Descent on elliptic curves

TL;DR

This paper discusses the method of descent on elliptic curves over number fields, focusing on computing Selmer groups and explicit representations to understand the Mordell-Weil group and Shafarevich-Tate group.

Contribution

It provides a detailed approach to explicitly computing Selmer groups and representing their elements as covering spaces, facilitating rational point searches and higher descents.

Findings

Explicit methods for computing Selmer groups

Representation of Selmer elements as covering spaces

Enhanced techniques for higher descent computations

Abstract

Let E be an elliptic curve over Q (or, more generally, a number field). Then on the one hand, we have the finitely generated abelian group E(Q), on the other hand, there is the Shafarevich-Tate group Sha(Q,E). Descent is a general method of getting information on both of these objects - ideally complete information on the Mordell-Weil group E(Q), and usually partial information on Sha(Q,E). What descent does is to compute (for a given n > 1) the n-Selmer group Sel^(n)(Q,E); it sits in an exact sequence 0 --> E(Q)/nE(Q) --> Sel^(n)(Q,E) --> Sha(Q,E)[n] --> 0 and thus contains combined information on E(Q) and Sha(Q,E). The main problem I want to discuss in this ``short course'' is how to actually do this explicitly, with some emphasis on obtaining representations of the elements of the Selmer group as explicit covering spaces of E. These explicit representations are useful in two respects - they allow a search for rational points (if successful, this proves that the element is in the image of the left hand map above), and they provide the starting point for performing ``higher'' descents (e.g., extending a p-descent computation to a p^2-descent computation).