Monodromy and the tate conjecture: Picard numbers and mordell-weil ranks in families

We use results of Deligne on ℓ-adic monodroray and equidistribution, combined with elementary facts about the eigenvalues of elements in the orthogonal group, to give upper bounds for the average "middle Picard number" in various equicharacteristic families of even dimensional hypersurfaces, cf. 6.11, 6.12, 6.14, 7.6, 8.12. We also give upper bounds for the average Mordell-Weil rank of the Jacobian of the generic fibre in various equicharacteristic families of surfaces fibred over P¹, cf. 9.7, 9.8. If the relevant Tate Conjecture holds, each upper bound we find for an average is in fact equal to that average.