Variation of the Swan conductor of an $\mathbb{F}_{\ell}$-sheaf on a rigid annulus

Abstract: Let $C$ be a closed annulus of radii $r < r' \in \mathbb{Q}_{\geq 0}$ over a complete discrete valuation field with algebraically closed residue field of characteristic $p>0$. To an étale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on $C$, ramified at most at a finite set of rigid points of $C$, one associates an Abbes-Saito Swan conductor function ${\rm sw}_{\mathcal{F}}: [r, r']\cap \mathbb{Q}_{\geq 0} \to \mathbb{Q}$ which, for a radius $t$, measures the ramification of $\mathcal{F}_{\lvert C^{[t]}}$ — the restriction of $\mathcal{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-thickness — along the special fiber of the normalized integral model of $C^{[t]}$. This function has the following remarkable properties: it is continuous, convex and piecewise linear outside the radii of the ramification points of $\mathcal{F}$, with finitely many integer slopes whose variation between radii $t$ and $t'$ can be expressed as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$. In this talk, I will explain the construction of ${\rm sw}_{\mathcal{F}}$ and the key nearby cycles formula in establishing the aforementioned properties of ${\rm sw}_{\mathcal{F}}$.

algebraic geometrynumber theory

Audience: advanced learners

( paper | slides )

VIASM Arithmetic Geometry Online Seminar