Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and $\rho = \rho_E : G_{\mathbb{Q}_p} \to \operatorname{GL}_2(\mathbb{C})$ its induced Galois representation.
We can define the characteristic polynomial of $\rho$ to be $$P(\rho,T) = \det(1-\operatorname{Frob}_K^{-1}T \, | \, \rho^{I_K} ).$$ Here, $\operatorname{Frob}_K \in G_{\mathbb{Q}_p}$ is an Frobenius element (i.e. a lift of $\operatorname{Frob}_p: x \mapsto x^p \in G_{\mathbb{F}_p}$ to $G_{\mathbb{Q}_p}$) and $\rho^{I_K} = \{ x\in \mathbb{C}^2 \, | \, \rho(x) = x \quad \forall x \in I_K\}.$
There is this result which I always used but never proved:
If $E$ has good reduction, it is $P(\rho,T) = pT^2 - a_p(E) T + 1$ where $a_p(E) = p+1-|\bar{E}(\mathbb{F}_p)|$.
Could you give me any reference where I could find the proof and all necessary definitions and intermediate results to understand this proof? I have no orientation on how to find such a reference, especially since I lack background knowledge despite being in research for a while now.
Could you please help me with this request?
If $E$ has good reduction then the reduction is an isomorphism on $E[l^\infty]\to \widetilde{E}[l^\infty]$
Those two torsion subgroups are $\cong (\Bbb{Z[l^{-1}]/Z})^2$.
$Aut((\Bbb{Z[l^{-1}]/Z})^2)\cong GL_2(\Bbb{Z}_l)$.
$Frob_p$ and $\phi$ (the true Frobenius on the reduction) act as matrices of $Gl_2(\Bbb{Z}_l)$ on those two torsion subgroups.
The characteristic polynomial of those two matrices defining the action of $Frob_p$ and $\phi$ are the same.
Because non-zero isogenies have finite kernel the characteristic polynomial of $\phi$ acting on $\widetilde{E}[l^\infty]$ doesn't depend on $l$.
We have that $$|\widetilde{E}(F_p)|= |\ker(\phi-1)|= N(\phi-1)$$ $$=(\phi^*-1)(\phi-1)=\phi^*\phi+1-Tr(\phi)=p+1-Tr_l(\phi)$$ where $Tr_l(\phi)$ is the trace of the matrix of the action of $\phi$ on $\widetilde{E}[l^\infty]$
and so $$|\widetilde{E}(F_p)|=p+1-Tr_l(Frob_p)$$ From there define your Galois representation, its trace should relate easily to the integer $Tr_l(Frob_p)$