Sometimes theorems over an infinite field can be proved by proving associated results on “reductions” over finite fields.
As the title indicates, the purpose of the present lecture is to show how to use finite fields for solving problems on infinite fields. This can be done on two different levels: the elementary one uses only the fact that most algebraic geometry statements involve only finitely many data, hence come from geometry over a finitely generated ring, and the residue fields of such a ring are finite; the examples we give in §§1-4 are of that type. A different level consists in using Chebotarev’s density theorem and its variants, in order to obtain results over non-algebraically closed fields; we give such examples in §§5-6. The last two sections were only briefly mentioned in the actual lecture; they explain how cohomology (especially the étale one) can be used instead of finite fields; the proofs are more sophisticated, but the results have a wider range.
Michael Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005) 271–318 numdam
J. S. Milne, The point on a Shimura variety modulo a prime of good reduction, in: The zeta functions of Picard modular surfaces, 1992 pdf
Marius van der Put?, Reduction modulo p of differential equations, Indagationes mathematicae N.S. pdf
An application to Jacobian conjecture,
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