We develop, for an -adic étale sheaf on a complete trait of characteristic \( p > 0 \), the notion of characteristic variety. Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramiﬁcation theory for local ﬁelds with general residue ﬁelds. We formulate the main property that should be satisﬁed by the characteristic variety (the isogeny conjecture), and prove it for rank one sheaves unconditionally on the trait, or unconditionally on the sheaf if the residue ﬁeld of the trait is perfect.
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Ahmed Abbes, Takeshi Saito, Analyse Micro-Locale -Adique en Caractéristique \( p > 0 \) le Cas d’un Trait. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, pp. 25–74DOI 10.2977/PRIMS/1234361154