My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ **(automorphism of an algebraic variety)** defined over a finite field which does not lift to an automorphism defined over a field in characteristic zero?

Let us first consider finite fields of the form $\mathbb{F}_p$ for some prime $p$. If you take an automorphism $f$ of the affine space that is defined over $\mathbb{Q}$ and such that $p$ does not divide any of the denominators of $f$ and of $f^{-1}$, you may consider the restriction of $f$ and $f^{-1}$ to automorphisms defined over $\mathbb{F}_p$, by considering the coefficients modulo $p$, and get automorphisms defined over $\mathbb{F}_p$. Is every automorphism of $\mathbb{A}^n$ over $\mathbb{F}_p$ obtained by this way? (one can do similar constructions for other finite fields and other fields of caracteristic zero).

For each prime $p$, every linear automorphism of $\mathbb{A}^n$, given by $(x_1,\ldots,x_n)\mapsto (a_{11}x_1+\cdots +a_{1n}x_n,\ldots,a_{n1}x_1+\cdots +a_{nn}x_n)$ for some matrix $(a_{ij})\in \mathrm{GL}_n(\mathbb{F}_p)$ comes from an element of $\mathrm{GL}_n(\mathbb{Q})$ whose entries are integers and whose determinant is not divisible by $p$. Its inverse in $\mathrm{GL}_n(\mathbb{Q})$ has thus all denominators that are not multiple of $p$. Hence, every linear automorphism of $\mathbb{A}^n$ over $\mathbb{F}_p$ comes from a linear automorphism of $\mathbb{A}^n$.

Similarly, if you take an elementary automorphism of $\mathbb{A}^n$ defined over $\mathbb{F}_p$, i.e. given by $$(x_1,\dots,x_n)\mapsto (x_1+a(x_2,x_3,\ldots,x_n),x_2,\ldots,x_n)$$ for some polynomial $a\in \mathbb{F}_p[x_2,\ldots,x_n]$, it comes from an automorphism of $\mathbb{A}^n$ defined over $\mathbb{Q}$ (here even over $\mathbb{Z}$). In particular, every "tame" automorphism (generated by linear and elementary automorphisms) of $\mathbb{A}^n$ over $\mathbb{F}_p$ comes from a tame automorphism of $\mathbb{A}^n$ over $\mathbb{Q}$. In dimension $n=1$ and $n=2$, every automorphism is tame, so every automorphism of $\mathbb{A}^n$ over $\mathbb{F}_p$ comes from an automorphism defined over $\mathbb{Q}$. The question is then more interesting for $n\ge 3$. It would give examples of non-tame automorphisms, a fact not know until now in positive characteristic (see the famous article Shestakov and Umirbaev - The tame and the wild automorphisms of polynomial rings in three variables for examples in characteristic zero).

This seems hard to answer in general, but can we then replace the variety $\mathbb{A}^n$ by an affine (or projective) algebraic variety, defined over $\mathbb{F}_p$ and find some automorphisms over $\mathbb{F}_p$ that do not come from automorphisms in characteristic zero?

EDIT: Will Sawin gave a nice answer for elliptic curves, where some given automorphism do not lift to $\mathbb{Q}$. Are there examples where some given automorphisms do not lift to any field of caracteristic zero?

isa restriction (suitably interpreted) of an automorphism over a field of characteristic zero, namely $\mathbb{Q}$, which is what you originally wrote. So it would be good to clarify what you mean here.) $\endgroup$31more comments