Models of genus one curves

Abstract

In this thesis we give insight into the minimisation problem of genus one curves defined by equations other than Weierstrass equations. We are interested in genus one curves given as double covers of P1, plane cubics, or complete intersections of two quadrics in P3. By minimising such a curve we mean making the invariants associated to its defining equations as small as possible using a suitable change of coordinates. We study the non-uniqueness of minimisations of the genus one curves described above. To achieve this goal we investigate models of genus one curves over Henselian discrete valuation rings. We give geometric criteria which relate these models to the minimal proper regular models of the Jacobian elliptic curves of the genus one curves above. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Then we use these computations to count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. Finally, we consider the minimisation problem of a genus one curve defined over Q.