Frobenius structure for rank one $p-$adic differential equations

Abstract : According to a criterion of B. Chiarellotto and G. Christol [Compositio Math. 100 (1996), no. 1, 77-99; MR1377409 (97b:14021)], a solvable rank one p-adic differential operator d/dx−g, with g=∑ni=1a−ixi, has a Frobenius structure if and only if a−1 is p-integral. Using natural estimates on tensor products, the author here generalizes this criterion to all g's in the Robba ring. As a corollary, he extends to the case p=2 the qualitative part of Matsuda's theorem [S. Matsuda, Duke Math. J. 77 (1995), no. 3, 607-625; MR1324636 (97a:14019)], according to which the Dwork-Robba twisted Artin-Hasse exponentials have Frobenius structures.
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Andrea Pulita. Frobenius structure for rank one $p-$adic differential equations. Eighth international conference on p-adic functional analysis, July 5-9 2004, Université Blaise Pascal, Clermont-Ferrand, France, Jul 2004, France. pp.247-258, ⟨10.1090/conm/384/0713⟩. ⟨hal-00804859⟩



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