Canonical heights and vector heights in families
Alex Carney - University of Rochester
Canonical heights are a standard tool of arithmetic dynamics over global fields. When studying families of dynamical systems, or systems over larger fields, however, there are significant geometric obstacles to constructing canonical heights. Either Northcott fails to hold, or the construction requires a model for the family with such strict properties (good reduction, minimality, extensions of rational maps,…) that such a model is unlikely to exist outside of very special settings. Instead, I’ll show how to resolve this problem using vector-valued heights, first introduced in characteristic zero by Yuan and Zhang. These generalize R-valued heights, produce canonical heights for any polarized dynamical system, and exhibit Northcott finiteness conditional on a strong non-isotriviality condition, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. This is achieved by working simultaneously over a system of models with much weaker requirements. As part of ongoing work, I’ll show how these arithmetic methods can produce results that hold over any field, and discuss how this can extend to quasi-projective varieties as well as projective varieties.
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