Explicit 8 - Descent on Elliptic Curves by Sebastian

In this thesis I will describe an explicit method for performing an 8-descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of n-coverings, which suits the needs of higher descent. Then I will sketch the classical method of 2-descent, and the two methods that are known for doing a second 2-descent, also called 4-descent. Next I will locate the starting position for 8-descent and supplement the exposition of 4-descent by some more detailed geometric information. In Chapter 3, I will describe the construction of the descent map. It is very similar to Cassels’ method for doing a 4-descent, however there are some differences that make our situation more complicated. This descent map can be used to give an explicit description of a subset of the 8-Selmer group. However, the set we get is only close to the right one, so I call it the fake Selmer set. The methods for computing the fake Selmer set are described in Chapter 4. The elements of the fake Selmer set are algebraic objects. One would like to have them represented by geometric objects, for example by n-coverings. Finding a method for representing elements of the fake Selmer set as ncoverings is one of my main results. The description of this method is the content of Chapter 5. The most important result I achieved is the Galois cohomological interpretation of 4and 8-descent. The relation of Merriman, Siksek, and Smart’s method of 4-descent to Galois cohomology has been open for almost ten years now. The methods with which I could solve that problem could immediately be transferred to the Galois cohomology of 8-descent. I guess that these methods, which I will explain in Chapter 6, can be used for giving the Galois cohomological interpretation of most higher descent that might be developed in future. Being able to compute new examples was the driving force for developing this method of 8-descent. With the program I wrote I was able to find explicit equations for curves of order 8 in the Shafarevich-Tate group of an elliptic curve—the first of such high order—and I was able to prove the Birchand Swinnerton-Dyer conjecture at the prime 2 for several elliptic curves, where previous methods could not succeed. I will present some examples which illustrate the methods nicely. Finally, I will conclude by giving some directions for further work related to 8-descent and beyond.