# CM abelian variety from an algebraic Hecke character?

Hi,

Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a "rank 1 CM-motive" $M$ with $\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the general theory of the Taniyama group and its quotient, the Serre group. My question is if there is a simpler way of constructing the motive $M$ starting from $\chi$...

Thanks!

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See Schappacher's book, Periods of Hecke characters (chapter un on Motives). dx.doi.org/10.1007/BFb0082094 See also the thread mathoverflow.net/questions/33269/fontaine-mazur-for-gl-1 –  Junkie Mar 10 '12 at 3:19