Unramified Geometric Iwasawa theory of function fields

For an algebraically closed field $k$ of characteristic $p>0$, we study unramified ${\mathbf{Z}}_p$-towers $\{X_n{\}}_{n\ge 0}$ of smooth, projective and connected curves over $k$. Our main result provides an Iwasawa-theoretic description of the Newton polygons of their Jacobians, establishing that the slopes of the new part at level $n$ are eventually affine functions of $p^{-n}$ with rational coefficients.