Abstract
For curves over the field of padic numbers, there are two notions of padic integration: BerkovichColeman integrals which can be performed locally, and Vologodsky integrals with desirable numbertheoretic 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.
BerkovichColeman 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 BerkovichColeman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by basic wide open spaces, we reduce the computation of BerkovichColeman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the BerkovichColeman integrals into Vologodsky integrals using our formula.
As an application, we provide an algorithm for computing ColemanGross padic 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.
BerkovichColeman 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 BerkovichColeman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by basic wide open spaces, we reduce the computation of BerkovichColeman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the BerkovichColeman integrals into Vologodsky integrals using our formula.
As an application, we provide an algorithm for computing ColemanGross padic 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.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  1Oct2021 
Place of Publication  [Groningen] 
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DOIs  
Publication status  Published  2021 