Abelian varieties with CM

Question

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:

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

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

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

Answer 1

If you take a modular forms $f$ of weight $2$ and $A_f$ is the abelian variety associated to $f$, and if we denote by $\rho_f$ the $p$-adic representation associated to $f$. Then $\rho_f$ splits at $p$ if and only if $f$ has CM and in this case $\rho_f$ is an induced representation.

And if you take a CM hida family, any specialization in weight two rise to a CM ordinary form such that the abelian variety associated to this form has CM.

See the paper of E.Ghate and V.Vatsal '' on the local behaviour of ordinary adic representations''