# Talk:Klein quartic

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## Affine version

Title added. —Nils von Barth (nbarth) (talk) 00:52, 17 April 2010 (UTC)

The curve discussed here is the affine version of the Klein quatric. For many applications, the compact Riemann surface (the smooth completion of the affine curve, to be precise) is more interesting. Here the arithmetic set up is more complicated. One needs a cubic extension of Q by \eta which is the real part of \zeta_7. Then 7 factors as the cube of \eta-2 modulo units. The ideal <\eta-2> defines a principal congruence subgroup in a suitable order in a quaternion algebra, which acts cocompactly. That's how one gets the Fuchsian group of the compact Klein quartic. Katzmik 12:24, 1 August 2007 (UTC)

Certainly the claim about seven regular heptagons is incorrect as stated, since the affine curve is not compact. Katzmik 12:26, 1 August 2007 (UTC)
You're right, there's confusion here. It seems to me that: the article focuses on the affine curve; my note about the congruence subgroup correctly applies to the affine curve; the claim about covering by regular hexagons applies to the compact curve. Is this correct? Tesseran 02:33, 2 August 2007 (UTC)
A reasonably complete discussion of an affine curve would include also the number of punctures of the complete hyperbolic Riemann surface of finite area. Also, I am not sure about the uniqueness assertion in this context. At any rate, certainly the most interesting case is the compact surface. All the pictures on J. Baez's site are certainly of the compact surface. I don't know what the focus is of the current version of the article. I find it rather poorly written. Eventually I would like to rewrite it when I find time, unless somebody else gets to it first. As far as the arithmetic background for the compact surface, it is described in detail by Elkies in his contribution to the "Eightfold" volume (with one small error that he corrected in another paper of his). Katzmik 07:23, 2 August 2007 (UTC)

Just to clarify a small point: the quartic equation in the article IS that of the compact curve, as it is given in homogeneous coordinates. Katzmik 10:56, 5 August 2007 (UTC)

The Fuchsian group of the compact Klein quartic is not a subgroup of the modular group PSL(2,Z), contrary to a claim that appears on Baez's homepage. I wrote to him twice about this, but the error has not been corrected. Thus, the current wiki site Klein quartic sends the visitor to a site containing erroneous information. I will therefore remove the link to Baez's page for now. Once the error is corrected, there is no harm in reinstating the link. Katzmik 08:41, 7 August 2007 (UTC)

## Affine case

The affine curve can indeed be represented by the congruence mod 7 in the modular group. This information can also be found in Elkies' papers. Katzmik 07:24, 8 August 2007 (UTC)

## Baez' error

The link to Baez's homepage was reinstated prematurely. His piece on the Klein quartic, while stimulating, contains a serious arithmetical error. It is contained in the paragraph starting with the words "Now I should break down". I have therefore removed the link. When the error is corrected, one could reintroduce the link. Katzmik (talk) 08:27, 15 April 2008 (UTC)

Many textbooks contain errors, but this is hardly a reason to avoid citing them if they contain useful information. Baez's article contains much that isn't available here (and includes fancy graphics to boot). As long as the error is corrected here I see no harm in linking the article.
As an aside, I have to admit I don't fully understand the nature of the error. Baez's article makes no mention of Fuchsian groups (at least by name). Are you saying that H/Γ(7) is not the Klein quartic? Or it is not the compact curve? If the confusion is simply between the affine curve and the compact curve, it hardly seems like a serious error to me. -- Fropuff (talk) 16:32, 15 April 2008 (UTC)
That's precisely what I am saying, H/Gamma(7) is not the Klein quartic. Namely, it is not a compact curve, while the Klein quartic is a compact curve. The arithmetic set-up needed to describe the Klein quartic is considerably more elaborate than Baez's page leads one to believe, as I already mentioned earlier on this wikipage. For instance, it cannot be defined over Q, but only over a cubic extension thereof. Katzmik (talk) 14:31, 16 April 2008 (UTC)
It seems to be more a case of not being completely rigorous, than an outright error. One can still obtain the compact curve from H/Γ(7) by appropriately filling in the cusp points, no? It should hopefully be clear to the reader that Baez's article is not meant to be completely rigorous. In any case, the correct construction is mentioned here, thanks to you. -- Fropuff (talk) 16:31, 16 April 2008 (UTC)
Baez is a first rate mathematician and would be shocked to be characterized as unrigorous. I assume this particular item on his homepage was written by a graduate student and he has not gotten around to correcting it. One can certainly fill in the cusp points but the hyperbolic metric on H/Gamma(7) has little relation to that of the Klein quartic, and one certainly loses the connection between the Klein quartic and the other Hurwitz surfaces which cannot be constructed (even up to filling in the cusps) out of the modular group. Katzmik (talk) 17:04, 16 April 2008 (UTC)
I meant no slight to Baez. On the contrary, I found his article quite helpful. I think others will as well, despite this shortcoming. Hence my desire to relink it from here. -- Fropuff (talk) 17:52, 16 April 2008 (UTC)

The page being discussed above is (apparently) http://math.ucr.edu/home/baez/klein.html. --Archelon (talk) 16:31, 21 April 2008 (UTC)

that's the right page. The best solution of course would be to have the error on his homepage corrected. Did you try sending him an email message? Katzmik (talk) 13:41, 28 April 2008 (UTC)

I agree with Fropuff. As long as the mistake is clearly explained, there is no harm in linking to Baez's page. There is much worthwhile material on Baez's webpage, and we should link to it. --C S (talk) 15:55, 30 April 2008 (UTC)

## Illustrations

Is it possible to use Klein's original illustration (a tessellated regular hyperbolic 14-gon with instructions for gluing) or the redrawing found in the MSRI volume? I don't know whether either of these would violate copyright. It would also be nice to show Helaman Ferguson's sculpture "The Eightfold Way." Ishboyfay (talk) 17:25, 20 September 2008 (UTC)

I've substituted a high-res scan of the cover of "The Eightfold Way" (the book) for the low-res Amazon file that was there. I've checked with the publisher that it's OK. Regarding Klein's illustration, it's not under copyright anymore, so anyone can make a scan. I can also provide a Postscript file of the figure in the book, which is my redrawing of Klein's illustration. levy@msri.org Silvio Levy (talk) 23:31, 13 June 2011 (UTC) (editor of the book)

## The Klein quartic has just one valid definition

The article states:

"The Klein quartic can be viewed as an algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates: x^3 y + y^3 z + z^3 x = 0."

This is *misleading*, because the locus of that equation in the complex projective plane is the definition of the Klein quartic, not merely a way that it "can be viewed".

Later in the article, the Klein quartic is described as being tilable by 24 congruent regular hyperbolic heptagons. This statement is false, because the geometry of the Klein quartic does not have constant negative curvature.

Rather, what is tilable with 24 congruent regular hyperbolic heptagons is a surface that is *conformally equivalent* to the Klein quartic.

This is an important distinction, since it is a common mathematical error to confuse the two concepts.Daqu (talk) 04:59, 17 November 2009 (UTC)

## THE affine quartic?

The text uses a definite article to describe an affine version of the compact quartic. Is an affine curve unique in any sense? Tkuvho (talk) 07:47, 7 June 2010 (UTC)

## The affine quartic?

I'm also uncomfortable with the terminology affine quartic. If you start from the usual projective equation x3y+ y3z+ z3x of the quartic and try to restrict it to an affine chart of P2C (example: z=1) then you will remove no more than 4 points. However H2/Gamma(7) has 24 punctures. Or maybe affine means that there is a (natural) local chart system on the punctured quartic for which the coordinate changes belong to the group group of affine transformations az+b of C? This should be investigated. Citations needed? Arnaud Chéritat (talk) 20:41, 25 February 2014 (UTC)

I thought again about it: there is indeed an chart system with affine coordinate changes: map each triangle of the order 7 triangle tessellation to a Euclidean triangle of the order 6 tesselation of the plane. The mapping does not close properly if one turns around a puncture but the affine structure does. Yet, I think one should either point to a reference where the terminology affine quartic is used or use another name. Arnaud Chéritat (talk) 08:51, 9 March 2014 (UTC)

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