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SUMMARY:Joe Kramer-Miller (UC Irvine)
DTSTART:20210211T170000Z
DTEND:20210211T182000Z
DTSTAMP:20260423T021249Z
UID:RAMpAGe/34
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/RAMpAGe/34/"
 >The ramification of p-adic representations coming from geometry</a>\nby J
 oe Kramer-Miller (UC Irvine) as part of Recent Advances in Modern p-Adic G
 eometry (RAMpAGe)\n\n\nAbstract\nThe purpose of this talk is to explain a 
 geometric analogue of Sen's classical theorem\, which describes the close 
 relationship between $p$-adic Lie filtrations and ramification filtrations
  for $p$-adic fields. Let $X$ be a smooth variety over a perfect field $k$
  with characteristic $p>0$\, let $D\\subset X$ be a reduced divisor with s
 mooth normal crossings\, and let $U=X\\backslash D$. Consider a continuous
  representation $\\rho:\\pi_1(U) \\to GL(\\Z_p)$\, which gives rise to an 
 $p$-adic Lie tower of \\'etale covers $U_n \\to U$. We may associate to ea
 ch cover a Swan divisor $sw(U_n/U)$\, supported on $D$\, using the ramific
 ation filtration of Abbes-Saito. In general\, the growth of these divisors
  can be arbitrarily wild. Instead\, we restrict ourselves to representatio
 ns that are ordinary geometric (e.g. $\\rho$ arises as the $p$-adic Tate m
 odule of a family of ordinary Abelian varieties). Our main result states t
 hat for $\\rho$ ordinary geometric\, there exists integers $c_1>c_0>0$ suc
 h that $c_1p^{2n} D > sw(U_n/U) > c_0 p^{2n}D$. This says that even though
  $\\rho$ has infinite monodromy\, the Swan conductors $sw(U_n/U)$ grow as 
 `slowly as possible'.\n
LOCATION:https://researchseminars.org/talk/RAMpAGe/34/
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