Mahler's measure and elliptic curves with potential complex multiplication

Abstract

Given an elliptic curve $E$ defined over $\mathbb{Q}$ which has potential complex multiplication by the ring of integers ${\mathcal{O}}_K$ of an imaginary quadratic field $K$ we construct a polynomial $P_E\in \mathbb{Z}[x,y]$ which is a planar model of $E$ and such that the Mahler measure $m(P_E)\in \mathbb{R}$ is related to the special value of the $L$-function $L(E,s)$ at $s=2$.

24 pages. Comments are very welcome!