Ramification in Iwasawa Theory and Splitting Conjectures

We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalized class groups and generalized Jacobians. We state some “splitting conjectures” that are equivalent to Leopoldt's conjecture.