# Abelian varieties with CM

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?

• I'm not sure what you mean by $K_{\mathfrak{p}}$ above. – Jeff Yelton Jan 13 '16 at 11:39
• At any rate, if in (2) you are asking whether "most" higher-dimensional Jacobians have CM, the answer is certainly "no" (as is the case for elliptic curves). – Jeff Yelton Jan 13 '16 at 11:43
• (In any case, in (2), please clarify what you mean precisely by "often".) – Todd Trimble Jan 27 '16 at 16:44

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.