Here you can see my PhD thesis, Computations on Diagonal Quartic Surfaces, for which I studied at the University of Cambridge under Sir Peter Swinnerton-Dyer. My thesis was examined on 11th November 2002 by Prof. Victor Flynn (Liverpool) and Dr Tom Fisher (Cambridge).
Erratum: Lemma 4.5 is incorrect. The proof given relies on Lang's theorem, the reference cited being Milne's lecture notes: in fact, the statement there is missing the hypothesis that the ring should be one-dimensional, and Lang's theorem does not apply here. The following corollary, 4.6, and subsequent statements, are nonetheless true; but the proof relies on the purity theorem for the Brauer group.
Various bits of data and implementations of algorithms described in my thesis may be downloaded from my diagonal quartic surfaces page.