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