A couple of conjectures in arithmetic dynamics over fields of positive characteristic

Abstract: 

The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety VV of a variety XX defined over a field KK of characteristic 00 with the orbit of a point in X(K)X(K) under an endomorphism ΦΦ of XX. The Zariski dense conjecture provides a dichotomy for any rational self-map ΦΦ of a variety XX defined over an algebraically closed field KK of characteristic 00: either there exists a point in X(K)X(K) with a well-defined Zariski dense orbit, or ΦΦ leaves invariant some non-constant rational function ff. For each one of these two conjectures we formulate an analogue in characteristic pp; in both cases, the presence of the Frobenius endomorphism in the case XX is isotrivial creates significant complications which we will explain in the case of algebraic tori.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

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Organisers:

Mike Bennett (University of British Columbia)

Philipp Habegger (University of Basel)

Alina Ostafe (UNSW Sydney)