Tate conjecture for products of Fermat varieties over finite fields
type: Item
instance of: scholarly article
title: Tate conjecture for products of Fermat varieties over finite fields
zbMATH Open document ID: 1298.14026
author: R. Sugiyama
published in: Manuscripta Mathematica
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.4207
review text: Let \(p\) be a prime number and let \(k\) be a finite field of characteristic \(p\). For a nonnegative integer \(r\) and a positive integer \(m\) prime to \(p\), let \(X_m^r=X_m^r(a_0, \cdots, a_{r+1})\subset\mathbb{P}^{r+1}\) denote the projective variety defined by the equation \(a_0x_0^m+a_1x_1^m+\cdots +a_{r+1}x_{r+1}^m=0\) with \((a_0, \cdots, a_{r+1})\in (k^*)^{r+2}\). Let \(X\) be the product \(X_{m_1}^{r_1}\times\cdots\times X_{m_d}^{r_d}\). In this paper the author proves that the Tate conjecture holds for \(X\) under certain assumptions. His proof is based on a combinatorial property of eigenvalues of Frobenius acting on \(\ell\)-adic cohomology of \(X_m^r\), and uses an argument similar to that \textit{M. Spieß} used in [Math. Ann. 314, No. 2, 285--290 (1999; Zbl 0941.11026)].
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 6336316
zbMATH Keywords: Tate conjecture
zbMATH Keywords: Fermat varieties
zbMATH Keywords: Frobenius map
MaRDI profile type: Publication
OpenAlex ID: W2115425214
Wikidata QID: Q123285301
arXiv ID: 1201.4207
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DOI: 10.1007/S00229-013-0652-8
description: scientific article; zbMATH DE number 6336316