Analogue of Tate's conjecture for fibrations in varieties

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Let f:X→Yf:\mathcal{X}\to\mathcal{Y} be a fibration in varieties over a number field KK. Let RR be the finite set of excluded prime ideals, let qpq_{\mathfrak p} denote the residue-field cardinality at p\mathfrak p, and let Bp(X)\mathfrak B_{\mathfrak p}(\mathcal X) be the average degree-two Frobenius trace defined in the source. Let NS⁡(X/K)\operatorname{NS}(X/K) be the Néron–Severi group. Analogue of Tate's conjecture for fibrations in varieties.

Res⁡s=2(∑p∉RBp(X)log⁡(qp)qps)=rank⁡(NS⁡(X/K)).\operatorname*{Res}_{s=2}\left(\sum_{\mathfrak p\notin R}\mathfrak B_{\mathfrak p}(\mathcal X)\frac{\log(q_{\mathfrak p})}{q_{\mathfrak p}^s}\right)=\operatorname{rank}\left(\operatorname{NS}(X/K)\right).

This is the proposed fibration analogue of the divisor case of Tate's conjecture. The source does not state whether it is known or resolved.

References

Primary source

Marc Hindry, Amilcar Pacheco and Rania Wazir, “Fibrations et conjecture de Tate”, arXiv:math/0311417 (2004).

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