Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree 0 divisors up to equivalence on $C$). Let the image of this map in $Ch_1(J)$ be denoted as [C.]
Instead of the standard Abel-Jacobi map, I'm interested in "doubling" it. Consider the map $b:C\rightarrow J$ given by $Q\rightarrow 2Q-2P.$ Notice that this map arises as the composition $$C\rightarrow J\times J\rightarrow J$$
where the first map is just $a$ in each component and the latter map is multiplication. We can denote the image of $b$ by the cycle $[W]$ .
Question: Is $[C]\sim [W]$ in the chow ring?
I don't have a particular reason to believe that this is true, but as a student that is fairly new to working with cycles, it would be good to see an argument or counterexample.
Thanks!
By definition of pushforward we have that $4^g[W]=f_*[C]$, where $f$ denotes the isogeny multiplication by 2 and $g$ is the genus of $C$. Now, at least over $\mathbb{C}$, one knows that $(C\cdot\Theta)=g$, where $\Theta$ is a symmetric theta divisor of $J$ (I haven't used symmetry yet). By the projection formula, $$f_*[C]\cdot[\Theta]=f_*([C]\cdot f^*[\Theta])=f_*([C]\cdot[4\Theta])=4f_*([C]\cdot[\Theta]),$$ and so $\deg(f_*[C]\cdot[\Theta])=g4^{g+1}$. If $[W]=[C]$ then $$g4^{g+1}=\deg(f_*[C]\cdot[\Theta])=g4^g,$$ a contradiction.