TY - JOUR
T1 - Monodromy and the tate conjecture
T2 - Picard numbers and mordell-weil ranks in families
AU - De Jong, A. Johan
AU - Katz, Nicholas M.
PY - 2000
Y1 - 2000
N2 - 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.
AB - 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.
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U2 - 10.1007/s11856-000-1271-0
DO - 10.1007/s11856-000-1271-0
M3 - Article
AN - SCOPUS:0009879853
SN - 0021-2172
VL - 120
SP - 47
EP - 79
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -