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SUMMARY:Rong Zhou (University of Cambridge)
DTSTART:20221214T160000Z
DTEND:20221214T170000Z
DTSTAMP:20260418T064041Z
UID:LNTS/82
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/82/">In
 dependence of $\\ell$ for $G$-valued Weil--Deligne representations associa
 ted to abelian varieties</a>\nby Rong Zhou (University of Cambridge) as pa
 rt of London number theory seminar\n\nLecture held in Huxley 139\, Imperia
 l.\n\nAbstract\nLet $A$ be an abelian variety over a number field $E$ of d
 imension $g$ and $\\rho_\\ell:\\mathrm{Gal}(\\overline{E}/E)\\rightarrow \
 \mathrm{GL}_{2g}(\\mathbb{Q}_\\ell)$ the Galois representation on the $\\e
 ll$-adic Tate module of $A$. For a place $v$ of $E$ not dividing $\\ell$\,
  upon fixing an isomorphism $\\overline{\\mathbb{Q}}_\\ell\\cong \\mathbb{
 C}$\, Grothendieck’s $\\ell$-adic monodromy theorem associates to $\\rho
 _\\ell$ a $\\mathrm{GL}_{2g}(\\mathbb{C})$-valued Weil-Deligne representat
 ion $\\rho_{\\ell\,v}^{WD}$. Then it is known that the conjugacy class of 
 $\\rho_{\\ell\,v}^{WD}$ is defined over $\\mathbb{Q}$ and independent of $
 \\ell.$ When $v$ is a place a good reduction\, this is just the result tha
 t the characteristic polynomial of Frobenius is defined over $\\mathbb{Z}$
  and independent of $\\ell$.\n\nWe consider a refinement of this result. A
  Theorem of Deligne implies that upon replacing $E$ by a finite extension\
 , the representations $\\rho_{\\ell\,v}^{WD}$ can be refined to a $G(\\mat
 hbb{C})$-valued Weil-Deligne representation $\\rho^{WD\,G}_{\\ell\,v}$\, w
 here $G$ is the Mumford--Tate group of $A$. We prove that for $p>2$ and $v
 |p$ a place of $E$ where $A$ has semistable reduction\, the conjugacy clas
 s of $\\rho^{WD\,G}_{\\ell\,v}$ is defined over $\\mathbb{Q}$ and independ
 ent of $\\ell$. This is joint work with Mark Kisin.\n
LOCATION:https://researchseminars.org/talk/LNTS/82/
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