2
$\begingroup$

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.

If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to endomorphism $F$ of $E$ (over field extension, so that $E$ has CM). Does characteristic polynomial of $F_p$ acting on Tate module $T_\ell E_p$ equal characteristic polynomial of $F$ acting on Tate module of $E$?

If yes, is this true in higher dimensions?

$\endgroup$

1 Answer 1

2
$\begingroup$

Yes. See Milne's text on Abelian Varieties: it shows that deg(F-n) is a polynomial in ℤ[n], and degree is preserved by reduction mod p. For elliptic curves, the characteristic polynomial has coefficients det(F)=deg(F) and Tr(F)=1+deg(F)-deg(1-F).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.