Mochizuki’s work on torally crys-stable bundles  has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: ﬁrst, one can conclude almost immediately certain results classifying Frobenius-unstable vector bundles on curves; and second, it follows from the results of  that one also obtains results on rational functions with prescribed ramiﬁcation in positive characteristic.
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Brian Osserman, Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications. Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, pp. 95–119DOI 10.2977/PRIMS/1199403809