Special values of $j$-invariant
Let $j(tau)$ be Klein's absolute invariant defined for $tau in mathbb{H}$ by
$$j(tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + cdots$$
with $q := e^{2pi itau}$.
Are there any known special values of $j(tau)$ for which $textrm{Re}(tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $textrm{Re}(tau)$ is rational.
modular-forms
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
add a comment |
Let $j(tau)$ be Klein's absolute invariant defined for $tau in mathbb{H}$ by
$$j(tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + cdots$$
with $q := e^{2pi itau}$.
Are there any known special values of $j(tau)$ for which $textrm{Re}(tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $textrm{Re}(tau)$ is rational.
modular-forms
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
add a comment |
Let $j(tau)$ be Klein's absolute invariant defined for $tau in mathbb{H}$ by
$$j(tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + cdots$$
with $q := e^{2pi itau}$.
Are there any known special values of $j(tau)$ for which $textrm{Re}(tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $textrm{Re}(tau)$ is rational.
modular-forms
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
Let $j(tau)$ be Klein's absolute invariant defined for $tau in mathbb{H}$ by
$$j(tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + cdots$$
with $q := e^{2pi itau}$.
Are there any known special values of $j(tau)$ for which $textrm{Re}(tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $textrm{Re}(tau)$ is rational.
modular-forms
modular-forms
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
asked Mar 3 '17 at 1:54
Ethan AlwaiseEthan Alwaise
6,136517
6,136517
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
Since it wasn't specified that $tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(tau)$ where $Re(tau)$ is irrational. However, $tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(tau) = n$$
can be solved for $tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$tau = frac{_2F_1big(tfrac16,tfrac56,1,1-alphabig)}{_2F_1big(tfrac16,tfrac56,1,alphabig)}sqrt{-1}$$
where
$$alpha=frac{1+sqrt{1-frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$tau approx -0.273239 + 0.6868913,i$$
such that,
$$j(tau)=-2017$$
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
add a comment |
The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $phi$, meaning the golden ratio $phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).
Why integer values are special is a much more difficult question!
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2169550%2fspecial-values-of-j-invariant%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Since it wasn't specified that $tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(tau)$ where $Re(tau)$ is irrational. However, $tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(tau) = n$$
can be solved for $tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$tau = frac{_2F_1big(tfrac16,tfrac56,1,1-alphabig)}{_2F_1big(tfrac16,tfrac56,1,alphabig)}sqrt{-1}$$
where
$$alpha=frac{1+sqrt{1-frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$tau approx -0.273239 + 0.6868913,i$$
such that,
$$j(tau)=-2017$$
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
add a comment |
Since it wasn't specified that $tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(tau)$ where $Re(tau)$ is irrational. However, $tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(tau) = n$$
can be solved for $tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$tau = frac{_2F_1big(tfrac16,tfrac56,1,1-alphabig)}{_2F_1big(tfrac16,tfrac56,1,alphabig)}sqrt{-1}$$
where
$$alpha=frac{1+sqrt{1-frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$tau approx -0.273239 + 0.6868913,i$$
such that,
$$j(tau)=-2017$$
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
add a comment |
Since it wasn't specified that $tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(tau)$ where $Re(tau)$ is irrational. However, $tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(tau) = n$$
can be solved for $tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$tau = frac{_2F_1big(tfrac16,tfrac56,1,1-alphabig)}{_2F_1big(tfrac16,tfrac56,1,alphabig)}sqrt{-1}$$
where
$$alpha=frac{1+sqrt{1-frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$tau approx -0.273239 + 0.6868913,i$$
such that,
$$j(tau)=-2017$$
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
Since it wasn't specified that $tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(tau)$ where $Re(tau)$ is irrational. However, $tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(tau) = n$$
can be solved for $tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$tau = frac{_2F_1big(tfrac16,tfrac56,1,1-alphabig)}{_2F_1big(tfrac16,tfrac56,1,alphabig)}sqrt{-1}$$
where
$$alpha=frac{1+sqrt{1-frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$tau approx -0.273239 + 0.6868913,i$$
such that,
$$j(tau)=-2017$$
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
edited Dec 12 '17 at 9:43
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
answered Dec 12 '17 at 9:25
Tito Piezas IIITito Piezas III
26.9k365169
26.9k365169
add a comment |
add a comment |
The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $phi$, meaning the golden ratio $phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).
Why integer values are special is a much more difficult question!
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
add a comment |
The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $phi$, meaning the golden ratio $phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).
Why integer values are special is a much more difficult question!
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
add a comment |
The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $phi$, meaning the golden ratio $phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).
Why integer values are special is a much more difficult question!
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $phi$, meaning the golden ratio $phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).
Why integer values are special is a much more difficult question!
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
edited Nov 28 '18 at 22:54
Marni Sheppeard
33
33
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
answered Oct 29 '17 at 22:37
Marni Dee SheppeardMarni Dee Sheppeard
11
11
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
add a comment |
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
Hm, $tau$ must have a non-zero imaginary part...
– Tito Piezas III
Dec 12 '17 at 9:26
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2169550%2fspecial-values-of-j-invariant%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
