Number Theory Seminar

Speaker: Thomas Gauthier (Ecole Polytechnique)

Title: Unlikely Intersections for polynomial dynamical pairs

Abstract: Let (Pt)t∈C be an algebraic family of degree d ≥ 2 polynomials parametrized by an affine curve C and let a,b : C → C be two marked points. Assume there exists infinitely many parameters t ∈ C for which a(t) and b(t) are simultaneously preperiodic under iteration of Pt. Baker and DeMarco conjectured that, under this assumption, there exists a persistent dynamical relation between the orbits {Ptn(a(t))}n≥1 and {Ptn(b(t))}n≥1. In the particular case when Pt(z) = z2 + t with t ∈ C and a, b ∈ C are constants, they proved this actually implies a2 = b2.
The aim of this talk is to present a joint work with Charles Favre, where we prove this conjecture in the case when the curve is defined over a number field. The proof follows the lines of the general strategy that Baker and DeMarco have used. Nevertheless, thee are big difficulties to overcome and every step of the proof requires a completely non-trivial input. The two main new ingredients of our proof are the following:
• a rigidity property for marked points with a smooth bifurcation locus,
• a continuity property concerning dynamical adelic metrizations of a specific ample line bundle on the curve C.

The talk will take place on Zoom. If you would like to receive the Zoom link and are not part of our current NT Seminar mailing list, please contact the organizer.