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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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