Common preperiodic points for quadratic polynomials

Let \begin{document}$ f_c(z) = z^2+c $\end{document} for \begin{document}$ c \in {\mathbb C} $\end{document}. We show there exists a uniform upper bound on the number of points in \begin{document}$ {\mathbb P}^1( {\mathbb C}) $\end{document} that can be preperiodic for both \begin{document}$ f_{c_1} $\end{document} and \begin{document}$ f_{c_2} $\end{document}, for any pair \begin{document}$ c_1\not = c_2 $\end{document} in \begin{document}$ {\mathbb C} $\end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in \begin{document}$ \overline{\mathbb{Q}} $\end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.