I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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."

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motive is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especially when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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."

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motive is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank 1 motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

I am trying to get some insight into Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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"

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motive is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

Thanks a lot for your help

I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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."

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motive is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

I am trying to get some insight into Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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"

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motives is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

Thanks a lot for your help

I am trying to get some insight into Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex multiplication. Unfortunately in his paper "Motives for modular forms" Scholl says

"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"

Question 0: Could anyone explain this or give a readable reference?

More precisely, I am interested in the following question. If $E$ is an elliptic curve over a number field $K$, one can look at the Chow motive with rational coefficients $$ h^1(E) $$ cut off from the total motive $(E, \nabla, 0)$ by $\nabla$ minus the two projectors associated to some rational point (over an extension of $K$ if necessary). In general, this motive is indecomposable.

However, when $E$ has complex multiplication by a quadratic imaginary field $F=\mathbb{Q}(\sqrt{-d})$, one gets a decomposition of $h^1(E)_F$ into two rank motives with coefficients in $F$.

On the other hand, one can look to the weight two modular form $f$ attached to $E$ (modularity in the CM case, which is much easier!). Thanks to the construction quoted by Scholl one gets a rank two motive with coefficients in the field generated by the Fourier coefficients of $f$.

Question 1: (General elliptic curve) What is the relation between these two motives? One would like to say there are the same, but the coefficient field are different, aren't?

Question 2 (CM case): Does the geometric construction using Jacobians of modular forms also give the decomposition into two pieces of rank one? If so, how?

Thanks a lot for your help