Iwasawa theory for class group schemes in characteristic p
Algebra and Number Theory Seminar
In a landmark 1959 paper, Iwasawa studied the growth of class groups in Z_p-towers of number fields, establishing a remarkable formula for the exact power of p dividing the order of the class group of the n-th layer of the tower. Iwasawa's work was inspired by a profound analogy between number fields and function fields over finite fields. In this setting, the direct analogue of Iwasawa theory is the study of class groups in Z_p-towers of global function fields over finite fields k of characteristic p, and an analogous formula for the order of p dividing the class group was established by Mazur and Wiles in 1983. An extraordinary feature of this function field setting is that the class group can be realized as the k-rational points of an algebraic variety---the Jacobian. We will briefly survey some of this history, and introduce a novel analogue of Iwasawa theory for function fields by studying not just the k-points of these Jacobians, but their full p-torsion group schemes, which are much richer, geometric objects having no analogue in the number field setting.