Applications of inverse of Klein's $j$-invariant
$begingroup$
The Klein $j$-invariant
$$j(tau) = q^{-1} + 744 + 196884q + cdots$$
is a weight $0$ modular function holomorphic for $tau$ in the upper-half plane $mathbb{H}$. I understand that $j$ is important, given that two elliptic curves are isomorphic over $mathbb{C}$ if and only if they have the same $j$-invariant.
We know that $j$ is a bijection from the fundamental domain $operatorname{SL}_2(mathbb{Z}) backslash mathbb{H}$ to $mathbb{C}$.
I am wondering, what are some interesting applications of the inverse $j^{-1}$? What are some situations in which computing $j^{-1}$ of some complex number arises?
motivation modular-function
edited Jan 9 at 18:08
Namaste
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
$endgroup$
add a comment |
$begingroup$
The Klein $j$-invariant
$$j(tau) = q^{-1} + 744 + 196884q + cdots$$
is a weight $0$ modular function holomorphic for $tau$ in the upper-half plane $mathbb{H}$. I understand that $j$ is important, given that two elliptic curves are isomorphic over $mathbb{C}$ if and only if they have the same $j$-invariant.
We know that $j$ is a bijection from the fundamental domain $operatorname{SL}_2(mathbb{Z}) backslash mathbb{H}$ to $mathbb{C}$.
I am wondering, what are some interesting applications of the inverse $j^{-1}$? What are some situations in which computing $j^{-1}$ of some complex number arises?
motivation modular-function
edited Jan 9 at 18:08
Namaste
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
$endgroup$
$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15
add a comment |
$begingroup$
The Klein $j$-invariant
$$j(tau) = q^{-1} + 744 + 196884q + cdots$$
is a weight $0$ modular function holomorphic for $tau$ in the upper-half plane $mathbb{H}$. I understand that $j$ is important, given that two elliptic curves are isomorphic over $mathbb{C}$ if and only if they have the same $j$-invariant.
We know that $j$ is a bijection from the fundamental domain $operatorname{SL}_2(mathbb{Z}) backslash mathbb{H}$ to $mathbb{C}$.
I am wondering, what are some interesting applications of the inverse $j^{-1}$? What are some situations in which computing $j^{-1}$ of some complex number arises?
motivation modular-function
edited Jan 9 at 18:08
Namaste
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
$endgroup$
The Klein $j$-invariant
$$j(tau) = q^{-1} + 744 + 196884q + cdots$$
is a weight $0$ modular function holomorphic for $tau$ in the upper-half plane $mathbb{H}$. I understand that $j$ is important, given that two elliptic curves are isomorphic over $mathbb{C}$ if and only if they have the same $j$-invariant.
We know that $j$ is a bijection from the fundamental domain $operatorname{SL}_2(mathbb{Z}) backslash mathbb{H}$ to $mathbb{C}$.
I am wondering, what are some interesting applications of the inverse $j^{-1}$? What are some situations in which computing $j^{-1}$ of some complex number arises?
motivation modular-function
motivation modular-function
edited Jan 9 at 18:08
Namaste
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
edited Jan 9 at 18:08
Namaste
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
edited Jan 9 at 18:08
Namaste
1
edited Jan 9 at 18:08
Namaste
1
edited Jan 9 at 18:08
Namaste
1
1
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
asked Oct 25 '16 at 23:24
Ethan AlwaiseEthan Alwaise
6,471717
6,471717
$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15
add a comment |
$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15
$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15
add a comment |
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$begingroup$
I'm trying to understand the proof of the little picard theorem using $lambda^{-1}(tau)$. Given that $lambda$ is closely related to $j$, it could mean you that you get a proof of the little Picard theorem using $j^{-1}$. Anyway $j^{-1}$ should be useful in the context of the modularity theorem.
$endgroup$
– reuns
Oct 26 '16 at 0:02
$begingroup$
And also, if $f(tau)$ is modular (i.e. meromorphic $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$) then $f(j^{-1}(tau))$ is meromorphic $mathbb{C}to mathbb{C}$, and you can apply the Liouville theorem that a bounded entire function is constant, obtaining that $j$ generates the function field $SL_2(mathbb{Z})setminus mathbb{H}to mathbb{C}$.
$endgroup$
– reuns
Oct 26 '16 at 0:15