I have two questions about pure Galois representations. Let $K$ be a number field and consider a continuous $\ell$-adic Galois representation $\rho$. Assume that we have a finite Galois representation $L/K$ such that $\rho\vert_{G_L}$ is pure of some weight, i.e. that $\rho$ is potentially pure. What can we say about the weights of Frobenii on $K$, i.e. if $\mathfrak{q}$ lies over $\mathfrak{p}$, what can be said about the eigenvalues of $\text{Frob}_{\mathfrak{p}}$ in terms of those of $\text{Frob}_{\mathfrak{q}}$. If I'm parsing the definition of weight correctly, then the eigenvalues of Frobenii ought to be of the same weight. So would it be reasonable to say that "potential pureness implies pureness"?
If a representation is pure of weight zero, then there are only finitely many Frobenii which do not act via the identity. Can we conclude that the representation has finite image? Or in what other way is the representation "simple"?
Best Answer
Potential purity implies purity. Fix a prime $\mathfrak p$ of $K$ and let $\mathfrak q$ be a prime of $L$ above it. Assume that $\mathfrak q/\mathfrak p$ is unramified, with residue degree $f$.
Then $\mathrm{Frob}_{\mathfrak q}\in G_L$ is exactly $\mathrm{Frob}_{\mathfrak p}^f$. Indeed, by definition, $\mathrm{Frob}_{\mathfrak p}$ is a pre-image of $x\mapsto x^{N\mathfrak p}$ in $\mathrm{Gal}(\overline{\mathbb F}_{\mathfrak p}/\mathbb F_{\mathfrak p})$ and $\mathrm{Frob}_{\mathfrak q}$ is a pre-image of $x\mapsto x^{N\mathfrak q}$ in $\mathrm{Gal}(\overline{\mathbb F}_{\mathfrak p}/\mathbb F_{\mathfrak q})$. But $N\mathfrak q = (N\mathfrak p)^f$ .
Now, by assumption, $\rho|_{G_L}$ is pure of some weight $w$. That means that the eigenvalues of $\rho(\mathrm{Frob}_{\mathfrak q})$ have absolute value $(N\mathfrak q)^{w/2}$.
But that means that the eigenvalues of $\rho(\mathrm{Frob}_{\mathfrak p}^f)$ have absolute value $(N\mathfrak p)^{fw/2}$. Hence the eigenvalues of $\rho(\mathrm{Frob}_{\mathfrak p})$ have absolute value $(N\mathfrak p)^{w/2}$, i.e. $\rho$ is pure of weight $w$.
For your second question, I think you've misunderstood what pure of weight $0$ means - it does not mean that Frobenii act as the identity. For example, let $\rho$ be the $\ell$-adic Galois representation attached to a weight $3$ modular form, and let $\epsilon$ be the $\ell$-adic cyclotomic character. Then $\rho$ is pure of weight $2$, and $\rho\otimes\epsilon^{-1}$ is pure of weight $0$. But it has infinite image, and its Frobenii do not act trivially.