This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
In the beginning of the article j(i)=1728, and in the end j(i)=1. One has something to do with it. — Preceding unsigned comment added by 95.84.228.116 ( talk) 10:41, 22 June 2013 (UTC)
There is a notational problem in this and related articles; standard notation seems to be that and j=1728 J. I'll try to review correctness/fix this when I have the chance. (where linas 03:54, 20 Jun 2005 (UTC)
If It can be of any help I think I remember there are two notations for Eisenstein series too: or , correspoing to different conventions (leading coefficient one, integer coefficients, norm 1 etc...)
Why is the elliptic modular function attributed to Klein? The fundamental domain of the modular group was known to Gauss. It was studied by Kummer, and by Hermite (1858) in connection with the solution of 5-ics, and defined by Dedekind in a remarkable paper in Crelle ?1878. Klein became involved after Dedekind. He defined j in terms of absolute invariants of Int dx/y, y^2 = 4-ic in x. After a fw pages he reverts to Dedekind's definition. [John McKay] May be Atkin should be consulted?]
"the Fourier coefficients for the positive exponents of q are the dimensions of the grade-n part" - It's not clear what n is. Does it mean the coefficient corresponding to q^n in the q-expansion?
"the rate of growth of ln(cn) is" - cn is never defined in the article. Does it mean the coefficient corresponding to q^n? Also, the asymptotic formula shows the growth of ln(cn), not its rate of growth.
"there are exactly 6486 of them [functions]... (see here for the complete list)" - The link is to a document which appears to not define a single function. For example, "14A0 14B 28B 1 1 1 2C0 71B0 14" does not seem to be a function. —Preceding unsigned comment added by 209.67.107.10 ( talk • contribs) 17:54, 6 July 2009
I have removed the doubtful material in the other two cases: if anyone can restore and clarify them please do so. JamesBWatson ( talk) 10:33, 14 December 2009 (UTC)
Elliptic modulus redirects here but is not mentioned.-- JohannesBuchner ( talk) 14:04, 4 November 2014 (UTC)
The section Special values begins as follows:
"The j-invariant vanishes at the "corner" of the fundamental domain at
Here are a few more special values (only the first four of which are well known; in what follows, j means J/1728 throughout):
BUT: There is no capital-J function defined anywhere in the article.
I see that someone already began a section about notation problems, but that was 2.5 years ago and this problem still persists.
Would someone knowledgeable in this subject please fix this? E.g., by defining a new function that is j(τ)/1728 instead of using the same notation both for the standard j and a nonstandard j ???
Or else please use the simpler idea of just stating that the list of Special values is a listing of j(τ) /1728 ??? Daqu ( talk) 19:40, 19 January 2016 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on J-invariant. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 13:51, 18 November 2017 (UTC)
The section Special values suddenly seems to use capital J instead of what the rest of the article uses: a lowercase J ("j").
Without any explanation. If the article defines the capital J function somewhere, I haven't found it (and it would not be easy to find).
Can someone knowledgeable on this subject please make this much clearer? 50.203.182.230 ( talk) 17:37, 31 October 2019 (UTC)
The section Class field theory and the j-invariant contains this sentence:
"If τ is any CM point, that is, any element of an imaginary quadratic field with positive imaginary part (so that j is defined), then j(τ) is an algebraic integer."
But the meaning of "CM" has not been explained anywhere in the article before this usage.
I hope someone knowledgeable about the subject can fix this bad writing. 2601:200:C000:1A0:3998:89DD:A33:8E64 ( talk) 03:31, 25 September 2021 (UTC)
The first sentence of the section Definition contains the phrase
"with the third definition implying j(𝝉) can be expressed as a cube".
It may be possible for some readers to guess what kind of cube is intended by this statement.
But since this is an encyclopedia article, it would be much better to state explicitly what kind of "cube" is referred to here.
Is it "a cube in the field of meromorphic functions on the upper half plane" ?
Of course, the answer is Yes, but I mean: Is this the kind of "cube" referred to in the phrase I quoted? 2601:200:C000:1A0:F4CF:60FC:DCAC:F9C5 ( talk) 21:43, 5 June 2022 (UTC)
The section Definition begins as follows:
"The j-invariant can be defined as a function on the upper half-plane H = {τ ∈ C, Im(τ) > 0},
with the third definition implying can be expressed as a cube, also since 1728."
But it is not at all clear what "expressed as a cube" means.
A cube of what ???
If 1728 were not the cube of an integer but instead were replaced by b3 for some non-integer complex number b, then of course we could still express the j-invariant as a cube ... of something.
I hope someone knowledgeable about this subject can fix this and make it clear what this means.
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
In the beginning of the article j(i)=1728, and in the end j(i)=1. One has something to do with it. — Preceding unsigned comment added by 95.84.228.116 ( talk) 10:41, 22 June 2013 (UTC)
There is a notational problem in this and related articles; standard notation seems to be that and j=1728 J. I'll try to review correctness/fix this when I have the chance. (where linas 03:54, 20 Jun 2005 (UTC)
If It can be of any help I think I remember there are two notations for Eisenstein series too: or , correspoing to different conventions (leading coefficient one, integer coefficients, norm 1 etc...)
Why is the elliptic modular function attributed to Klein? The fundamental domain of the modular group was known to Gauss. It was studied by Kummer, and by Hermite (1858) in connection with the solution of 5-ics, and defined by Dedekind in a remarkable paper in Crelle ?1878. Klein became involved after Dedekind. He defined j in terms of absolute invariants of Int dx/y, y^2 = 4-ic in x. After a fw pages he reverts to Dedekind's definition. [John McKay] May be Atkin should be consulted?]
"the Fourier coefficients for the positive exponents of q are the dimensions of the grade-n part" - It's not clear what n is. Does it mean the coefficient corresponding to q^n in the q-expansion?
"the rate of growth of ln(cn) is" - cn is never defined in the article. Does it mean the coefficient corresponding to q^n? Also, the asymptotic formula shows the growth of ln(cn), not its rate of growth.
"there are exactly 6486 of them [functions]... (see here for the complete list)" - The link is to a document which appears to not define a single function. For example, "14A0 14B 28B 1 1 1 2C0 71B0 14" does not seem to be a function. —Preceding unsigned comment added by 209.67.107.10 ( talk • contribs) 17:54, 6 July 2009
I have removed the doubtful material in the other two cases: if anyone can restore and clarify them please do so. JamesBWatson ( talk) 10:33, 14 December 2009 (UTC)
Elliptic modulus redirects here but is not mentioned.-- JohannesBuchner ( talk) 14:04, 4 November 2014 (UTC)
The section Special values begins as follows:
"The j-invariant vanishes at the "corner" of the fundamental domain at
Here are a few more special values (only the first four of which are well known; in what follows, j means J/1728 throughout):
BUT: There is no capital-J function defined anywhere in the article.
I see that someone already began a section about notation problems, but that was 2.5 years ago and this problem still persists.
Would someone knowledgeable in this subject please fix this? E.g., by defining a new function that is j(τ)/1728 instead of using the same notation both for the standard j and a nonstandard j ???
Or else please use the simpler idea of just stating that the list of Special values is a listing of j(τ) /1728 ??? Daqu ( talk) 19:40, 19 January 2016 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on J-invariant. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 13:51, 18 November 2017 (UTC)
The section Special values suddenly seems to use capital J instead of what the rest of the article uses: a lowercase J ("j").
Without any explanation. If the article defines the capital J function somewhere, I haven't found it (and it would not be easy to find).
Can someone knowledgeable on this subject please make this much clearer? 50.203.182.230 ( talk) 17:37, 31 October 2019 (UTC)
The section Class field theory and the j-invariant contains this sentence:
"If τ is any CM point, that is, any element of an imaginary quadratic field with positive imaginary part (so that j is defined), then j(τ) is an algebraic integer."
But the meaning of "CM" has not been explained anywhere in the article before this usage.
I hope someone knowledgeable about the subject can fix this bad writing. 2601:200:C000:1A0:3998:89DD:A33:8E64 ( talk) 03:31, 25 September 2021 (UTC)
The first sentence of the section Definition contains the phrase
"with the third definition implying j(𝝉) can be expressed as a cube".
It may be possible for some readers to guess what kind of cube is intended by this statement.
But since this is an encyclopedia article, it would be much better to state explicitly what kind of "cube" is referred to here.
Is it "a cube in the field of meromorphic functions on the upper half plane" ?
Of course, the answer is Yes, but I mean: Is this the kind of "cube" referred to in the phrase I quoted? 2601:200:C000:1A0:F4CF:60FC:DCAC:F9C5 ( talk) 21:43, 5 June 2022 (UTC)
The section Definition begins as follows:
"The j-invariant can be defined as a function on the upper half-plane H = {τ ∈ C, Im(τ) > 0},
with the third definition implying can be expressed as a cube, also since 1728."
But it is not at all clear what "expressed as a cube" means.
A cube of what ???
If 1728 were not the cube of an integer but instead were replaced by b3 for some non-integer complex number b, then of course we could still express the j-invariant as a cube ... of something.
I hope someone knowledgeable about this subject can fix this and make it clear what this means.