zbMATH — the first resource for mathematics

Arithmetic and differential Swan conductors of rank one representations with finite local monodromy. (English) Zbl 1198.12004
Let \(k\) be some field of characteristic \(p\) and let \(E=k((X))\) and let \(G_E\) be the absolute Galois group of \(E\) and let \(I_E\) be the inertia subgroup of \(G_E\). If \(k\) is perfect, then there is an equivalence of categories between the category of representations of \(G_E\) whose restriction to an open subgroup of \(I_E\) is trivial (the “finite local monodromy” condition), and the category of “overconvergent \((\varphi,\nabla)\)-modules”. These objects are \(p\)-adic differential equations with a Frobenius structure, and coefficients in a field of bounded power series. One associates to a representation \(V\) the differential equation \(\mathrm{D}^\dagger(V)\).
The ramification of \(V\) is measured by a numerical invariant, the Swan conductor \(\mathrm{Sw}(V)\) and likewise, the ramification of a \(p\)-adic differential equation \(\mathrm{D}^\dagger\) is measured by the irregularity \(\mathrm{Irr}(\mathrm{D}^\dagger)\). It is a theorem of N. Tsuzuki [Compos. Math. 111, No. 3, 245–288 (1998; Zbl 0926.12004)] that \(\mathrm{Sw}(V) = \mathrm{Irr}(\mathrm{D}^\dagger(V))\).
The purpose of this article is to extend this result to the case where \(k\) is not perfect. The equivalence between the category of representations of \(G_E\) with finite local monodromy and a suitable category of “overconvergent \((\varphi,\nabla)\)-modules” is then due to K. S. Kedlaya [Algebra Number Theory 1, No. 3, 269–300 (2007; Zbl 1184.11051)]. The definition of the Swan conductor \(\mathrm{Sw}(V)\) is due to A. Abbes, T. Saito, [Am. J. Math. 124, No. 5, 879–920 (2002; Zbl 1084.11064)], and the generalized definition of the irregularity is due to K. S. Kedlaya [Zbl 1184.11051]. In the article under review, the authors prove that \(\mathrm{Sw}(V) = \mathrm{Irr}(\mathrm{D}^\dagger(V))\) if \(V\) is of dimension \(1\). In this case, the Swan conductor had actually been defined by [K. Kato, Algebraic K-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 101–131 (1989; Zbl 0716.12006)]. The idea is to use the explicit description of \(\mathrm{D}^\dagger(V)\) afforded by the work of the second author [Math. Ann. 337, No. 3, 489–555 (2007; Zbl 1125.12001)].
Liang Xiao has more recently given a proof that \(\mathrm{Sw}(V) = \mathrm{Irr}(\mathrm{D}^\dagger(V))\) in general [L. Xiao, “On ramification filtrations and \(p\)-adic differential equations, I: equal characteristic case”, to appear in Algebra and Number Theory)].

12H25 \(p\)-adic differential equations
11F80 Galois representations
11S15 Ramification and extension theory
11S20 Galois theory
11S31 Class field theory; \(p\)-adic formal groups
14F30 \(p\)-adic cohomology, crystalline cohomology
Full Text: DOI arXiv