Articles of elliptic functions

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus (0,0)} \frac{1}{(m+\tau n)^{2k}}.$$ Motivation Exercise 6.6 of Silverman’s “The Arithmetic of Elliptic Curves” asks to compute the special value: $$j(i)= j(\mathbb{Z} \oplus i\mathbb{Z})=1728.$$ Where $j(\tau)=j(\mathbb{Z} \oplus \tau \mathbb{Z}) = 1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ […]

When does $\wp$ take real values?

Whittaker and Watson mention that when the invariants $g_2, g_3$ of Weierstrass $\wp$ function are real and such that $g_2^3 – 27g_3^2 > 0$, and if $2\omega_1$ and $2\omega_2$ are its periods then the function takes real values on the perimeter of the parallelogram with vertices $0,\omega_1,-\omega_2,-\omega_1-\omega_2$. I am not able to understand why $\wp$ […]

Equation Similar to Lamé Differential Equation

I have an equation similar to the Lamé differential equation in the Jacobi form, defined as $$\frac{d^2y}{dx^2} + (a\,\mathrm{sn}(x)^2+b)y(x)=0$$ where the function $\mathrm{sn}(x)$ is one of the Jacobi elliptic functions. My equation takes the form $$\frac{d^2y}{dx^2} = \left[k^2-\alpha_1\left(\frac{1-\gamma_1\mathrm{sn}^2(\gamma_2 x,\beta)}{r-\gamma_1\mathrm{sn}^2(\gamma_2 x,\beta)}\right)-\alpha_2\left(\frac{r-\gamma_1\mathrm{sn}^2(\gamma_2 x,\beta)}{1-\gamma_1\mathrm{sn}^2(\gamma_2 x,\beta)}\right)^2\right]y(x)$$ where $k,r,\alpha_1,\alpha_2,\beta,\gamma_1,\gamma_2$ are all real constants. I recognize this could also be viewed […]

The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested in any resources that may give the history of the Weierstrass function and its derivation. I do understand the basics of […]

Questions aobut Weierstrass's elliptic functions

from the wikipedia: == In terms of the two periods, Weierstrass’s elliptic function is an elliptic function with periods ω1 and ω2 defined as $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 \ne 0} \left\{ \frac{1}{(z+m\omega_1+n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\}. $$ Then $\Lambda=\{m\omega_1+n\omega_2:m,n\in\mathbb{Z}\}$ are the points of the period lattice, so that $$ \wp(z;\Lambda)=\wp(z;\omega_1,\omega_2) $$ == so i really like these elliptic functions. […]

Translation of Weber's Lehrbuch der Algebra vol 1, 2, 3

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 […]

Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ is the hypergeometric function. There are formulas for $m = 2,3,4,6$, so I was wondering if there are for other m as well. However, one thing I observed was that, let, $$q = \exp\left(\frac{-\,\pi\sqrt{n}}{\sin(\pi/m)}\right)$$ Conjecture: $$\lim_{n\to \infty}\frac{x}{q} = \text{constant}$$ namely, […]

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K’}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page cites several cases $\frac{K’}{K}=\sqrt{r}$, when $r$ is integer. Update: This question has been answered here.

Addition theorems for elliptic functions: is there a painless way?

The Weierstrass $\wp$ function satisfies the addition formula $$\wp(z+Z)+\wp(z)+\wp(Z) = \left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^2.$$ Of course, this is just the $x$-coordinate of the sum of the points $(\wp'(z), \wp(z))$ and $(\wp'(Z), \wp(Z))$ on the Weierstrass elliptic curve $y^2=4x^3-g_2x-g_3$. If one has an a priori knowledge of this fact, the computation of the addition formula is absolutely trivial. However, […]

The importance of modular forms

I’m studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called $E(\Lambda)=\mathbb C(\wp,\wp’)$ and they “represent” all meromorphic functions on the torus $T=\mathbb C/\Lambda$. Is demonstrated that two tori $\mathbb C/\Lambda$ and $\mathbb C/\Lambda’$ are conformally equivalent iff […]