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Rigidity and Frobenius structure. (English) Zbl 1391.14038
This paper studies rigidity of overconvergent isocrystals on an open subset $$U$$ of the projective line. Under some natural and necessary conditions, the author proves that rigid overconvergent isocrystals over $$U$$ admits Frobenius structure. This can be viewed as certain $$p$$-adic analogue of N. M. Katz’s work on rigid local systems [Rigid local systems. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0864.14013)].
We summarize the main results as follows. Let $$K$$ be a complete discrete valuation field of mixed characteristic $$(0,p)$$ with residue field $$k$$. Let $$U \subseteq \mathbb P^1_k$$ denote an open subset with finite complement $$S$$. The first theorem of the paper says that, for an irreducible overconvergent isocrystal $$M^\dagger$$ on $$U$$ such that $$\mathrm{End}(M^\dagger)$$ satisfies the non-Liouville condition of G. Christol and Z. Mebkhout [Ann. Math. (2) 146, No. 2, 345–410 (1997; Zbl 0929.12003)] at points in $$S$$, if the Euler characteristic $$\chi(\mathrm{End}(M^\dagger)) = 2$$, then $$M^\dagger$$ is $$p$$-adically rigid in the sense that $$M^\dagger$$ is the unique isocrystal with the given local monodromy at $$S$$. The proof is essentially the same as Katz’s except using Christol and Mebhout’s characteristic formula.
Next, the author points out that if $$M^\dagger$$ comes from an algebraic local system $$M$$ with regular singularity, satisfying some mild hypotheses including the convergence of the connection and the non-Liouville condition, the rigidity of the algebraic local system $$M$$ implies that the $$p$$-adic rigidity of $$M^\dagger$$.
Finally, the author proves that if in addition that the exponents of $$M^\dagger$$ are rational, and if $$M^\dagger$$ is irreducible and rigid, then $$M^\dagger$$ admits a $$q$$-Frobenius structure for some $$p$$-power $$q$$. By the rigidity result above, the proof of this result follows from that the restriction of $$M^\dagger$$ at points in $$S$$ admits Frobenius structure, which comes down to a local computation.
The paper is well-organized and well-written.
##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 12H25 $$p$$-adic differential equations
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##### References:
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