# Arithmetic stability in p-adic towers of global function fields.

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Given a global function field K of characteristic p>0, the fundamental arithmetic invariants include the genus, the class number, the p-rank and more generally the slope sequence of the zeta function of K. In this expository lecture, we explore possible stability of these invariants in a p-adic Lie tower of K. Strong stability is expected when the tower comes from algebraic geometry, but this is already sufficiently interesting and difficult in the case of Zp towers.