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Session 3. Arithmetic Geometry
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Galois representations attached to étale cohomology |
Sebastian Petersen, Universität der Bundeswehr, Neubiberg, Germany
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Let \(K\) be a field and \(X/K\) a separated algebraic scheme.
Grothendieck and Artin constructed
for every prime number \(\ell\) and every \(i\in\bf{N}\) a cohomology group \(H^i(X_{\overline{K}}, \bf{Q}_\ell)\) which
comes with a natural action of the Galois group \(\mathrm{Gal}(\overline{K}/K)\); one thus obtains a Galois representation
\[\rho_\ell: \mathrm{Gal}(\overline{K}/K)\to \mathrm{Aut}_{\bf{Q}_\ell}(H^i(X_{\overline{K}}, \bf{Q}_\ell)).\]
Denote by \(\rho: \mathrm{Gal}(\overline{K}/K)\to \prod_{\ell} \mathrm{im}(\rho_\ell)\) the homomorphism induced by the
\(\rho_\ell\).
Serre proved recently that in the case where \(K\) is a number field the family \((\rho_{\ell})_{\ell}\)
is almost independent in the following sense: There exists a finite Galois extension \(K'/K\) such that
\(\rho(\mathrm{Gal}(\overline{K}/K'))=\prod_\ell\rho_\ell(\mathrm{Gal}(\overline{K}/K'))\). This information is quite useful when
working with such families of \(\ell\)-adic representations attached to schemes, and it ties in well with the adelic openness
conjecture. There are analogous results in the case
where \(K\) is an arbitrary finitely generated field of characteristic zero, and where \(K\) is a geometric
function field of arbitrary characteristic.
Following a suggestion of Illusie and making strong use of results of Orgogozo this was used to
establish a quite general independence theorem for families \((F_\ell)_\ell\) of étale sheaves of \(\bf{F}_\ell\)-vector
spaces over an arithmetic scheme which satisfy a uniform constructability and a potential semistability condition.
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