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Let $f_c(z) = z^2+c$ for $c \in \mathbb{C}$. We show there exists a uniformbound on the number of points in $\mathbb{P}^1(\mathbb{C})$ that can bepreperiodic for both $f_{c_1}$ and $f_{c_2}$ with $c_1\not= c_2$ in$\mathbb{C}$. The proof combines arithmetic ingredients with complex-analytic;we estimate an adelic energy pairing when the parameters lie in$\bar{\mathbb{Q}}$, building on the quantitative arithmetic equidistributiontheorem of Favre and Rivera-Letelier, and we use distortion theorems in complexanalysis to control the size of the intersection of distinct Julia sets. Theproof is effective, and we provide explicit constants for each of the results.

Common preperiodic points for quadratic polynomials