Variations on a theme of Minkowski and Serre
A result of Serre says that if an automorphism of finite order of a semi-abelian variety induces the identity on the scheme-theoretic kernel of multiplication by \(n\), then it is the identity if \(n\geq 3\), and its square is the identity if \(n=2\). This result is useful in the study of moduli spaces of abelian varieties with full level \(n\geq 3\) structure. Serre's lemma relies on the fact that if \(n\geq 3\) then every root of unity which is congruent to 1 modulo \(n\) is 1. This idea dates back to Minkowski, who proved that an integral matrix of finite order, which is congruent to the identity modulo \(n\), is the identity if \(n\geq 3\). In this paper we give generalizations and variations of the Serre-Minkowski results. For example, if \(k\) and \(n\) are positive integers, \(\alpha\) is a root of unity in an integral domain \({\mathfrak O}\) of characteristic zero, \((\alpha-1)^k\) is divisible by \(n\), and no prime divisor of \(n\) is a unit in \({\mathfrak O}\), then \(\alpha=1\) if \(n\) is outside of a certain finite set of prime powers determined by \(k\). (The case \(k=1\) is the Serre-Minkowski case.) The proof relies on the arithmetic of cyclotomic integers. Although the ideas are simple, the result is useful and does not seem to have been noticed before. We give best-possible restrictions on \(\alpha\) when \(n\) is in the finite exceptional set. We have results for rings that are not necessarily integral domains (section 4), and we have applications to matrix rings, eigenvalues, projective modules, and quasi-unipotent elements (see sections 6 and 7). We provide additional information when \(n\) is in the exceptional finite set and give examples which show that our results are sharp. In section 8 we give applications to endomorphisms of semi-abelian varieties which generalize Serre's result. In section 9 we give applications to compatible systems of \(\ell\)-adic representations and the cohomology of projective varieties.
scientific article; zbMATH DE number 920036