Counting rational points on genus one curves

Licentiate thesis, 2017

This thesis contains two papers dealing with counting problems for curves of genus one. We obtain uniform upper bounds for the number of rational points of bounded height on such curves. The main tools to study these problems are descent and various refined versions of Heath-Brown’s p-adic determinant method. In the first paper, we count rational points on smooth plane cubic curves. In the second paper, we count rational points on non-singular complete intersections of two quadrics. The methods are different for curves of small height and large height and descent is only used for curves of small height.