This book originated in the idea that open problems act as crystallization points in mathematical research. Mathematical books usually deal with fully developed theories. But here we present work at an earlier stage—when challenging questions can give new directions to mathematical research.

In mathematics, significant progress is often made by looking at the underlying structures of open problems and discovering new directions that are developed to find solutions. In that process, the search for finding the “true” nature of the problem at hand is the impetus for our thoughts. It is only much later, in retrospect, that we see the “flow of mathematics”—from problem to theory and new insight. This is the gist of the present volume.

The origin of this volume lies in a collection of nineteen problems presented in 1995 to the participants of the conference Arithmetic and Geometry of Abelian Varieties.