Canonical extensions, irregularities and the Swan conductor
\(k\) is a perfect field of characteristic \(p>0\) and \(X/k\) is an affine smooth, absolutely irreducible curve with smooth compactification \(\overline{X}\). One fixes a finite field extension \(K\) of the field of fractions of \(W(k)\) and an action of Frobenius on \(K\) which has fixed field \(K_0\). According to N.~Tsuzuki, the category of the continuous representations of \(\pi _1(X)\) in finite dimensional vector spaces over \(K_0\) with finite local monodromy at the points of \(\overline{X}-X\) is equivalent to the category of the overconvergent unit-root \(F\)-isocrystals on \(X\) over \(K\). Moreover N.~Tsuzuki proved that the Swan conductor of the local monodromy representation at a point \(z\) of \(\overline{X}- X\) is equal to the irregularity of the connection of the corresponding crystal at \(z\). In this paper, a geometric proof of the last statement is presented. The basic tool is Katz's theory of the canonical extension of a local Galois representation which leads to an \(\ell\)-adic construction of the Swan representation. Let a representation of \(\pi _1(X)\), as above, be given. Consider a point \(z\in \overline{X}-X\). Let \(\eta _{\infty}\) be the field \(k((T^{-1}))\), equal to the completion of the function field of \(X\) at the place \(z\). The finite group \(G\) is the local monodromy group of the representation at \(z\) and gives rise to an étale covering \(Y\rightarrow X\). Let \(\eta\supset \eta _{\infty}\) denote the induced Galois extension with group \(G\). Then the \(p\)-adic version of Katz's construction states that the Swan representation is isomorphic to \(H^1_c({\mathbb{P}}^1- \{0\},Rj_*\text{Reg}_\eta)\), where \(j:{\mathbb{G}}_m\rightarrow {\mathbb{P}}^1-\{0\}\) is the inclusion and \(\text{Reg}_\eta =f_*K\) is a unit-root \(F\)-isocrystal endowed with a \(G\)-action. Here \(K\) denotes the trivial \(F\)-isocrystal on the étale covering \(Y\) of \(X\). The final statement which completes the proof is that for any overconvergent unit-root \(F\)-isocrystal \(M\) on \(\eta /K\) with corresponding representation \(V\) of \(\pi _1(\eta)\) one has that the Swan conductor of \(V\) is equal to the irregularity of \(M\). The author notes that Robba's definition and formula for the irregularity has been avoided in this proof.
scientific article; zbMATH DE number 1406569