  We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and
consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of
curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total
Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of
this tower of curves. This reduces to the study of the Newton slopes of
L-functions associated to characters of the Galois group of this tower. We
prove that, when the conductor of the character is large enough, the Newton
slopes of the L-function form arithmetic progressions which are independent of
the conductor of the character. As a corollary, we obtain a result on the
behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt
tower, analogous to the result of Buzzard and Kilford.
