p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

Abstract

For curves over the field of p-adic numbers, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally, and Vologodsky integrals with desirable number-theoretic properties. These integrals have the advantage of being insensitive to the reduction type at p, but are known to coincide with Coleman integrals in the case of good reduction. Moreover, there are practical algorithms available to compute Coleman integrals.Berkovich-Coleman and Vologodsky integrals, however, differ in general. In this thesis, we give a formula for passing between them. To do so, we use combinatorial ideas informed by tropical geometry. We also introduce algorithms for computing Berkovich-Coleman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by basic wide open spaces, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the Berkovich-Coleman integrals into Vologodsky integrals using our formula.As an application, we provide an algorithm for computing Coleman-Gross p-adic heights on Jacobians of bad reduction hyperelliptic curves, whose definition relies on Vologodsky integration. This algorithm, for instance, can be used in the quadratic Chabauty method to find rational points on hyperelliptic curves of genus at least two.Finally, I owe an enormous debt of gratitude towards my family and friends. The times I have shared with them make my life worthwhile. Teşekkürler! Some of the work in this thesis was carried out during my two visits to the Ohio State University. I thank the mathematics cluster DIAMANT for partially supporting the visit.During my PhD, I have been partially supported by the Dutch Research Council (NWO) grant 613.009.124.