Let $C$ be a smooth projective curve with a $k-$rational point $x_0$ and $J$ its Jacobian variety. Let us consider the (almost) canonical embedding $j:C \to J$ that sends $x_0$ to the identity $e \in J(k)$.
There is a decomposition of the Chow Motive of $C$ that we call $h(C)$, depending on $x_0$, namely if we put $p_0(C) = x_0\times C,\, p_2(C)= C\times x_0$ and $p_1(C)= \Delta - p_0(C) - p_2(C)$ then $$h(C) = h^0(C) \oplus h^1(C) \oplus h^2(C) \quad \text{with }\quad h^i(C) = (C,p_i(C),0)$$
On the other hand, for every abelian variety $A$ of dimension $g$ there is a canonical decomposition $$h(A) = \bigoplus_{i=0}^{2g} h^i(A)$$ where $h^i(A) = (A,\pi_i(A),0)$. The projectors $\pi_i$ are unique in such a way that $$(id_A\times n)^*(\pi_i(A)) = n^i\pi_i(A)$$
I should prove that the induced map $h(j):h(J)\to h(C)$ is such that restricted to $h^n(J)$ for every $n\in \{0,1,2\}$ it induces a morphism of motives $h^n(J)\to h^n(C)$, while for $n >2$ is the zero map. Moreover, it should be an isomorphism for $n=0,1$.
For $n=0$ it should be true for any variety $X$ with a rational point that $h^0(X)$ is isomorphic to the motive of a point, and since $i$ sends $x_0$ to $e$ it should be trivially true that the induced map is an isomorphism of motives.
On the other hand I tried to prove it for $n=1$ but without much results. I know that it should be an "easy" exercise, but I started working on motives quite recently and I am still a bit confused on the techniques I should apply. I think that the proof should be quite clean since I don't think is necessary to use an explicit description of the $\pi_i$'s but rather the property of being "eigenvectors" for the multiplication by $n$.
Any suggestion will be appreciated.
