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Reading Serre's letter to Gray , I wonder if now modern expositions of the themes in Klein's book exist. Do you know any?

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The link should go to page 550 of the book, presumably. – Mariano Suárez-Alvarez Dec 21 '09 at 14:42
up vote 28 down vote accepted

"Geometry of the Quintic" is available for free at my website.

Jerry Shurman

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Wonderfull! Thanks! – Thomas Riepe Jan 23 '10 at 8:56
Thanks. What a beautiful book. – Ben McKay Aug 6 '13 at 21:08
To save other people a few clicks, Jerry's webpage is – James Cranch Jun 10 '14 at 10:59

I got interested in this subject last year and just got round to writing up some notes which I hope may be of use.

I also have a python script which implements the Klein's icosahedral solution of the quintic linked from this page.

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Great! Thanks! – Thomas Riepe Feb 5 '12 at 21:31

I covered Klein's "Lectures on the Icosahedron" in a modern way in my doctoral thesis:

Elliptic Curves and Icosahedral Galois Representations, Stanford University (1999)

A much shorter and more direct exposition is my publication in IMRN:

Icosahedral $\mathbb Q$-Curve Extensions, Mathematical Research Letters 10, 205–217 (2003)

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In Glimpses of algebra and geometry by Gabor Toth, chapter 25 is devoted to Klein's main result.

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There is a (german) new edition of Klein's "Vorlesungen über das Ikosaeder ..." by Peter Slodowy (1993) with a large (about 80 pages) section of comments and remarks about new developments.

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Chapter 5 of McKean and Moll's "Elliptic Curves" explores the circle of ideas around Ikosaeder.I'm not sure if you'd consider this sufficiently "modern" - it's certainly a contemporary book but it doesn't use, say, scheme-theoretic language.

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It looks like our identical answers crossed paths! You beat me by one minute, so I'll delete my answer. – Andy Putman Dec 21 '09 at 19:18
lol, you're right - I caught it just before you deleted. Very gallant of you! and hey, in math, every minute counts. :-) – Alon Amit Dec 21 '09 at 19:58
As an aside to everyone else -- McKean and Moll's book is really beautiful! I read large chunks of it as an undergraduate, and I still go back to it periodically. – Andy Putman Dec 21 '09 at 20:12

There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and Doyle-McMullen approaches (and then some more).

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"expensive", yes. – lhf Dec 21 '09 at 22:51
since it's softcover with horrible print quality - yes it is. – David Lehavi Dec 22 '09 at 5:07
See Jerry Shurman's answer for a link to a free copy ;) – Dr Shello Jan 12 '11 at 8:33
Re: Dr. Shello - yes, Shurman remarked a few months ago on MO he made the text publicly available. This answer predates his release. – David Lehavi Jan 12 '11 at 9:03
Is it really too expansive, or did you mean expensive? – Ben McKay Aug 6 '13 at 21:10

In my PH Thesis work

In my papier I have asociatted a Riemann Surface to each Schwarz function triangle. After that ,I´ve got genus and geometric density of spherical tesselation expresions in a new method. Then I see Their Poincare Groups ( of each above Riemann Surfaces) as index normal finite subgroup of Г(2) (Thanks to Modular Function Lambda). Then I calculate signature of these fuchisian Groups. Finally I see there are only nine ( of above Riemann Surfaces) more Dihedrical cases.

I think my idea is a new interpretation of Schwarz triangles , different one to the Famous Schwarz Classification based on 14 Schwarz triangles +Dihedrical cases.

Alfonso García

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You could also take a look at Section 1.6 of Finite Mobius Groups, Immersion of Spheres, and Moduli, by Gabor Toth.

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