Effective obstruction to lifting Tate classes from positive characteristic

Abstract

We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from $p$-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds.

35 pages, 2 figures; v3 typos corrected and a new example

References in corpus (3)

• The Tate conjecture over finite fields (AIM talk)
• A numerical transcendental method in algebraic geometry
• On reconstructing subvarieties from their periods

Cited by in corpus (2)

• Deep Learning Gauss-Manin Connections
• On reconstructing subvarieties from their periods