closely to a phenomenon generally referred to as "visualising Sha"