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Functional Equation $f(mn)=f(m)f(n)$.

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.

- Prove $\int_0^{\pi/2}{\frac{1+2\cos 2x\cdot\ln\tan x}{1+\tan^{2\sqrt{2}} x}}\tan^{1/\sqrt{2}} x~dx=0$
- What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

- How to transform a general higher degree five or higher equation to normal form?
- Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$
- elliptic functions on the 17 wallpaper groups
- Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?
- Prove $\int_0^{\pi/2}{\frac{1+2\cos 2x\cdot\ln\tan x}{1+\tan^{2\sqrt{2}} x}}\tan^{1/\sqrt{2}} x~dx=0$
- Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
- Closed form solutions of $\ddot x(t)-x(t)^n=0$
- The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?
- Any even elliptic function can be written in terms of the Weierstrass $\wp$ function
- Addition theorems for elliptic functions: is there a painless way?

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.

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.

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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