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**How to use finite fields for problems concerning infinite fields.**
*(English)*
Zbl 1199.14007

Lachaud, Gilles (ed.) et al., Arithmetic, geometry, cryptography and coding theory. Proceedings of the 11th international conference, CIRM, Marseilles, France, November 5–9, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4716-9/pbk). Contemporary Mathematics 487, 183-193 (2009).

The paper under review arose from the lecture delivered at AGCT-11 (Luminy, 2007). As it is clear from the title, the author considers several situations when a statement formulated for algebraic varieties defined over an algebraically closed field \(k\) can be proved using arithmetic properties of varieties defined over finite fields. Here are some results of this kind: every involutive automorphism of the complex affine space has a fixed point (Theorem 1.1); or, more generally, an algebraic action of a finite \(p\)-group on the affine space \(\mathbb A^n_k\) over an algebraically closed field \(k\) of characteristic different from \(p\) has a fixed point (Theorem 1.2); or, generalizing Theorem 1.1 in a different way, if \(a\) and \(a'\) denote the numbers of fixed points with respect to actions of a finite group of order \(m\), acting on a \(k\)-variety \(X\) with finite number of fixed points and freely outside these points, then \(a\equiv a' \pmod m\) (Theorem 2.1).

The author describes several different approaches for proving these results. Apart from arguments based on reduction to the finite field case, which use the “spreading out” techniques (replacing a scheme over \(k\) with a scheme over a ring \(\Lambda \subset k\) finitely generated over \(\mathbb Z\) and then reducing modulo a maximal ideal of \(\Lambda\)), he provides proofs based on using cohomological techniques (Sections 7 and 8). Namely, he explains various tools such as the Cartan–Leray spectral sequence (for proving Theorem 1.1), Euler–Poincaré characteristics (for proving Theorems 1.2 and 2.1), Lefschetz’s trace formula (for proving Theorem 1.2), and Smith’s theory (for proving statements more precise than Theorem 1.2, namely, for showing that under the hypotheses of this theorem the number of fixed points is either infinite or 1).

It is worth noting that this topological approach was recently used for getting (partial) answers to a question posed in the paper under review in connection with Theorem 1.2: assuming that the action is defined over some subfield \(k_0\) of \(k\), does there exist a fixed point which is rational over \(k_0\)? Namely, H. Esnault and J. Nicaise [arXiv.math/1009.1281] showed that this question can be answered in the affirmative in the following cases: \(n\geq 2\); \(k_0\) is a henselian discretely valued field of characteristic zero with algebraically closed residue field and with residue characteristic different from \(p\); the residue field is finite of cardinality \(q\) such that \(p\) divides \(q-1\).

Apart from theorems on fixed points described above, the author presents some more instances of the principle mentioned in the title. He shows how to use reduction to finite fields in order to prove the following results: the Ax–Grothendieck theorem (every injective endomorphism of an algebraic variety defined over an algebraically closed field is an automorphism); the Lazard theorem (every algebraic group defined over an algebraically closed field whose underlying variety is isomorphic to the affine space is nilpotent); the Minkowski theorem on bounding the order of a finite subgroup of \(GL_n(\mathbb Q)\) (as well as generalizations of this bound to other reductive algebraic groups and fields); a Sylow-type theorem for \(p\)-subgroups in \(GL_n(\mathbb Q)\) of biggest order allowed by the Minkowski bound.

For the entire collection see [Zbl 1166.11003].

The author describes several different approaches for proving these results. Apart from arguments based on reduction to the finite field case, which use the “spreading out” techniques (replacing a scheme over \(k\) with a scheme over a ring \(\Lambda \subset k\) finitely generated over \(\mathbb Z\) and then reducing modulo a maximal ideal of \(\Lambda\)), he provides proofs based on using cohomological techniques (Sections 7 and 8). Namely, he explains various tools such as the Cartan–Leray spectral sequence (for proving Theorem 1.1), Euler–Poincaré characteristics (for proving Theorems 1.2 and 2.1), Lefschetz’s trace formula (for proving Theorem 1.2), and Smith’s theory (for proving statements more precise than Theorem 1.2, namely, for showing that under the hypotheses of this theorem the number of fixed points is either infinite or 1).

It is worth noting that this topological approach was recently used for getting (partial) answers to a question posed in the paper under review in connection with Theorem 1.2: assuming that the action is defined over some subfield \(k_0\) of \(k\), does there exist a fixed point which is rational over \(k_0\)? Namely, H. Esnault and J. Nicaise [arXiv.math/1009.1281] showed that this question can be answered in the affirmative in the following cases: \(n\geq 2\); \(k_0\) is a henselian discretely valued field of characteristic zero with algebraically closed residue field and with residue characteristic different from \(p\); the residue field is finite of cardinality \(q\) such that \(p\) divides \(q-1\).

Apart from theorems on fixed points described above, the author presents some more instances of the principle mentioned in the title. He shows how to use reduction to finite fields in order to prove the following results: the Ax–Grothendieck theorem (every injective endomorphism of an algebraic variety defined over an algebraically closed field is an automorphism); the Lazard theorem (every algebraic group defined over an algebraically closed field whose underlying variety is isomorphic to the affine space is nilpotent); the Minkowski theorem on bounding the order of a finite subgroup of \(GL_n(\mathbb Q)\) (as well as generalizations of this bound to other reductive algebraic groups and fields); a Sylow-type theorem for \(p\)-subgroups in \(GL_n(\mathbb Q)\) of biggest order allowed by the Minkowski bound.

For the entire collection see [Zbl 1166.11003].

Reviewer: Boris Kunyavskii (Ramat Gan)

### MSC:

14G15 | Finite ground fields in algebraic geometry |

20G30 | Linear algebraic groups over global fields and their integers |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14L30 | Group actions on varieties or schemes (quotients) |

14R20 | Group actions on affine varieties |

11E57 | Classical groups |