CM abelian varieties and potential good reduction

Question

Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite extension $L$ of $F$ such that $A_L$ has good reduction over every prime of $L$.

And what about the inverse: if $A$ is known to be of potential good reduction everywhere, how far is it from having complex multiplication?

As the reduction behavior is determined by the Galois representations of the decompositon groups, one can reformulate the problem as follows: let $A$ be an abelian variety over $F$, $p$ a fixed rational prime, $V$ the p-adic Tate module of $A$; and for $\lambda$ primes of $F$, $\rho_\lambda$ is the $p$-adic representation on $V$ of the decomposition group $G_\lambda$ at $\lambda$. If $\rho_\lambda$ is potentially unramified for $\lambda$ not dividing $p$, and potentially cristalline for $\lambda$ dividing $p$, do we know that the global Galois representation $\rho$ on $V$ is potentially abelian, i.e. when shifting to some open subgroup, the image of $\rho$ is contained in a torus of $GL_{\mathbb{Q}_p}(V)$? What do we know about the Fontaine-Mazur conjecture in this case?

Thanks!

Answer 1

No, absolutely not

In fact, the hypotheses you discuss are rather weak. Take $F$ a totally real number field. If $A/F$ is the abelian variety attached to an eigenform $f$ of weight $2$ and level $N$, then the representation $\rho$ attached to the $p$-adic Tate module of $A$ is crystalline at $p$ for $p\nmid N$ . At $\ell≠p$, the representation $\rho$ is potentially unramified if and only if the automorphic representation $\pi(f)_{\ell}$ is principal series of supercuspidal (i.e not Steinberg). This certainly happens a lot. All such eigenforms do not have potential CM.

I'm not sure I understand your question about the Fontaine-Mazur conjecture but much is known in the set-up that you seem to be interested in. For instance, if $F$ is totally real and $p\geq 7$ then an odd $G_F$-representation in $\operatorname{GL}_{2}(\mathbb F_{p})$ which stays irreducible after restriction to $G_{F(\zeta_{p})}$ comes from the Tate module of an abelian variety $A/F$.

Answer 2

Potential good reduction everywhere is quite far from having complex multiplication.

For elliptic curves, the condition is equivalent to the $j$-invariant being an algebraic integer. For $F = \mathbb{Q}$ there are only finitely many $j$-invariants of CM elliptc curves (corresponding to the imaginary quadratic fields of class number $1$) but any element of $\mathbb{Z}$ is of course the $j$-invariant of an elliptic curve over $\mathbb{Q}$.