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Irreducibility results in dynamical moduli spaces.

Mathematics Colloquium | Niladri Patra | Oct 17th, 2023

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Consider the moduli space, $\\mathcal{M}_{d}$, of degree $d \\geq 2$ polynomials over $\\C$, with a marked critical point. Given $k \\geq 0,\\; p$ an odd prime, we show that the set $\\Sigma_{k,1,p}$ of conjugacy classes of degree $p$ polynomials, for which the marked critical point is strictly $(k,1)$-preperiodic, is an irreducible set under Zariski topology on $\\mathcal{M}_{p}$. Irreducibility of these sets was conjectured by Milnor, and has been proved for $p=3$ by Buff, Epstein and Koch.
We prove that the subspaces of $\\Sigma_{k,1,p}$, that arise by varying the ramification index of the marked critical point all the way up to the unicritical case, are all irreducible subvarieties. Finally, using the irreducibility of $\\Sigma_{k,1,p}$ we give a new and short proof of the fact that the set of all unicritical points of $\\Sigma_{k,1,p}$ form one Galois orbit under the action of absolute Galois group of $\\Q$.

About the speaker :

Niladri Patra is a Ph.D. student in Tata Institute of Fundamental Research, Mumbai. He is working under Prof. C. S. Rajan. His areas of interest include arithmetic dynamics, complex dynamics and number theory. He completed his B.Math(2017) and M.Math(2019) from Indian Statistical institute, Bangalore centre.

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