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

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: University of Zagreb

2017-11-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63722

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/