Common preperiodic points for quadratic polynomials

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ f_c(z) = z^2+c $\end{document}</tex-math></inline-formula> for id="M2">\begin{document}$ c \in {\mathbb C} $\end{document}</tex-math></inline-formula>. We show there exists a uniform upper bound on the number of points in id="M3">\begin{document}$ P}^1( C}) that can be preperiodic both id="M4">\begin{document}$ f_{c_1} and id="M5">\begin{document}$ f_{c_2} $\end{document}</tex-math></inline-formula>, any pair id="M6">\begin{document}$ c_1\not c_2 id="M7">\begin{document}$ The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when parameters lie id="M8">\begin{document}$ \overline{\mathbb{Q}} building quantitative equidistribution theorem Favre Rivera-Letelier, use distortion theorems complex analysis to control size intersection distinct Julia sets. proofs are effective, provide explicit constants each results.</p>