Intereting Posts

why is a simple ring not semisimple?
Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$
In what sense is the forgetful functor $Ab \to Grp$ forgetful?
finding the multiplicative inverse in a field
Formal proof that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.
Establishing a Basis for $F/(f(x))$
Extension and contraction of ideals in polynomial rings
Proof that operator is compact
Ramsey Number R(4,4)
Apparent Paradox in the Idea of Random Numbers
What is meant with unique smallest/largest topology?
Carmichael number factoring
Question about $L^1$-$L^2$ integrable functions
Quotient space and continuous linear operator.
Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$

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.

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

- Deriving the addition formula for the lemniscate functions from a total differential equation
- The importance of modular forms
- What does | mean?
- Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$
- elliptic functions on the 17 wallpaper groups
- Can you recommend some books on elliptic function?
- Closed form solutions of $\ddot x(t)-x(t)^n=0$
- The derivation of the Weierstrass elliptic function
- Prove $\int_0^{\pi/2}{\frac{1+2\cos 2x\cdot\ln\tan x}{1+\tan^{2\sqrt{2}} x}}\tan^{1/\sqrt{2}} x~dx=0$
- The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

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.

- Can anyone explain the intuitive meaning of 'integrating on both sides of the equation' when solving differential equations?
- A proper local diffeomorphism between manifolds is a covering map.
- Choosing the correct subsequence of events s.t. sum of probabilities of events diverge
- Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$
- Do circles divide the plane into more regions than lines?
- Representing IF … THEN … ELSE … in math notation
- L-functions identically zero
- “Probability” of a map being surjective
- Given $h(\mathcal{O}_{\mathbb{Q}(\sqrt{d})}) = 1$, what is the longest possible run of inert primes in that ring?
- Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods
- Intuition for dense sets. (Real analysis)
- Why is that the extended real line $\mathbb{\overline R}$ do not enjoy widespread use as $\mathbb{R}$?
- How original RS codes and the corresponding BCH codes are related?
- Permutations to satisfy a challenging restriction
- Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?