On Primes of Ordinary and Hodge-Witt Reduction

Abstract

Jean-Pierre Serre has conjectured Conj. 3.2.1, in the context of abelian varieties, that there are infinitely primes of good ordinary reduction for a smooth, projective variety over a number field. We prove this conjecture for K3 surfaces Thm 3.3.1 (this is unpublished joint result with C. S. Rajan which was also independently established by Fedor Bogomolov and Yuri Zarhin by a different method). Any prime of ordinary reduction is also a prime of Hodge-Witt reduction but not conversely. Conj. 4.1.2 (of Joshi-Rajan) asserts the existence of infinitely many primes of Hodge-Witt reduction. The two conjectures are related but not equivalent (Thm 4.1.4). We prove the latter conjecture for abelian threefolds Thm 4.3.1 (joint with C. S. Rajan), and smooth Fano threefolds (Thm 4.4.9) and in Thm 4.5.1 for abelian varieties with complex multiplication. We show that the set of primes of ordinary and Hodge-Witt reduction can have different densities (Thm 4.6.11, Example 7.3.1). Thm 5.1.1, Thm 5.1.3 establish the existence of ordinary reductions for certain wonderful compactifications and a large class of configuration spaces. Thm 5.2.2 deals with the relationship between Conj. 4.1.2, Conj. 3.2.1 and the Musta\c{t}\v{a}-Srinivas conjectures and Thm 5.2.3 establishes this conjecture in some cases. Section 6 deals with existence of primes of non Hodge-Witt reductions and establishes this in a number of cases (Theorem 6.4.3), and Thm 7.2.1 asserts that for Fermat hypersurfaces of dimension $\geq$ 3 and degrees $\geq$ 211, at least 98% of the primes are of non Hodge-Witt (and hence non-ordinary) reduction and in the degree $\rightarrow\infty$ limit, almost all primes are of non Hodge-Witt (and hence non-ordinary) reduction. Section 7 provides a number of examples illustrating densities of ordinary, Hodge-Witt and non Hodge-Witt reductions.