Proofs in the same vein as Ax-Grothendieck

Question

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the corresponding wikipedia article:

This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic $0$ translate into algebraic relations over finite fields with large characteristic.Thus, one can use the arithmetic of finite fields to prove a statement about $\mathbb{C}$ even though there is no non-trivial homomorphism from any finite field to $\mathbb{C}$.The proof thus uses model theoretic principles to prove an elementary statement about polynomials.The proof for the general case uses a similar method.

Edit(after the comment by Angelo to Martin's answer below): answers along the line of "using finite fields for problems concerning infinite fields" are also welcome.

Answer 1

There is a fantastic paper by Serre: How to use finite fields for problems concerning infinite fields, arXiv.

Answer 2

Mori's bend and break technique which is used to show the existence of rational curves in higher dimensional varieties, is a famous example of this.

Answer 3

A not so famous example is that Mazorchuk and I used model theory to prove that if a finite semigroup S has a faithful representation of degree d over $\mathbb C$ then it must have a faithful representation of degree d over some finite field. The proof is that there is a first order sentence saying that S has a faithful representation of degree d. Thus it follows by model theory that if S has a faithful representation of degree d over $\mathbb C$ then S has a faithful representation over an algebraically closed field of characteristic p. But the image of S under the rep will live in matrices over a finite field.

This allowed us to use results about finite matrix monoids to obtain lower bounds for the minimum faithful degree of a complex representation over $\mathbb C$.

Answer 4

Since in your edit you say that you don't insist on the use of model theory:

A very nice and famous example is Deligne-Illusie's proof of the degeneration of the Hodge to deRahm spectral sequence. Deligne, P.; Illusie, L. Relèvements modulo p^2 et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247–270.

Answer 5

See the papers:"Transfer Principle for the Fundamental Lemma" by Loeser, Cluckers and Hales http://www.math.jussieu.fr/~loeser/transfer_fl_2010_10_14.pdf (with an application to the fundamental lemma), "Constructible exponential functions, motivic Fourier transform and transfer principle" by Loeser and Cluckers http://www.math.jussieu.fr/~loeser/aom_cl.pdf and "Transfer principles for integrability and boundedness conditions for motivic exponential functions" by R. Cluckers, J. Gordon, I. Halupczok http://arxiv.org/pdf/1111.4405v1.pdf which all use and devleop transfer principle from finite characteristic to zero characteristic and vice versa.

Answer 6

I saw a talk two weeks ago by Michel Brion that uses reduction to finite fields to prove that every algebraic semigroup (not necessarily affine) has an idempotent. Over a finite field it is clear because finite semigroups have idempotents.