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Galois module structure of Tate modules
Mathematische Zeitschrift, 1997If \(Y\to X\) is a \(G\)-covering of smooth projective curves over an algebraically closed field \(k\), then the Tate module \(T_\ell(Y)=\text{projlim}_n\text{Pic}^0(Y)[\ell^n]\) is naturally a module over \(\mathbb Z_\ell[G]\). The subject of the present paper is to determine this module for the case where \(G\) is a cyclic \(\ell\)-group, and \(\ell\)
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A single general formula is given for the weak approximation in algebraic tori over global fields. We calculate the first cohomology group for the torus of an embedding problem of fields with Abelian kernel, the coefficients being the Picard group of a nonsingular projective model of the torus.
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