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I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawsky, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimics Langlands in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exist to deduce the existence of $\pi(\rho)$.

But a few paragraphs before, Rogawski introduces the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exists for cyclic extension.

What I naturally want to do is: show that $\rho$ is induced from $\xi$ and then use automorphic induction to prove the existence of $\pi(\rho)$.

Is there a reason for Ragowski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explains why he goes around.

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawski, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimics Langlands in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exist to deduce the existence of $\pi(\rho)$.

But a few paragraphs before, Rogawski introduces the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exists for cyclic extension.

What I naturally want to do is: show that $\rho$ is induced from $\xi$ and then use automorphic induction to prove the existence of $\pi(\rho)$.

Is there a reason for Rogawski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explains why he goes around.

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawsky, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimics Langlands in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exist to deduce the existence of $\pi(\rho)$.

But a few paragraphs before, Rogawski introduces the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exists for cyclic extension.

What I naturally want to do is after, showing that $\rho$ is induced form $\xi$ use automorphic induction to show the existence of $\pi(\rho)$.

Is there a reason for Ragowski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explains why he goes around.

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawsky, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimics Langlands in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exist to deduce the existence of $\pi(\rho)$.

But a few paragraphs before, Rogawski introduces the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exists for cyclic extension.

What I naturally want to do is: show that $\rho$ is induced from $\xi$ and then use automorphic induction to prove the existence of $\pi(\rho)$.

Is there a reason for Ragowski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explains why he goes around.

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawsky, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimic Langlangs in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exists to deduce the existence of $\pi(\rho)$.

But a few paragraph before, Rogawski introduce the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exist for cyclic extension.

What I naturally want to do is after, showing that $\rho$ is induced form $\xi$ use automorphic induction to show the existence of $\pi(\rho)$.

Is there a reason for Ragowski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explain why he goes around.

I am trying to understand the proof of existence of $\pi(\rho)$ for tetrahedral representation (Galois representation of dim 2 having image $A_4$ in $PGL_2(\mathbb{C})$) explained by Rogawsky, Functoriality and the Artin conjecture.

The first step is to show that the representation $\rho$ is induced from a character $\xi$ of an extension $E/F$ of degree 3.

At this point, I guess Rogawski mimics Langlands in showing that $\pi(Ad(\rho))$ and $\pi(\rho_E)$ exist to deduce the existence of $\pi(\rho)$.

But a few paragraphs before, Rogawski introduces the concept of automorphic induction and the result of Arthur-Clozel showing that automorphic induction exists for cyclic extension.

What I naturally want to do is after, showing that $\rho$ is induced form $\xi$ use automorphic induction to show the existence of $\pi(\rho)$.

Is there a reason for Ragowski to not use automorphic induction ? I guess Langlands didn't have the result of Arthur-Clozel that explains why he goes around.