The density of rational points and invariants of genus one curves

Abstract: The present thesis contains three papers dealing with two arithmetic problems on curves of genus one, which are closely related to elliptic curves. The first problem is to study the density of rational points presented in Papers I and II. We give uniform upper bounds for the number of rational points of bounded height on smooth curves of genus one given by ternary cubics or complete intersections of two quadratic surfaces. The main tools used in these two papers are descent on elliptic curves and determinant methods. While working with the rational points counting problem, one need to deal with the smoothness of geometric objects and the bad reduction of polynomials. To characterize these properties, there is a classical object called the discriminant which naturally appears. The above discriminant gives an inspiration to the study of the next problem in the thesis concerning invariants of models of genus one curves presented in Paper III. Here an invariant of a genus one curve is a polynomial in the coefficients of the model defining the curve that is stable under certain linear transformations. The discriminant is a classical example of an invariant. Besides that, there are two more important invariants which generate the ring of invariants of genus one models over a field. Fisher considered these invariants over the field of rational numbers and normalized them such that they are moreover defined over the integers. We provide an alternative way to express these normalized invariants using a natural connection to modular forms. In the case of the discriminant of ternary cubics over the complex numbers, we also present another approach using determinantal representations. This latter approach produces a natural connection to theta functions. The common idea in the thesis is to link a smooth genus one curve to a Weierstrass form, which is a more well-understood object.