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I have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would like to study the arithmetical part related to imaginary quadratic fields and their relation to elliptic functions. Having very limited knowledge of Group Theory and Galois Theory (and things covered under Modern/Abstract Algebra) I tried to find classical references and stumbled upon Weber's Algebra, but unfortunately it is in German. Does anyone know if an English translation exists and if so how can it be obtained?

If there is no English translation available can someone point out any good references for "Imaginary quadratic fields and their relation to elliptic functions" keeping in view my very limited knowledge of Abstract Algebra.

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  • $\begingroup$ This book was pivotal for my thesis (all the stuff of the $\,j-$invariant) and I had to swallow it in german. If you're already at a stage you need Weber then you may as well begin to work your way through german. OTOH, being with a "very limited" knowledge of group theory and Galois theory getting into elliptic functions seems to be not a very good idea...hopefully your complex analysis is better, but anyway I think you must first try to cover these three subjects thoroughly. $\endgroup$ – DonAntonio Jul 8 '13 at 7:58
  • $\begingroup$ In the past I have tried to study Group Theory / Galois Theory but somehow it looks so different from other mathematical fields like "Analysis", "Number Theory". The final results in Group Theory look too deep compared to the starting axioms and elementary results. I am just unable to believe that such simple axioms of a group lead to such complex and deep results. That's one of the major stumbling blocks. To give an example I read construction of regular polygons from Disquisitiones Arithmeticae and found it to be much simpler than the corresponding treatment from Galois Theory. $\endgroup$ – Paramanand Singh Jul 8 '13 at 8:07
  • $\begingroup$ First, there are no "final results" in most mathematical fields, and not in group theory. OTOH, you're really going to need bad all this stuff if you really want to understand elliptic functions in some depth. $\endgroup$ – DonAntonio Jul 8 '13 at 8:11
  • $\begingroup$ I got your point DonAntonio and I believe it will require some reasonable amount of time and hard work before I can appreciate elliptic functions and quadratic fields. Will definitely put effort regarding this. BTW, by "final" results I meant results presented in later (last) chapters on books of Group theory. Sorry if that term offended. $\endgroup$ – Paramanand Singh Jul 8 '13 at 8:16
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    $\begingroup$ Dear Paramanand, If your aim is to quickly reach into modern arithmetical theory, the best reference for you at this stage might be E. Hecke, Lectures on the Theory of Algebraic Numbers. It is hard to beat it, and it has no prerequisites at all. It does not contain the proof of Kronecker-Weber, unfortunately. It does contain a great many interesting things, though. Definitely have a look. $\endgroup$ – user96815 Oct 31 '13 at 16:00
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This is a very deep subject which continues to be pivotal in contemporary number theory. I've never read Weber carefully, but my impression is that his arguments are difficult to follow and regarded as somewhat incomplete. For example, in his solution of the class number one problem, Heegner cited some of Weber's results, and my sense is that one of the reasons that people originally rejected Heegner's argument was that these resuls of Weber were regarded as unproved.

There is a survey by Brian Birch from the late 60s where he goes over the results of Weber that Heegner uses and shows why they are all valid (and hence why Heegner's argument is valid); his arguments use class field theory.

In general, I'm not sure that you can learn all that much about this subject without learning some Galois theory and algebraic number theory; indeed, proving the relationships between quad. imag. fields and elliptic functions was one of the driving forces in the invention of algebraic number theory and class field theory.

You could try the book of Cox, Primes of the form $x^2 + n y^2$, which surveys some of this material. I would guess that it uses more algebraic number theory than you would be comfortable with, but perhaps it will give you some hints.

There is also the book $\pi$ and the AGM by the Borwein brothers, which gives a very eclectic survey of some this material. The proofs are both elementary and (often) quite unusual from a modern, systematic point of view. But they may be more accessible to you, and it is an amazing book with a lot packed into it.

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  • $\begingroup$ Thanks Matt E for the references. Fortunately I have both the books with me. And I love the way Pi and the AGM is written. Based on it I have written some posts in my blog. But the other book from David Cox deals with too many algebraic topics and it seems I will need to study these concepts in detail. I also have David Cox "Galois Theory" with me. Do you think it will be worthwhile to study this first and then go to "Primes of the form ..."? $\endgroup$ – Paramanand Singh Jul 8 '13 at 7:43
  • $\begingroup$ Also I was hoping that in Weber's time these algebra concepts were not developed, so his book might have treated the subject in a more accessible way. But after your response I am not very sure of that. $\endgroup$ – Paramanand Singh Jul 8 '13 at 7:45
  • $\begingroup$ @ParamanandSingh: Dear Paramanand, Well, Galois theory goes back to Galois's work almost a century before Weber's books were written, and his book treats Galois theory (but in a form that is less familiar to me than modern treatments, and, in my opinion, harder to learn). And algebraic number theory is one of the subjects that Weber helped to develop. So I'm not sure you can escape these topics if you want to learn more! Fortunately, there are many good texts on both. (I don't know Cox's text in particular, but I'm sure it's as good a place to start as any.) Regards, $\endgroup$ – Matt E Jul 8 '13 at 12:13
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A translation of the original might not exist as such. However edification might be found in alternate sources. Some books that might of interest to you are:

A. Weil, Elliptic Functions According to Eisenstein and Kronecker.

I do not consider Weil to be the best introduction; but in principle you should be able to read this book with some background in complex analysis/algebra.

The other book that might interest you is

G. Shimura, Introduction to the arithmetic theory of automorphic functions.

Again I do not know if this is the most optimal book; for example the author uses a very old language for algebraic geometry. But in principle you will be able to read it. At least at the start.

Much more accessible than Shimura will be the last chapter of the book,

J.-P. Serre, A course in Arithmetic.



Another very relevant reference is S. G. Vladut's "Kronecker's Jugendtraum and Modular Functions". Parts of this may be too advanced for your present state of knowledge, unfortunately. Still you might find some introductory parts advantageous.

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  • $\begingroup$ Thanks Doldrums, I will try to get hold of these references. $\endgroup$ – Paramanand Singh Nov 1 '13 at 6:34
  • $\begingroup$ @Paramanand: You do not seem to have gone much into complex variables in your blog posts. This whole topic might require some familiarity with it. After all even the name of the method/phenomenon is "complex multiplication". For an introduction to complex analysis with a quicker(than is otherwise) route to modular and abelian function, the first two volumes of Carl Ludwig Siegel on complex function theory will be appropriate. $\endgroup$ – user96815 Nov 3 '13 at 4:29
  • $\begingroup$ Doldrums, I know basic theorems like cauchy integrals, liouvilles, residue calculus, but I don't feel as comfortable with complex analysis as I am with real analysis. Even after studying proofs of various theorems in complex analysis, it looks mysterious whereas theorems of real analysis look intuitive. I will see if I can get hold of Siegel's book. Also the name complex multiplication is bit of a misnomer at least to me. It should mean multiplication of complex numbers, but it means studying when $F(nz)$ can be expressed rationally in terms of $F(z)$ and $F'(z)$ where $n \in \mathbb{C}$ $\endgroup$ – Paramanand Singh Nov 3 '13 at 5:20
  • $\begingroup$ Dear Paramanand, Did you study those theorems in some engineering course? If so, then the motivation and rigor might not have been sufficient for a proper understanding. The term "complex multiplication" becomes clearer in more modern terminology. This is however the best place to chat about such matters, since it requires more explanation than would fit in here. If you wish, you can ask separate questions for each of your dilemmas, or contact me via my blog link in my profile. $\endgroup$ – user96815 Nov 3 '13 at 13:33
  • $\begingroup$ A short attempt: Note that being differentiable wrt a complex variable is a much stronger criterion than just the real and imaginary parts of the function being real differentiable. This imposes very strong rigidity conditions. Every holomorphic function can be represented as a power series. Power series are close to being polynomials this connection is deeper in certain topics. This closeness to polynomials make it a very ideal thing for Kronecker's dictum to try to do everything using "finite" constructions; refer K.'s quote, "God created the integers and everything else is man's handywork". $\endgroup$ – user96815 Nov 3 '13 at 14:07
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There apparently is no English translation, but there is a French translation:

Traité d'algebre supérieure

Author: Heinrich Weber Publisher: Paris : Gauthier-Villars, 1898.

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  • $\begingroup$ Oops!! Sorry to say, but for me both French and German are the same to me as I don't know an iota of these languages. $\endgroup$ – Paramanand Singh Jul 8 '13 at 7:49
  • $\begingroup$ I think it may be easier for someone who knows English to learn to read mathematics in French than in German. Once I was a student in a math course in which I heard only English in the classroom, but the textbook was in French. Some people who had never received any instruction in French were alarmed, but I think they got by. I had never received any instruction in French and I found I didn't need to know much French to understand the book. $\endgroup$ – Michael Hardy Oct 1 '13 at 17:48

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