關於這個星期宣佈的演講，時間有些更動，Abelian Varity那場改至12/13(星期四)的四 點至五點半。 ------------------------------------------------------------------------------ Speaker: Frans Oort (University of Utrecht) Time: December 7 (Friday), 4 - 5:30 pm Place: Room 101 Title: Prime numbers Abstract: In my talk I will pose several questions about prime numbers. We will see that on the one hand some of them allow an answer with a proof of just a few lines, on the other hand, some of them lead to deep questions and conjectures not yet understood. This seems to represent a general pattern in mathematics: your curiosity leads to a study of easy questions related with quite deep structures. I will give many examples, suggestions and references for further study. ------------------------------------------------------------------------------ Speaker: Frans Oort (University of Utrecht) Time: December 13 (Thursday), 4 - 5:30 pm Place: Room 102 Title: Abelian varieties over finite fields Abstract: In 1948 André 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. --

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