An application of difference fields to algebraic dynamics.
Location: MSRI: Simons Auditorium
Please note that due to techincal difficulties the video was compromised for the first 11 minutes.
Let k be a field, t an indeterminate, and K=k(t). Assume given a variety V defined over K, together with a birational map f:V \to V, and fix an embedding of V into projective space. There is a notion of Weil height on points of V(K): if P is represented by (x_0:...:x_n), with the x_i in k[T] without common divisor, then h(P)=max deg(x_i).
Assume that for some large N, for all m, the set of points P of V(K) such that P, f(P),..., f^m(P) all have height < N, is Zariski dense in V.
Then we would like to conclude something about (V,f). In general, it is not true that (V,f) will "descend to k", but what we can prove, is that for some r>0, (V,f^r) has a (positive dimensional) quotient defined over k. (Here, f^r denotes the r-th iterate of f).
The hypotheses imply that for some r, (V,f^r) is dominated by an algebraic dynamics (W,g) defined over k. We translate the problem in terms of difference fields, and types (in some existentially closed difference field).
The problem boils down to the study of types over the fixed field which are internal to the fixed field. Use of definable Galois theory comes into play, as well as certain rigidity results. After giving all definitions, I will give some details on the structure of the proof.
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