We fix a monic polynomial f (x) ∈ F q [x] over a finite field and consider the Artin-Schreier-Witt tower defined by f (x); this is a tower of curves · · · → C m → C m−1 → · · · → C 0 = A 1 , with total Galois group 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.

We fix a monic polynomialf (x) ∈ Fq[x] over a finite field of characteristic p of degree relatively prime to p, and consider the Z p ℓ -Artin-Schreier-Witt tower defined byf (x); this is a tower of curves · · · → Cm → C m−1 → · · · → C 0 = A 1 , whose Galois group is canonically isomorphic to Z p ℓ , the degree ℓ unramified extension of Zp, which is abstractly isomorphic to (Zp) ℓ 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 Z p ℓ -Artin-Schreier-Witt tower (over a large subdomain of the weight space). This extends the main result in [DWX] from rank one case ℓ = 1 to the higher rank case ℓ ≥ 1.

Let F be a totally real field and p be an odd prime which splits completely in F . We prove that the eigenvariety associated to a definite quaternion algebra over F satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the Up-slopes of points and the p-adic valuations of the p-parameters are bounded by explicit numbers, for all primes p of F over p. Applying Hansen's p-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its Up slope goes to zero. In the case of eigencurves, this completes the proof of Coleman-Mazur's 'halo' conjecture. Contents 1. Introduction 1 2. Newton-Hodge decomposition over certain noncommutative rings 7 3. Automorphic forms for definite quaternion algebras and completed homology 26 4. A filtration on the space of integral p-adic automorphic forms 40 5. Proof of the main theorem 50 6. Application to Hilbert modular eigenvarieties 57 References 60