Torsion points and the Lattès family

We give a dynamical proof of a result of Masser and Zannier: for any $a\neq b\in{\overline{\Bbb Q}}\setminus\{0,1\}$, there are only finitely many parameters $t\in\Bbb{C}$ for which points $P_a=(a,\sqrt{a(a-1)(a-t)})$ and $P_b=(b,\sqrt{b(b-1)(b-t)})$ are both torsion on the Legendre elliptic curve $E_t=\{y^2= x(x-1)(x-t)\}$. Our method also gives the finiteness of parameters $t$ where both $P_a$ and $P_b$ have small N\'eron-Tate height. A key ingredient in the proof is an arithmetic equidistribution theorem on ${\Bbb P}^1$. For this, we prove two statements about the degree-4 Latt\`es family $f_t$ on ${\Bbb P}^1$: (1) for each $c\in{\Bbb C}(t)$, the bifurcation measure $\mu_c$ for the pair $(f_t,c)$ has continuous potential across the singular parameters $t=0,1,\infty$; and (2) for distinct points $a,b\in{\Bbb C}\setminus\{0,1\}$, the bifurcation measures $\mu_a$ and $\mu_b$ cannot coincide. Combining our methods with the result of Masser and Zannier, we extend their conclusion to points $t$ of small height also for $a,b\in{\Bbb C}(t)$.

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