Slopes for higher rank Artin-Schreier-Witt Towers

We fix a monic polynomial $\bar{f}(x)\in {\mathbb{F}}_q[x]$ over a finite field of characteristic $p$, and consider the ${\mathbb{Z}}_{p^{\mathrm{\ell }}}$-Artin-Schreier-Witt tower defined by $\bar{f}(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =$\mathbb{A}$^1$,whoseGaloisgroupiscanonicallyisomorphicto$$\mathbb{Z}$_{p^\ell}$,thedegree$\ell$unramifiedextensionof$$\mathbb{Z}$_p$,whichisabstractlyisomorphicto$($\mathbb{Z}$_p)^\ell$ as a topological group. 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 asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the ${\mathbb{Z}}_{p^{\mathrm{\ell }}}$-Artin-Schreier-Witt tower. This extends the main result in [DWX] from rank one case $\mathrm{\ell }=1$ to the higher rank case $\mathrm{\ell }\ge 1$.