Questions tagged [elliptic-functions]

Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.

If $f:\mathbb{C} \to \mathbb{C}$ is elliptic, then there exists nonequal $t_0,t_1 \in \mathbb{C}$ such that $$ f(z) = f(z + t_0), f(z) = f(z + t_1) $$ the parallelogram $0,t_0,t_0+t_1,t_1$ is called the fundamental parallelogram of $f$. $f$'s value is entirely determined by its values on the fundamental parallelogram. By Liouville's theorem, any holomorphic elliptic function must be constant, so the usual elliptic functions are meromorphic. In fact, they must have at least two poles (with multiplicity) on the fundamental parallelogram.

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Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?

I tried to solve the following problem from Ahlfors' text, please verify my solution: Where does the function $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ take the values $0$ and $1$, and with what…
user1337
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Relation Between Involutions on an Elliptic Curve and the Corresponding Complex Torus

This is a question from the book Elliptic Curves by Henry McKean and Victor Moll: Consider the cubic $X_1: y^2=x^3-x$. It admits the involution $(x,y) \mapsto (\frac{-1}{x}, \frac{y}{x^2})$. It is not the involution $j: (x,y) \mapsto (x,-y)$ so it…
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How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$

I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$ where $q=e^{2\pi i \tau}$ is referred to as the nome. and the Gamma Function…
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Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough time with that portion of the book right now. Let…
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How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding Weierstrass $P$ function) but I don't know how to get…
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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…
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An Elliptic Curve has 3 real roots if and only if $\frac{\omega_2}{\omega_1}$ is purely imaginary

Define $\Lambda=\{n\omega_1+m\omega_2 \mid n,m\in \mathbb{Z}\}$ for $\omega_1, \omega_2\in \mathbb{C}$ and $\wp(z)$ is the corresponding Weierstrass elliptic function. I want to show that $e_1=\wp(\frac{\omega_1}{2}), e_2 = \wp(\frac{\omega_1 +…
Masoud
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Addition formula for elliptic integral of second kind

Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one of the Jacobian elliptic functions. When the…
Paramanand Singh
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Klein's j-invariant and Ford circles

Klein's j-invariant has structure which seems to resemble Ford circles: The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.) Can someone explain the connection?
AndrewG
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Pi, the Lemniscatic elliptic functions, and the Dixonian elliptic functions

$n=2$ Circle: $x^2+y^2=c^2$. We all know the role of $\pi$ in the circle and the trigonometric functions, $$\pi_2 = \color{brown}{B\big(\tfrac12,\tfrac12\big)}=3.1415\dots$$ with the beta function $B(x,y)$ and circumference for $D=1$ $$C=\pi_2 =…
Tito Piezas III
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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…
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Solving 5th degree or higher equations

According to this, there is a way to solve fifth degree equations by elliptic functions. Some related questions that came to mind: Besides use of elliptic functions, what other (known) methods are there for solving 5th degree or higher…
jimjim
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Showing Weierstrass elliptic function has periods

Let $$\wp(z)=\frac{1}{z^2}+\sum_{w \in \Lambda^*} \left[\frac{1}{(z+w)^2}-\frac{1}{w^2}\right] $$ be the Weierstrass elliptic function with $\Lambda=\Bbb{Z}+\Bbb{Z}\tau$, $\Lambda^*=\Lambda-0$. I want to show that $\wp(z+w)=\wp(z)$ whenever $w \in…
Gobi
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Closed form of $\sum_{n=-\infty}^\infty \frac{(-1)^n}{\sinh (z+n)}$?

I considered the following function: $$f:\, \mathbb{C}\mapsto\mathbb{C}_{\infty},\, z\mapsto \left(\sum_{n=-\infty}^\infty \frac{(-1)^n}{\sinh (z+n)}\right)^{-1}.$$ It can be seen that $f(z)=0$ at every integer (division of a non-zero complex number…
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