Multiplication polynomials and relative Manin-Mumford

Abstract

After the introduction we prove in chapter 2 that the resultant of the standard multiplication polynomials $A_n,B_n$ of an elliptic curve in the form $y^2 = x^3+ax+b$ is $(16\Delta)^{{n^2(n^2-1) \over 6}}$, where $\Delta=-(4a^3+27b^2)$ is the discriminant of the curve. In the appendix we give an application to good reduction of an associated Latt\`es map. We also prove a similar result for the discriminant of the largest square free factor of $B_n$. In the third chapter we prove a Manin-Mumford type result for additive extensions of elliptic families over the field of all complex numbers. We show in the appendix that there are finiteness consequences for Pell's equation over polynomial rings and integration in elementary terms. Our work can be made effective because we use counting results only for analytic curves.