On the Tate conjecture for divisors on varieties with $h^{2,0} = 1$ in positive characteristics

TL;DR

This paper proves the Tate conjecture for divisors on certain varieties with specific Hodge numbers in positive characteristic, and applies the result to cases of the BSD conjecture and general type surfaces.

Contribution

It introduces a new approach to the Tate conjecture for varieties with $h^{2,0} = 1$ in positive characteristic, and derives new cases of the BSD and Tate conjectures.

Findings

Tate conjecture is generically true for mod p reductions of certain complex varieties.

New cases of the BSD conjecture over global function fields are established.

Tate conjecture is proved for a class of general type surfaces with geometric genus 1.

Abstract

We prove that the Tate conjecture for divisors is ''generically true'' for mod p reductions of complex projective varieties with $h^{2, 0} = 1$, under a mild assumption on moduli. By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1.