TITLE: Elliptic factors in the Jacobian and zeta function
QUESTION [7 upvotes]: Consider a hyperelliptic curve $\mathcal{C}$ over $\mathbb{Q}$ and its Jacobian $J(\mathcal{C})$. Assume that $J(\mathcal{C})$ admits an elliptic factor $\mathcal{E}$.
For almost all primes, we can reduce $\mathcal{C}$ modulo $p$, and consider its zeta function $Z_p$. Can the elliptic factor property be seen as a special property of $Z_p$?
Conversly, if such property is satisfied for almost all primes, can we prove that a hyperelliptic curve possess an elliptic factor in its Jacobian?

REPLY [5 votes]: This is an incomplete answer to the second question (Jackson has indicated the answer to the first question in the comments).
If you ask the same question over a general number field $F$, then you will find counterexamples by considering "false elliptic curves" i.e. principally polarized abelian surfaces with quaternionic multiplication. These are automatically  Jacobians of hyperelliptic curves of genus two. They have the property that for almost all primes their reduction is a product of elliptic curves, but over $F$ (and even over its algebraic closure) the Jacobian may very well be simple (hence no elliptic factors).
Over $\mathbb{Q}$, you might look at abelian surfaces with "potential" quaternion multiplication, but actually I would guess that these don't give counterexamples--  just a guess.  In genus greater than 2, it's even less clear to me.