Effective obstruction to lifting Tate classes from positive characteristic

Abstract

A recent result of Bloch-Esnault-Kerz describes the obstruction to formally lifting algebraic classes from positive characteristic to characteristic zero. We use their result to give an algorithm that takes a smooth hypersurface 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 it relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more.

Sato-Tate groups of abelian threefolds

“…(ii) The monodromy software repository [Zyw21] provides a Magma implementation of the algorithm described in [Zyw20], which uses a given set of L-polynomials at good primes p ≤ B to compute an upper bound on the geometric endomorphism algebra End(A Q ) Q (and a lower bound on the Mumford-Tate group) that is guaranteed to be correct for sufficiently large B (but without an effective bound on such a B). (iii) The crystalline obstruction software repository [Cos21] provides a SageMath implementation of the algorithm described in [CS20], which uses a p-adic computation at a prime of good reduction to compute upper bounds on the rank r of the Néron-Severi group of A Q and the dimension d of the geometric endomorphism algebra End(A Q ) Q ; by varying the choice of p and taking minima one expects to eventually obtain tight bounds. As shown in [CFS19], the pair (r, d) coincides with the 2-simplex of moments of ST(A Q ), which by Theorem 6.22 determines ST(A) 0 and End(A Q ) R .…”