ncts

Schmidt's Subspace Theorem has been a key tool in Diophantine approximation since its appearance in the early1970s. In 1994 Faltings and Wustholz introduced a new geometric method of applying the Subspace Theorem, called the filtration method, which involved working with "many" sections of a line bundle and producing many linear combinations of them vanishing along appropriate divisors. This was further developed by Evertse and Ferretti. Independently, Corvaja and Zannier also worked with filtrations of the same kind, which was further refined and developed by Levin and Autissier, etc. Recently, Ru and Vojta formulated a general version of the Subspace Theorem that unify many known results with filtration methods. We will introduce these developments and state some applications of Ru-Vojta's theorem in the study of integral points, gcd theorem, and divisibility of polynomial values. This talk includes joint works with Erwan Rousseau and Amos Turchet and a joint work with Yu Yasufuku.