in this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a k3 surface by studying their frobenius polynomials. in the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers. similarly, by studying the variation of the geometric picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a k3 surface of even geometric picard rank.