In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case posted on his homepage (he made a mistake three years ago for full BSD). But, I am interested in this present paper because he uses the fact that the Tate module over $K_{\wp}$ equipped with $Gal(\overline{K}/K)$ splits if the elliptic curve $E$ has CM by $K$ and relates the L-function of $E$ and Artin L-functions of algebraic number fields. I think that this can be generalized to the higher dimensional Abelian varieties with CM. In particular, it is intereting that it applies to the Jacobian of a curve with higher genus. Now, my questions are:

the Tate module of such Abelian variety also splits? (under some assumptions)

the Jacobian of a curve with higher genus often has CM?

the L-function of the Jacobian of a curve with higher genus has anything to do with rational points on that curve?