Computing the trace of the image of Frobenius

Question

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ and $q$ distinct primes and $e$ a positive integer. Fixing a basis for the $p^e$ torsion, we get a natural Galois representation

$$\rho_{E,p^e}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}_2(\mathbb{Z}/p^e\mathbb{Z}).$$

If $\sigma_q\in {\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ is the $q$-Frobenius, then it is true that:

1. ${\rm Tr}(\rho_{E,p^e}(\sigma_q))\equiv 1 + q - \#E(F_q)\mod p^e$, and
2. ${\rm det}(\rho_{E,p^e}(\sigma_q))\equiv q \mod p^e$.

I know this is true if $e=1$, but I am wondering if it generalizes to prime powers.

Answer 1

Yes, this should be true by the Lefschetz trace formula.

The action of Frobenius on $H^2$ is given by multiplication with q. That proves 2).

Then you just need that the trace of Frobenius (on $H^*$) is given by the number of fixed points. For this you use the Lefschetz trace formula proven by Deligne in SGA 4.1/2