Tropical Elliptic Curves and J-invariants

In this bachelor's thesis, the j-invariant for elliptic curves over the field of Puiseux series, P, will be discussed. For elliptic curves over any algebraically closed field, for instance C or P, we have that elliptic curves have the same j-invariant if and only if they are isomorphic to each other. Therefore every such j-invariant in P will correspond to a class of isomorphic elliptic curves. Katz, Markwig and Markwig showed in a paper of theirs that this j-invariant is related to the cycle length in the tropical world. In this thesis we show that for every j-invariant we can explicitly find an elliptic curve and its corresponding tropical counterpart such that they obey the above relation. This tropical counterpart can be found by the tropicalisation process. In this thesis we study the tropicalisation of a plane curve C over P. We take such a curve C and then apply a logarithm map to every point in this curve. This will result in the amoeba of C. This new object will contain several tentacles and an eye. By scaling the logarithm map by a factor t, we can make these tentacles and eyes arbitrarily thin in a limiting process. The limit version will be called the tropicalisation of C. This limit process can be generalised to any field k by means of a valuation. This valuation is a mapping from your field k to the field of rational numbers Q. In the limit process, the behaviour of the logarithm is determined precisely by the valuation at every point. Thus we can replace an analytic tool with a purely algebraic one, the valuation. The field of Puiseux series has a natural valuation on it, which is a generalisation of the usual degree of a polynomial. Applying this to an elliptic curve over the field P, we obtain a piece-wise linear curve known as the tropical elliptic curve. Most tropical elliptic curves will contain a bounded complex, also known as a cycle. It can be shown that the length of such a cycle is something that is shared by tropical elliptic curves which are related by morphisms. We therefore introduce a new invariant for tropical elliptic curves: j-trop. Katz, Markwig and Markwig showed in their paper that if we have an elliptic curve with v(j)<0, then under certain mild conditions v(j) will equal minus the cycle length. We show in this paper that for every j-invariant we can find an explicit elliptic curve with a tropicalisation having a cycle with length equal to -v(j). That is, if v(j)<0, then we can find an isomorphic elliptic curve with a tropicalisation having a cycle with length equal to -v(j). To show this we will use the method of reduction. The reduction of an elliptic curve over P to the field C deletes all powers of t in the coefficients and points (analogous to the reduction of an elliptic curve over Z to Zp). As in the analogous case, this requires the equation to be a so-called minimal Weierstrass equation. We show how to obtain such an equation and what properties these equations have. Afterwards we perform the actual reduction, resulting in a curve over C. This new curve might be singular over C however. An elliptic curve reducing to a singular curve is said to have bad reduction. If it reduces to a non-singular curve (which is then a smooth elliptic curve), we say that it has good reduction. These two types of reduction can be related to the valuation of the j-invariant. A curve with bad reduction will have v(j)<0. To find a curve with a proper tropicalisation, we take a curve with bad reduction and show that it has a cycle with length equal to $-v(j)$.