We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree for varieties with over a finitely generated field of characteristic , under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford–Tate conjectures for several classes of algebraic surfaces with .
"On the Tate and Mumford–Tate conjectures in codimension for varieties with ." Duke Math. J. 166 (4) 739 - 799, 15 March 2017. https://doi.org/10.1215/00127094-3774386