Mahler's measure and elliptic curves with potential complex multiplication

Abstract

Given an elliptic curve E de ned over Q which has potential complex multiplication by the ring of integers O K of an imaginary quadratic eld K we construct a polynomial P E ∈ Z[x, ] which is a planar model of E and such that the Mahler measure m(P E

Entanglement in the family of division fields of elliptic curves with complex multiplication

“…A modern proof of this formula can be obtained using [21,Theorem 3] and [26,Theorem 12]. This is detailed in [24,Appendix A].…”

“…The rst named author uses Theorem 1.1 in [6] to study, jointly with Stevenhagen, cyclic reduction of CM elliptic curves. The second named author uses Theorem 1.1 in [24] to provide explicit planar models of CM elliptic curves de ned over Q and to compute their Mahler measure.…”