The class group of the number field is one of the essential and mysterious objects in algebraic number theory. The divisibility properties of class numbers provide information to understand the class group's structure. Therefore, studying the divisibility properties of class numbers of the number fields becomes crucial. Tate-Shaferevich groups arise in one of the seven Millenium prize problem Birch-Swinnerton Dyer Conjecture. The study of the connection between the Tate-Shafarevich group of elliptic curves and class numbers provide new insight between arithmetic geometry and algebraic number theory. In this project, we would like study this connection. The researchers also establish various divisibility results of class numbers of number fields. They also study Iizuka's conjecture and will analyze the weaker version of the conjecture. One of the crucial ingredients which arise in Iizuka's conjecture is the study of solutions of certain Diophantine equation. We would study these equations.
Indian Institute of Science
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