The arithmetic of polynomial dynamical pairs

Abstract: We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andr\'e-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.