Tate conjecture for products of Fermat varieties over finite fields
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)].
scientific article; zbMATH DE number 6336316