Logarithmic Tate conjectures over finite fields

TL;DR

This paper proposes a logarithmic analogue of the Tate conjecture over finite fields, linking it to the weight-monodromy conjecture and suggesting new structures in log motives involving monodromy and ${\frak{sl}}(2)$ actions.

Contribution

It introduces a novel conjecture in log geometry over finite fields that connects algebraic cycles, monodromy, and log motives, extending classical Tate conjecture ideas.

Findings

Weight-monodromy conjecture follows from the new conjecture and Frobenius semi-simplicity.

Suggests existence of monodromy cycle and ${\frak{sl}}(2)$ action in log motives.

Provides a framework linking log geometry with classical conjectures.

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 ${\frak{sl}}(2)$ on the cohomology, and which lives in the world of log motives.