The main focus of this thesis is to study the equation A(x4+y4+z4+w4)+Bxyzw+C(x2y2+z2w2)+D(x2z2+y2w2)+E(x2w2+y2z2) = 0. To do so, we view this equation as a family of quartic K3 surfaces in P3[x,y,z,w], parametrised by points [A,B,C,D,E] E P4. We pursue two directions. First we look at 320 conics that such a K3 surface contains. In particular, we explore the field of definition of these 320 conics and the Monodromy group linked to these conics. In the other direction we explore the quartic K3 surfaces which contain lines. We list all subfamilies of K3 surfaces for which a very general member contains 8, 16, 24, 32 or 48 lines. We combine the two directions, by using the lines and conics found, to explore the Picard group of the various families found. In particular, not only do we work out the Picard rank of a very general member of a family, but we also decompose the Picard lattice into known lattices. This thesis has a secondary focus on hyperelliptic curves of genus two with complex multiplication (CM). At the end of the thesis, we design an algorithm to find CM curves of genus two which are defined over quadratic extensions of the rationals. To do so we also develop an algorithm which makes the coefficients of a curve smaller.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:693224 |
| Date | January 2016 |
| Creators | Bouyer, Florian |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/80893/ |
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