In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by , as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 90s), Ralph Greenberg has proposed an Iwasawa theory for motives.