I'll begin a short series of talks outlining the theory of the descent algorithms implemented in Magma, whose purpose is to determine the group of rational points on a given elliptic curve.

(For elliptic curves over ℚ, we have 2-descent, 4-descent and now full-fledged 8-descent; alongside this, we have 3-descent and now 6- and 12-descent. Complementary to 4- and 8-descent, we have the Cassels-Tate pairing for 2- and 4-coverings.)

The first talk will describe 2-descent from an elementary point of view, so it will be accessible and will lay the foundations for the remaining talks.