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# RTG Seminar on Number Theory Seminar

Variation of the Swan conductor of an $\mathbb{F}_{\ell}$-sheaf on a rigid disc
Monday, January 11, 2021
2:00-3:50 PM
Off Campus Location
Let $K$ be a complete discrete valuation field of residue characteristic $p>0$ and $\ell\neq p$ a prime number. To a finite dimensional $\mathbb{F}_{\ell}$-representation $M$ of the absolute Galois group $G_K$, the ramification theory of Abbes and Saito attaches a Swan conductor ${\rm sw}(M)$ and a characteristic cycle ${\rm CC}(M)$. Let $D$ be the rigid unit disc over $K$ and $\mathcal{F}$ a lisse sheaf of $\mathbb{F}_{\ell}$-modules on $D$. For $t\in \mathbb{Q}_{\geq 0}$, the normalized integral model $\mathcal{D}^{(t)}$ of the subdisc $D^{(t)}$ of radius $t$ is defined over some finite extension of $K$. The restriction $\mathcal{F}_{\lvert D^{(t)}}$ defines, at the generic point $\mathfrak{p}^{(t)}$ of the special fiber of $\mathcal{D}^{(t)}$, a Galois representation $M_t$ over a complete discrete valuation field, thus yielding a Swan conductor ${\rm sw}(M_t)$ and a characteristic cycle ${\rm CC}(M_t)$. The goal of the talk is to explain how we connect earlier works, of Lütkebohmert on a discriminant function attached to a cover of $D$, and of Kato on the ramification of valuation rings of height $2$, and prove that the function $t\mapsto {\rm sw}(M_t)$ is continuous and piecewise linear with finitely many slopes which are all integers, and that its right derivative is $t\mapsto -{\rm ord}_{\mathfrak{p}^{(t)}}({\rm CC}(M_t)) + \dim_{\mathbb{F}_{\ell}}(M_t/M_t^{(0)})$, where ${\rm ord}_{\mathfrak{p}^{(t)}}$ is a normalized discrete valuation at $\mathfrak{p}^{(t)}$ extended to differentials and $M_t^{(0)}$ is the tame part of $M_t$.