Effective Methods for Diophantine Problems

Description & Aim

The aim of this workshop is to discuss possible future directions in the study of effective methods in Diophantine number theory.

Classical effective methods in Diophantine number theory include Baker's method, Runge's method, Chabauty's method and the hypergeometric method. These famous methods have governed the direction of the research for decades. Around the end of the previous century it seemed that these methods had reached their limits. However, the past few years have seen important new developments by combining the existing approaches with methods from algebraic geometry. We mention a few of these developments: methods using modularity of elliptic curves; effective results of Shafarevich type; effective methods for finding integral points of curves and varieties over number fields and  finitely generated domains; effective methods for points on subvarieties of tori.

Each of these recent developments revitalized the research on effective results in Diophantine number theory. Further, in many cases the best results are obtained by combining new methods, especially insights from algebraic or arithmetic geometry, with classical methods, such as Baker's method.

The goal of the proposed workshop is to bring together researchers interested in using these new methods, to share their expertise and try to figure out how these new methods can be combined to provide even more powerful tools to prove effective results in Diophantine number theory.