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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Elliptic functions prove convergence on lattice

The problem statement, all variables and given/known data Hi I am looking at the proof attached for the theorem attached that: If $s \in R$, then $\sum'_{w\in\Omega} |w|^-s $ converges iff $s > 2$ where $\Omega \in C$ is a lattice with basis…
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Q on proof of periods of non-constant meromorphic function - either {0}, {nw_1} or {nw_1 + mw_2}

Theorem attached. I know the theorem holds for a discrete subgroup of ${\mathbb C}$ more generally, ${\mathbb C}$ the complex plane, and that the set of periods of a non-constant meromorphic function is a discrete subset. I have a question on part…
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set of periods of meromorphic function forms a discrete set - q on step in the proof

The problem statement, all variables and given/known data Hi, As part of the proof that : the set of periods $\Omega_f $ of periods of a meromorphic $f: U \to \hat{C} $, $U$ an open set and $\hat{C}=C \cup \infty $, $C$ the complex plane, form a…
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Find differential equation of ellipse

How can I find the differantial equation of one parametered ellipse family with the equation : $$t^{2}/c + y^{2} = 1/(c-1)$$
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Embedding $\mathbb{C}/\Lambda$ in $\mathbb{P}^{2}$

In the book of Jurgen Jost, Compact Riemann Surface, is written (page 261): " As Theorem 5.7.1, it can be shown that $z \rightarrow (1,\wp(z),\wp'(z))$ defines an embedding of $\mathbb{C}/\Lambda$ in $\mathbb{P}^{2}$. The image of…
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Geometry of Elliptic curves

I'm reading about group structures in Elliptic curves (Joseph H. Silverman The Arithmetic of Elliptic Curves; chapter III and VI). I can not understand the introduction of Weierstrass P-function for the case of elliptic curves over $\mathbb{C}$. I…
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Number of Poles in a Period Parallelogram

The Weierstrass $\wp$ function has a double pole on every period. Its derivative $\wp'$ then has a triple pole on each period. Can I conclude that the quotient function $\dfrac{\wp'}{\wp}$ has a simple pole on each period? Is there any other poles…
Sekots Reivan
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Construction of the Weierstrass $\wp$ function

In the construction of the Weierstrass $\wp$-function, why do we have to add a leading term $\frac{1}{z^2}$ instead of letting $\wp(z)=\sum_{\omega\neq0}\left\{\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\right\}$? I am guessing this is to take care of…
Sekots Reivan
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Why is $\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q \, \Pi_{n\geq1}(1+q^n)^{24}}$

Where the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ and $q = \exp(2\pi i\tau)$. I cant seem to get the equality, am I missing some identities? Thanks!
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Reference for transformation period integrals into elliptic integrals

The theory of elliptic functions tells us that any elliptic curve defined by a cubic $y^2 = 4(x - e_1(x - e_2)(x - e_3)$ with distinct roots is isomorphic to the quotient $\mathbb{C}$ by a lattice. The periods of this lattice can be found by…
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Find all the points where the tangent plane is perpendicular to a given vector.

Given an ellipsoid : x^2+4y^2+z^2=9 Find the tangent plane of the ellipsoid at (x_0,y_0,z_0) I found the equation for the tangent plane to be: 2x(x-x_0)+8y(y-y_0)+2z(z-z_0)=0 but I need to find every point where this plane is perpendicular to the…
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Is there a relation between Elliptic Curves and Frobenius Numbers?

A CS professor yesterday asked me this query. I think there is no direct relation if any. The frobenius number is the largest number that cannot be represented by $au+bv$ where $gcd(a,b)=1$ holds and $a,b,u,v\in\Bbb N$ holds. Elliptic curves are…
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Prove that the Weierstrass $\wp$- function satisfies $\wp''\left(\frac{\omega_1}{2}\right)=2(e_1-e_2)(e_1-e_3)$

I am reading Apostol's Modular Functions and Dirichlet Series in Number Theory. And I am stuck on exercise 8 of Chapter 1. The problem asks to show that $$\wp''\left(\frac{\omega_1}{2}\right)=2(e_1-e_2)(e_1-e_3),$$ where…
Sekots Reivan
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Elliptic curve, different forms of.

$y^2 = x^3 + mx + c$ An elliptic curve in the form defined in Wikipedia $y^2 = x(x-A)(x+B) = x^3 +(B-A)x^2 + ABx$ Frey's curve has no term in $x^2$, but $2$. does because from Fermat, $A=a^n$ not equal to $B=b^n$ Question: Is there a…
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Laurent Series Expansion Logic

Relative to the below image, I am curious about the progression from equation 3.2 to equation 3.3, then from equation 3.3 to equation 3.4. I understand the logic in 3.2. I understand that a Laurent expansion about a complex function $f(z)$ is…
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