LAUSR

Iwasawa theory is an area of number theory that emerged from the foundational work of Kenkichi Iwasawa in the late 1950s and onward. It studies the growth of arithmetic objects, such as class groups, in towers of number fields. Its key observation is that a part of this growth exhibits a remarkable regularity, which it aims to describe in terms of values of meromorphic functions known as L-functions, such as the Riemann zeta function. Through such descriptions, Iwasawa theory unveils intricate links between algebraic, geometric, and analytic objects of an arithmetic nature. The existence of such links is a common theme in many central areas within arithmetic geometry. So it is that Iwasawa theory has found itself a subject of continued great interest. This year’s Arizona Winter School attracted nearly 300 students hoping to learn about it! The literature on Iwasawa theory is vast and often technical, but the underlying ideas are possessing of an undeniable beauty. I hope to convey some of this, while explaining the original questions of Iwasawa theory and giving a sense of the directions in which the area is heading.