Monodromy and the Tate Conjecture-1
نویسنده
چکیده
Introduction We use results of Deligne on ...-adic monodromy 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 MordellWeil rank of the Jacobian of the generic fibre in various equicharacteristic families of surfaces fibred over @1, 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 The paper is organized as follows: 1.0 Review of the Tate Conjecture 2.0 The Tate Conjecture over a finite field 3.0 Middle-dimensional cohomology 4.0 Hypersurface sections of a fixed ambient variety 5.0 Smooth hypersurfaces in projective space 6.0 Families of smooth hypersurfaces in projective space 7.0 Families of smooth hypersurfaces in products of projective spaces 8.0 Hypersurfaces in @1≠@n as families over @1 9.0 Mordell-Weil rank in families of Jacobians References
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