Logarithmic Tate conjectures over finite fields

Abstract

We formulate an analogue of Tate conjecture on algebraic cycles, for the log geometry over a finite field. We show that the weight-monodromy conjecture follows from this conjecture and from the semi-simplicity of the Frobenius action. This conjecture suggests the existence of the monodromy cycle which gives the monodromy operator and an action of $\mathfrak{sl}(2)$ on the cohomology, and which lives in the world of log motives.