On Primes of Ordinary and Hodge-Witt Reduction
On Primes of Ordinary and Hodge-Witt Reduction
Jean-Pierre Serre has conjectured, in the context of abelian varieties, that there are infinitely primes of good ordinary reduction for a smooth, projective variety over a number field. Any prime of ordinary reduction is also a prime of Hodge-Witt reduction but not conversely. More generally, Conjecture 4.1.2 asserts the existence of infinitely many primes of Hodge-Witt reduction. The two conjectures are related but not equivalent (Theorem 4.1.4). We prove this conjecture for abelian threefolds Theorem 4.3.1 (joint with C. S. Rajan), and smooth Fano threefolds (Theorem 4.4.9) and in Theorem 4.5.4 for all simple abelian varieties with complex multiplication (this implies a conjecture of MustaÅ£Ä-Srinivas for such abelian varieties). We show that the set of primes of ordinary and Hodge-Witt reduction can have different densities (Theorem 4.5.4, Example 6.3.1). Theorem 6.2.1 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. We include here an unpublished joint result Theorem 3.3.1 with C. S. Rajan.
Kirti Joshi
数学
Kirti Joshi.On Primes of Ordinary and Hodge-Witt Reduction[EB/OL].(2025-06-27)[2025-07-16].https://arxiv.org/abs/1603.09404.点此复制
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