Computational Algebra Seminar
We have computed with divisors on curves for a decade. Analogous calculations are possible using curves on surfaces (rather than points on curves) with many new phenomena. Del Pezzo surfaces are the 2-dimensional analogues of conic curves. Their divisor class groups are lattices ℤd, where the bilinear form comes from the intersection of curves on a surface. The basic geometrical theory is available in magma, and I want to explain that and then show how it relates to some arithmetic properties of surfaces. This has plenty of relations to what Martin and Steve have been doing recently, to Mike's sheaf code and to John Cremona's code for conic curves over a function field; for comparison later, Andrew is working on similar theory but with an added finite group action.
We study three-point covers of the projective line whose Galois group is either PSL2(Fq) or PGL2(Fq). We construct these covers by isolating certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL2(ℤ), Hecke triangle groups, and 19 families of Shimura curves associated to arithmetic triangle groups. We determine the field of moduli of the curves associated to these groups and thereby realize the above groups regularly as Galois groups in many cases over explicitly given abelian number fields. This is joint work with Pete L. Clark.