# Questions tagged [elliptic-functions]

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

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### Is the Weierstrass $\wp$-function compatible with automorphisms of $\mathbb{C}$?

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $\sigma\in\mathrm{Aut}(\mathbb{C})$. Is it true that \begin{equation} \tag{$*$} \sigma(\wp(z; \Lambda))=\wp(\sigma(z); \sigma(\Lambda))? \end{equation} ...
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### Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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### Complete integral involving square of Jacobi elliptic functions

I've come across the following integral while studying the motion of a steady 2D fluid with stream function cos(x)cos(y), the flow of which involves Jacobi elliptic functions [sn, cn, dn]. I'm hoping ...
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1 vote
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### Biharmonic problem with boundary conditions on Laplacian

Let the problem, where $\Omega$ is an open set of $\mathbb{R}^3$ and $h_1$ and $h_2$ are regular given functions \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \left\{ \begin{array}[c]...
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### How to approach the problem of summation of Eisenstein series on shifted lattices?

This question is an attempt to complete the issues discussed in a previous question of mine (How did Gauss sum Eisenstein series?), since my updated question did not recieve any attention. In my ...
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### How did Gauss sum Eisenstein series?

Entry 61 in Gauss's diary states (this is a translation from latin): From the integer powers of $$\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ depends $$\sum_{m,n}(\frac{m^4 - 6m^2n^2 +n^4}{(m^2+n^2)^4})^k$$. ...
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### Jacobi elliption function values for complex-valued modulus

I have a program that can compute Jacobi elliptic function values for real-valued argument u and modulus m. I would now like to ...
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### Question about addition of points on elliptic curves in relation to elliptic integrals

I'm working on a long form piece on the development of elliptic curve cryptography, basically tracing the use of a point at infinity back to the discovery of the rules for linear perspective in the ...
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### Normalized periods of elliptic functions

I am a bit confused, and inexperienced. I have an elliptic function with periods $2\omega_1,2\omega_2$. I heard one can normalize these periods as $\tau=\frac{\omega_2}{\omega_1}$. I know, you can ...
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### Elliptic uniformization of $\sqrt{1+k+k^2}$

I have several elliptic polynomials $P_i(u)$ in Jacobi elliptic functions $sn(u|k)$, $cn(u|k)$ and $dn(u|k)$ with standard definitions $sn^2(u|k)+cn(u|k)^2=1$, $dn^2(u|k)+k^2sn^2(u|k)=1$. Coefficients ...
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### Given a meromorphic function $f$ on $\mathbb{C}/\Lambda$, there is indeed a standard proof that $f \in \mathbb{C}(\wp,\wp')$
This question is from Silverman's 'the arithmetic of elliptic curves', p167. Given a meromorphic function $f$ on $\mathbb{C}/\Lambda$, there is indeed a standard proof that $f \in \mathbb{C}(\wp,\wp')$... 