||In 1948 Andre Weil proved that the Frobenius morphism on an abelian variety over a finite field with q elements has eigenvalues with all absolute values equal to the square root of q. This was the first case of a long chain of beautiful conjectures and results.
In 1968 Honda and Tate proved that conversely such an algebraic integer can be realized as an eigenvalue of the Frobenius of an abelian variety over that finite field: a simple construction of an algebraic integer with some easy properties proves the existence of a complicated arithmetic-geometric object.
We sketch a modern proof of this deep theorem of Weil. We indicate what is used in the proof by Honda and Tate, and (open problem) we ask for a proof of this elegant result along lines of algebraic geometry, not using complex uniformization.
Material exposed in this talk is classical and well-understood by now. We give examples and we sketch some applications.