Properties of elliptic curves with a point of order n over number fields of degree d

Let $E$ be an elliptic curve over a number field $K$ of degree $d$ with a point of order $n$. What properties is $E$ forced to have? For appropriate choices of $(n,d)$ the answer can be: 1) have even rank; 2) be a $\mathbb{Q}$-curve; 3) the field $K$ over which $E$ is defined has to be of certain type; 4) be a base change of an elliptic curve defined over $\mathbb{Q}$; 5) have Tamagawa numbers of specific form. In this talk we will sketch why these kinds of results are true and show that they come from the geometric properties of modular curves and maps between modular curves and their moduli interpretations. We will describe in a bit more detail new results regarding Tamagawa numbers of elliptic curves with a point of order $13$ over quadratic fields.Non UBCUnreviewedAuthor affiliation: University of ZagrebFacult