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SUMMARY:Amadou Bah (IHES)
DTSTART:20201218T153000Z
DTEND:20201218T170000Z
DTSTAMP:20260423T022623Z
UID:CAFAS/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAFAS/14/">V
 ariation of the Swan conductor of an $\\mathbb{F}_{\\ell}$-sheaf on a rigi
 d disc</a>\nby Amadou Bah (IHES) as part of Columbia Automorphic Forms and
  Arithmetic Seminar\n\n\nAbstract\nLet $K$ be a complete discrete valuatio
 n field of residue characteristic $p>0$ and $\\ell\\neq p$ a prime number.
  To a finite dimensional $\\mathbb{F}_{\\ell}$-representation $M$ of the a
 bsolute Galois group $G_K$\, the ramification theory of Abbes and Saito at
 taches a Swan conductor ${\\rm sw}(M)$ and a characteristic cycle ${\\rm C
 C}(M)$. 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 subdis
 c $D^{(t)}$ of radius $t$ is defined over some finite extension of $K$. Th
 e restriction $\\mathcal{F}_{\\lvert D^{(t)}}$ defines\, at the generic po
 int $\\mathfrak{p}^{(t)}$ of the special fiber of $\\mathcal{D}^{(t)}$\, a
  Galois representation $M_t$ over a complete discrete valuation field\, th
 us 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 o
 f $D$\, 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 th
 at 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 $\\math
 frak{p}^{(t)}$ extended to differentials and $M_t^{(0)}$ is the tame part 
 of $M_t$.\n
LOCATION:https://researchseminars.org/talk/CAFAS/14/
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