On Primes of Ordinary and Hodge-Witt Reduction
On Primes of Ordinary and Hodge-Witt Reduction
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关于普通质数和霍奇维特约简
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发表时间：
2016
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通讯作者：
Kirti Joshi
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作者：
Kirti Joshi
Jean-Pierre Serre在阿贝尔簇的上下文中证明，对于数域上的光滑射影簇，存在无穷个好的普通约化素数。我们考虑这个猜想和它的自然变种。特别地，我们用C. S. Rajan）Hodge-Witt约化的无穷多个素数的存在性（任何普通约化的素数也是Hodge-Witt约化的素数）。这两种学说并不等同，而是有联系的。我们证明了两者之间的精确关系，我们证明了几个结果，这些结果提供了一些证据，我们表明，素数的普通和Hodge-Witt减少可以有不同的密度。证明了我们关于Hodge- Witt猜想和阿贝尔簇的普通约化.我们在这里包括一个未发表的联合结果与C。S. Rajan（也是由Fedor Bogomolov和Yuri Zarhin用不同的方法独立建立的）关于K3曲面的普通约化素数的存在性的证明;我们的证明还表明，对于数域上的三重交换，存在一组正密度的素数，在该素数处它具有Hodge-Witt约化（这也是与C. S. Rajan）。我们给出了一些例子，包括那些费马超曲面，我们所做的所有假设都成立。
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. We consider this conjecture and its natural variants. In particular we have conjectured (with C. S. Rajan) the existence of infinitely many primes of Hodge-Witt reduction (any prime of ordinary reduction is also a prime of Hodge-Witt reduction). The two conjectures are not equivalent but are related. We prove a precise relationship between the two; we prove several results which provide some evidence for these conjectures; we show that primes of ordinary and Hodge-Witt reduction can have different densities. We prove our conjecture on Hodge- Witt and ordinary reduction for abelian varieties with complex multiplication. We include here an unpublished joint result with C. S. Rajan (also independently established by Fedor Bogomolov and Yuri Zarhin by a different method) on the existence of primes of ordinary reductions for K3 surfaces; our proof also shows that for an abelian threefold over a number field there is a set of primes of positive density at which it has Hodge-Witt reduction (this is also a joint result with C. S. Rajan). We give a number of examples including those of Fermat hypersurfaces for which all the conjectures we make hold.