IWASAWA THEORY OF GRAPHS AND THEIR DUALS

Abstract

Abstract In this article, we study questions pertaining to ramified Z p d $\mathbb {Z}_p^d$ double struck upper Z Subscript p Superscript d -extensions of a finite connected graph X $X$ upper X . Let X ∞ $X_\infty $ upper X Subscript infinity denote a Z p d $\mathbb {Z}_p^d$ double struck upper Z Subscript p Superscript d ramified derived graph of X $X$ upper X (with d ≥ 1 $d\geq 1$ d greater than or equals 1 ) and Λ $\Lambda $ normal upper Lamda denote the associated Iwasawa algebra. We first prove an asymptotic growth formula in this general setting and show that it is possible to interpret the characteristic ideal of Pic Λ ( X ∞ ) $\mathrm {Pic}_{\Lambda }(X_\infty )$