Kedlaya's minimal slope conjecture for overconvergent F-isocrystals

Let XX be a smooth connected scheme separated of finite type over Speck\operatorname{Spec} k. Let M\mathcal M^\dagger and N\mathcal N^\dagger be irreducible overconvergent FF-isocrystals on X/KX/K, and let M\mathcal M and N\mathcal N be the associated convergent FF-isocrystals. Suppose they admit slope filtrations {Mi}\{\mathcal M_i\} and {Nj}\{\mathcal N_j\}, respectively. An isomorphism h:M1N1h:\mathcal M_1\to\mathcal N_1 is an isomorphism between their minimal-slope constituents. Kedlaya's minimal slope conjecture. If such an isomorphism hh exists, then there is a unique isomorphism g:MNg^\dagger:\mathcal M^\dagger\to\mathcal N^\dagger of overconvergent FF-isocrystals such that the induced diagram from hh and the associated isomorphism g:MNg:\mathcal M\to\mathcal N commutes in the category of convergent FF-isocrystals on X/KX/K. The conjecture concerns whether the overconvergent objects are determined by the minimal-slope constituent of their slope filtrations. It is trivially true when XX is proper; over a finite field, Ambrosi and D'Addezio proved a rank-one case under a nontriviality hypothesis on hh, while the stronger version without that hypothesis remains under study.

Sources & referencesView supporting material

Primary source

Nobuo Tsuzuki, “Minimal slope conjecture of F-isocrystals”, arXiv:1910.03871 (2021).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.