Given an étale double covering of curves $f: C\to C_0$, there is an induced norm map $Nm: J(C) \to J(C_0)$, which sends $\sum_i p_i$ to $\sum_i f(p_i)$. On page 285 of the book Geometry of algebraic curves, question 18 and question 19 are concerned about the kernel of the Nm.
Q.18 Show the kernel of Nm consists of a finite union of cosets of the image of $1-\tau$ by the points of order $2$ on $J_2(C)$. ($\tau$ is the induced map on the Jacobian of $C$ by the order two action. $J_2(C)$ is the subgroup of $J(C)$ which consists of the isomorphism classes of degree zero line bundles $L$ such that $L^{\otimes 2}\simeq O_C$).
Q.19 Show that the order of $\operatorname{ker}(Nm)\cap J_2(C)$ is $2^{(2g+1)}$ where $g$ is the genus of $C_0$ and conclude the kernel of Nm has two connected components.
Can anyone give some suggestions about how to solve this two problems??
