Algorithmic and Experimental Arithmetic Geometry

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This proposal contains two projects related in various ways to the arithmetic of algebraic varieties and to the study of their rational points in particular. The aim of the first project is to devise and implement an algorithm that produces a nice set of equations for a given variety; this is important for doing further computations with it. In many cases, further computations would be infeasible without such a nice model. The second project is more specifically concerned with rational points on curves of higher genus. It is known that on each such curve there can be only finitely many rational points, but so far there is no algorithm that determines this set explicitly. We will improve and extend existing algorithms covering certain cases. We will then use them to gather statistical information from many curves to get some more precise idea on the behavior of their rational points. On the other hand, we plan to study possible approaches to a general algorithm that finds the set of rational points on any given curve.