Abstract. For an algebraic variety X over a finite field of q elements, its zeta function is a rational function whose reciprocal zeros and poles are algebraic integers. These algebraic integers are part of a larger set of Frobenius eigenvalues acting on any Weil cohomology of X, and hence are Weil q-numbers by the Weil conjectures. In this introductory lecture, we report on recent progress on the q-divisibility of these Frobenius eigenvalues as algebraic integers and its Hodge theoretic analogue. The new results sharpen and extend previous q-divisibility theorems of Ax-Katz (71),  Deligne (73), Esnault-Katz (05) and Esnault-Wan (22). This is based on joint work with Dingxin Zhang.