Torsion Points on the Weierstrass Family of Elliptic Curves

Pairs of complex numbers $(a,b)$ outside the vanishing locus of $4a^3+27b^2$ parametrize elliptic curves in Weierstrass form $y^2=x^3+ax+b$. If we fix the first coordinate to be $1$, then the set of $(a,b)$ such that $(1,\sqrt{1+a+b})$ is torsion on the corresponding elliptic curve can be quite big. Indeed, the curve given by $1+a+b=0$ gives rise to infinitely many points of order 2. However, Masser and Zannier asked if there are only finitely $(a,b)$ where the three points $(1,*)$, $(2,*)$, and $(3,*)$ are simultaneously torsion. They had proved a finiteness result for two points on the (one parameter) Legendre family of elliptic curves.

The answer to Masser and Zannier's question is yes. I will discuss a proof in two steps. The first half uses the Pila-Wilkie Theorem as in the strategy proposed by Zannier to treat diophantine problems of Manin-Mumford type. The o-minimal structure involved is generated by restricted analytic functions. The second half is a height inequality on a class of abelian schemes which generalizes a result of Silverman.