The analytic subgroup theorem, logarithmic forms and periods. Lecture 3

In this series of lectures we shall give an overview about modern developments in transcendence theory originating in Hilbert's 7th solved independently by A. O. Gelfond and Th. Schneider on 1934. We shall in the first lecture give an exposition on the Analytic Subgroup Theorem and the theory of multip0licity estimates which are behind. The second lecture deals with applications in the theory of logarithmic forms. This will cover the isogeny theorem which gives an alternative transcendental approach to the Tate conjecture. As is well known that Tate conjecture implies the Shafarevich conjecture from which, by a beautiful device found by Kodaira and Parshin, a proof of the Mordell conjecture can be obtained, and this gives Falting's Theorem on rational points on curves. We shall also touch the André–Oort conjecture in the context of transcendence properties of hypergeometric functions. The last lecture is devoted to a conjecture of Leibniz as described by Arnold in his book on Huygens and Barrow, Newton and Hook. We shall formulate a modern version of the conjecture, relate it to a conjecture of Kontsevich on motives and to a generalization of the famous Schanuel conjecture.