Arithmetic and differential Swan conductors of rank one representations with finite local monodromy.

*(English)*Zbl 1198.12004Let \(k\) 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)].

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)].

Reviewer: Laurent Berger (Lyon)