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Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives:

(a) The Chow motive $h^1(E)$ appearing in the decomposition of the total motive

$h(E)=h^0(E) \oplus h^1(E) \oplus h^2(E)$

(b) The Grothendieck motive $M(f)$ attached by Scholl to the modular form

Are these motives known to be isomorphic?

If in addition $E$ has complex multiplication by $K$, the $L$ function is also the $L$-function of a Hecke character $\psi$ of $K$. In this case we also have

(c) The Grothendieck (or Chow?) motive $M(\psi)$ attached to the Hecke character. I am less familiar with this construction but, from what I understood, is something you cut off in the cohomology of some power of the elliptic curve (contrary to the case of the modular form, which comes from the cohomology of the universal elliptic curve).

Are (b) and (c) known to be isomorphic?

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    $\begingroup$ Chow motives (for rational equivalence) and Grothendieck motives (for numerical, or homological, equivalence) do not live in the same category. They cannot be isomorphic. $\endgroup$ – abx May 14 '14 at 17:25
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    $\begingroup$ Of course, but you are not answering the real question (and you know it). $h^1(E)$ exists also as homological motive. Is it isomorphic to the other two? $\endgroup$ – paL May 14 '14 at 18:50
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From the first page of Scholl's paper:

1.0.0. Consider integers $n \geq 3$, $k \geq 1$. (We do not treat here the case $k = 0$, which corresponds to cusp forms of weight $2$; the associated motives are then given by the Jacobians of modular curves, and are well understood.)

So (a) and (b) are indeed isomorphic, but this is basically just the modularity theorem.

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