Common preperiodic points for quadratic polynomials

Let document fc (z) = z²+c document for document c C document. We show there exists a uniform upper bound on the number of points in document P¹ (C) document that can be preperiodic for both document f₂䃑 document and document f₂䃒 document, for any pair document c₁ = c₂ document in document C document. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in document Q 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.