Iwasawa theory

Summary

Iwasawa theory studies arithmetic objects, such as ideal class groups, along an infinite tower of number fields $K = K_0 \subset K_1 \subset K_2 \subset \cdots$. The union of the tower has Galois group isomorphic to the additive group of p-adic integers $\Z_p$. Such a tower is a $\Z_p$-extension, and the main example is obtained from the fields $\Q(\zeta_{p^n})$ of $p$-power roots of unity. The objects attached to the whole tower are modules over the Iwasawa algebra $\Z_p[\![T]\!]$, whose structure theory gives regular growth formulas. The Iwasawa main conjecture links these modules to $p$-adic $L$-functions.