Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?

Question

$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link

Just so everyone is on the same page.

-Let $E$ be a number field, $\lambda$ a place over $l$ and $E_{\lambda}$ the corresponding complete local field.
-Let $\mathbb I$ be the group of ideles of $K$ and $\mathbb I_{\infty}$ the subgroup of those ideles whose only nontrivial components are on the infinite places.
-$T=\Res_{K/\mathbb Q}(G_{m/K})$ is the torus obtained by restrincting scalar from $K$ to $\mathbb Q$ of the multiplicative group $\mathbb G_{m/K}$.

On this lemma Ribet says that a locally algebraic character $\delta_{\lambda}:\Gal(\overline{\mathbb Q}/\mathbb Q)\longrightarrow E_{\lambda}$ is associated to a Grossencharacter of type $A_0$ or in modern lenguage an algebraic Hecke character.

By that I understand that what he means is that $\delta_{\lambda}\circ i:\mathbb I_{K}\longrightarrow E_{\lambda}$ is and algebraic Hecke character in particular ir takes values in $E$. My definition of algebraic Hecke character is a character of the idele class group $\chi:\mathbb I_{K} \longrightarrow \mathbb C$ such that when restricted to the infinite places it is of the form $\chi(x_{\infty})=\prod_{\nu \in S_{\infty}} x_{\nu}^{n_{\nu}}$ with $n_{\nu}\in \mathbb Z$.

For the proof of this fact Ribet cite page 761 of *"Galois action on division points of abelian varieties with real multiplications" JSTOR link behind paywall.

What says there is that all locally algebraic representations come from linear representations of a certain algebraic group $S_{\mathfrak m}$ called the Serre torus. Later on this same article it is stated that characters of the serre torus are given by a pair of homomorphisms $\chi:\mathbb I_{K}\longrightarrow \overline{\mathbb Q}$ and $f:T\longrightarrow S_{\mathfrak m}$ such that
1-$\chi(x)=1$ when $x\in U_{\mathfrak m}$
2-$\chi(x)=f(x)$ when $x\in K^*$

I suppose that using this two properties one can show that $\chi$ is an algebraic Hecke character but the things I have tried did not work and I can not found the proof anywhere. I am also wondering if in the case of a representation of dimension 1 we can skip the use of the Serre torus $S_{\mathfrak m}$ since the condition of being locally algebraic seem to me very close to conditions 1 and 2. I am not very used to work with ideles so do not worry about being too explicit.

Answer 1

For $K =\mathbb Q$, class field theory (or, in this case, Kronecker-Weber) says that the abelianization of the Galois group of $\mathbb Q$ is $\mathbb I/\mathbb I_\infty$. So a character $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q) \to E_\lambda$, which necessarily factors through the abelianization, is the same as a character $\mathbb I/\mathbb I_\infty \to E_\lambda$.

Thus we have a character $\chi\colon \mathbb I \to E_\lambda$ that is trivial on $\mathbb I_\infty$. Since the trivial representation is algebraic, why isn't $\chi$ already a Grossencharacter? The reason is Grossencharacters are required to have the image of the units of each non-archimidean local field be finite, i.e. in this case we want the image of $\mathbb Z_p^\times$ to be finite for all $p$. This is true for all $p\neq l $ but may not be true for $l$.

Instead, the restriction of $\chi$ to $\mathbb Z_l^\times$ is locally algebraic, i.e. there is some open subgroup $U$ and some integer $n$ such that $x\in U$ is sent by $\chi\mid_{\mathbb Z_\ell^\times}$ to $x^n$.

All we have to do to get a Grossencharacter is to modify the character $\chi$ by dividing $\chi(x)$ for an idele $x$ by the $n$th power of the $\mathbb Q_l$-component of $x$ and then multiplying by the $n$th power of the $\mathbb R$-component of $x$. For $x\in \mathbb Q$ the $\mathbb Q_p$ and $\mathbb R$ component are the same rational number so this doesn't do anything.

This preserves all local properties of the character at places other than $l$ and $\infty$. At $l$ we have made the character trivial on the open subgroup $U$, giving the desired finite image of the units. At $\infty$ we have made the character look like $x \to x^n$ and hence be algebraic.

We can send Grossencharacters to locally algebraic Galois characters by reversing this process.

For other fields $K$ it's not much more complicated, we just replace the formula $x^n$ by some product of powers of images of $x$ under different embeddings of $K$.