Analogue of Tate's conjecture for fibrations in varieties

Let f:XYf:\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.

Ress=2(pRBp(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.

Sources & referencesView supporting material

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.