Variation of the Swan conductors of a rigid disc's cyclic covers
Speaker: Đặng Quốc Huy (Viện Toán học)
Time: 9:30 - 11:00,  October 06, 2021
Venue: Room 302, A5, Institute of Mathematics.
Abstract: A generalization of the Riemann-Hurwitz formula to surfaces gives rise to the notation of (refined) Swan conductor. That is a differential form invariant defined by Kato and extensively studied by Saito, Abbes, Matsuda, Xiao, Hu, etc. This talk discusses the variation of the Swan conductor of a $mathbb{F}_{ell}$-modules $mathcal{F}$ of a rigid disc $D$ over a complete discrete valuation field $K$. When $mathcal{F}$ arises from a cyclic cover of $D$, we derive from its Swan conductor a ramification data. That is a finite sequence of pairs $(delta_i, omega_i)$, where $delta_i$ is a positive rational number and $omega_i$ a differential form on the residue of $K$. In a recent manuscript, we give some properties of this sequence when $K$ is an equal characteristic field. The result follows from a study of the extensions of complete discrete valuation rings with imperfect residue fields.

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