Mixing
Index of the congruence subgroups of $PGL_2(mathbb{Z}_p)$.
$begingroup$
Let $Gamma_n$ be the $n$-th congruence subgroup of $GL(2,mathbb{Z}_p)$. So $Gamma_n$ consists of matrices in $GL(2,mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(Gamma_n)$ the the center of $Gamma_n$.
My question is to compute the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$
I know that the index of $[Gamma_n:Gamma_{n+1}]=p^4$. I also know that if $G_n$ is the $n$-th congruence subgroup of $SL(2,mathbb{Z}_p)$, then the index of $[G_n:G_{n+1}]$ is probably $p^3$
(because the topological generators of $G_n$ are $1+p^nE_{2,1}, 1+p^nE_{1,2} $ and $ (1+p^n)E_{1,1}+(1+p^n)^{-1}E_{2,2}$).
Here $E_{i,j}$ is the elementary matrix having $1$ at $(i,j)$-th place and $0$ elsewhere.
I can't figure out the the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$, that is the index of the congruence subgroups of $PGL_2(mathbb{Z}_p)$.
Any help is welcome. Thanks in advance.
abstract-algebra group-theory number-theory lie-groups p-adic-number-theory
asked Jan 22 at 3:10
MathStudentMathStudent
593420
$endgroup$
add a comment |
$begingroup$
Let $Gamma_n$ be the $n$-th congruence subgroup of $GL(2,mathbb{Z}_p)$. So $Gamma_n$ consists of matrices in $GL(2,mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(Gamma_n)$ the the center of $Gamma_n$.
My question is to compute the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$
I know that the index of $[Gamma_n:Gamma_{n+1}]=p^4$. I also know that if $G_n$ is the $n$-th congruence subgroup of $SL(2,mathbb{Z}_p)$, then the index of $[G_n:G_{n+1}]$ is probably $p^3$
(because the topological generators of $G_n$ are $1+p^nE_{2,1}, 1+p^nE_{1,2} $ and $ (1+p^n)E_{1,1}+(1+p^n)^{-1}E_{2,2}$).
Here $E_{i,j}$ is the elementary matrix having $1$ at $(i,j)$-th place and $0$ elsewhere.
I can't figure out the the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$, that is the index of the congruence subgroups of $PGL_2(mathbb{Z}_p)$.
Any help is welcome. Thanks in advance.
abstract-algebra group-theory number-theory lie-groups p-adic-number-theory
asked Jan 22 at 3:10
MathStudentMathStudent
593420
$endgroup$
add a comment |
$begingroup$
Let $Gamma_n$ be the $n$-th congruence subgroup of $GL(2,mathbb{Z}_p)$. So $Gamma_n$ consists of matrices in $GL(2,mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(Gamma_n)$ the the center of $Gamma_n$.
My question is to compute the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$
I know that the index of $[Gamma_n:Gamma_{n+1}]=p^4$. I also know that if $G_n$ is the $n$-th congruence subgroup of $SL(2,mathbb{Z}_p)$, then the index of $[G_n:G_{n+1}]$ is probably $p^3$
(because the topological generators of $G_n$ are $1+p^nE_{2,1}, 1+p^nE_{1,2} $ and $ (1+p^n)E_{1,1}+(1+p^n)^{-1}E_{2,2}$).
Here $E_{i,j}$ is the elementary matrix having $1$ at $(i,j)$-th place and $0$ elsewhere.
I can't figure out the the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$, that is the index of the congruence subgroups of $PGL_2(mathbb{Z}_p)$.
Any help is welcome. Thanks in advance.
abstract-algebra group-theory number-theory lie-groups p-adic-number-theory
asked Jan 22 at 3:10
MathStudentMathStudent
593420
$endgroup$
Let $Gamma_n$ be the $n$-th congruence subgroup of $GL(2,mathbb{Z}_p)$. So $Gamma_n$ consists of matrices in $GL(2,mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(Gamma_n)$ the the center of $Gamma_n$.
My question is to compute the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$
I know that the index of $[Gamma_n:Gamma_{n+1}]=p^4$. I also know that if $G_n$ is the $n$-th congruence subgroup of $SL(2,mathbb{Z}_p)$, then the index of $[G_n:G_{n+1}]$ is probably $p^3$
(because the topological generators of $G_n$ are $1+p^nE_{2,1}, 1+p^nE_{1,2} $ and $ (1+p^n)E_{1,1}+(1+p^n)^{-1}E_{2,2}$).
Here $E_{i,j}$ is the elementary matrix having $1$ at $(i,j)$-th place and $0$ elsewhere.
I can't figure out the the index $[Gamma_n/Z(Gamma_n):Gamma_{n+1}/Z(Gamma_{n+1})]$, that is the index of the congruence subgroups of $PGL_2(mathbb{Z}_p)$.
Any help is welcome. Thanks in advance.
abstract-algebra group-theory number-theory lie-groups p-adic-number-theory
abstract-algebra group-theory number-theory lie-groups p-adic-number-theory
asked Jan 22 at 3:10
MathStudentMathStudent
593420
asked Jan 22 at 3:10
MathStudentMathStudent
593420
asked Jan 22 at 3:10
MathStudentMathStudent
593420
asked Jan 22 at 3:10
MathStudentMathStudent
593420
asked Jan 22 at 3:10
MathStudentMathStudent
593420
593420
add a comment |
add a comment |
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