## Rational Torsion Points on CM Elliptic Curves

Abbey Bourdon

September 18, 4-5 pm. Manchester 124.

Let $E$ be an elliptic curve defined over a number field $F$. By a classical theorem of Mordell and Weil, the collection of points of $E$ with coordinates in $F$ form a finitely generated abelian group. We seek to understand the subgroup of points with finite order. In particular, given a positive integer $d$, we would like to know precisely which abelian groups arise as the torsion subgroup of an elliptic curve defined over a number field of degree $d$. After providing a brief introduction to elliptic curves and summarizing prior results, I will discuss recent progress on this problem for the special class of elliptic curves with complex multiplication (CM).