$\ell$-adic analogue of Kedlaya-Mochizuki

Question

There is a well-known analogy between holonomic $\mathcal{D}$-modules on complex algebraic varieties and $\ell$-adic perverse sheaves on varieties over finite fields. Many theorems in one setting have natural analogues in the other. (Even if the proofs are sometimes completely different.)

A major breakthrough on the de Rham side is the Kedlaya–Mochizuki theorem ([Ked, Thm. 8.1.3; Moc, Thm. 1.3.3]), which morally states that any holonomic $\mathcal{D}$-module can, locally, be decomposed into a direct sum of regular holonomic pieces tensored with exponential twists. This result was instrumental in the proof of the irregular Riemann–Hilbert correspondence.

Question: What would be the appropriate $\ell$-adic analogue of the Kedlaya–Mochizuki theorem for $\ell$-adic perverse sheaves, and is such a result known to be true?

References:

• [Ked] K. Kedlaya - Good formal structures for flat meromorphic connections, II: Excellent schemes
• [Moc] T. Mochizuki - Wild Harmonic Bundles and Wild Pure Twistor $\mathcal{D}$-modules

Answer 1

In one dimension we have the following result:

Let $X$ be a curve over a finite field $k$ of characteristic $p$} and $x\in X$ a point. Let $K$ be a perverse sheaf on $X$ of generic rank less than $p$. Then the pullback of $X$ to the étale local ring of the point $x$ can be decomposed as a sum of pushforwards from tamely ramified covers of tamely ramified perverse sheaves tensored with Artin-Schreier-Witt sheaves.

However, the result is manifestly untrue if the generic rank is greater than $p$. For example, the rank $p$ Kloosterman sheaf is already a counterexample.

I do not know whether there is an analogue for low-rank sheaves in higher dimensions.

Proof sketch: We can first decompose the perverse sheaf into the direct sum of a part with unipotent monodromy and a part with no monodromy invariants, whose stalk at the point $x$ vanishes. The unipotent part is tame, allowing us to restrict attention to sheaves whose stalk at $x$ vanishes, which are equivalent to representations of the Galois group of the local field at $x$. Thus we have a representation of the Galois group of rank at least $p$. Restricted to the wild inertia group, which is a pro-$p$ group, it splits into representations of rank a power of $p$, hence splits into representations of rank at least $1$. The Galois group acts on the isomorphism classes of these representations by finitely many orbits, and the representation splits as a direct sum over orbits, so we may restrict attention to a single orbit. The stabilizer of the orbit defines a cover, necessarily tamely ramified, and we can write a representation whose characters all lie in that orbit as an induced representation from the stabilizer, i.e. a pushforward from the associated cover, of a representation where the wild inertia group acts by scalars. The order of the group of scalars that arise this way is necessarily a power of $p$. Since the dimension of the representation is still less than $p$, taking the determinant and raising to the inverse of the dimension modulo this power of $p$ produces a one-dimensional representation on which the wild inertia group has the same action. The one-dimensional representation can be decomposed into an Artin-Schreier-Witt character and a tame character, and tensoring with the inverse of the Artin-Schreier-Witt character produces a representation on which wild inertia acts trivially, i.e. a tamely ramified representation, as desired.