SOLVING EQUATIONS IN INTEGERS

DIOPHANTINE GEOMETRY AND TRANSCENDENCE THEORY

抄録

Mathematics, in large part, can be said to have originated from the study of equations in integers. Fermat, Euler, Gauss, Hilbert and many great mathematicians have contributed to the field, and numerous results have been obtained through the centuries to resolve particular instances. It is only in recent times, however, that a general method, based on transcendence theory, has been successfully developed in this context. The method has deep connections with the theory of elliptic curves, with class number questions, with p-adic L-functions and with many aspects of Diophantine geometry. The lecture will survey the current state of the subject.