The ABC conjecture and its consequences

The abc-conjecture is an integer analogue of Mason-Stothers theorem for polynomials which was proposed by J. Oesterle and D. Masser in number theory. It expresses a relation between multiplication and addition for integers. Stronger versions of this conjecture yield solutions of many important problems including Fermat's Last Theorem. Using his Inter-universal Teichmüller theory, S. Mochizuki obtained proofs of non-effective Vojta, Szpiro, and abc- conjectures over algebraic number fields. More recently, an effective abc-conjecture over Q or an imaginary quadratic number fields and the Szpiro conjecture have been proved by I. Fesenko, Y. Hoshi, A. Minamide ,S. Mochizuki and W. Porowski. In this thesis, the above developments are discussed in some detail.