Questions tagged [elliptic-curves]
For questions about elliptic curves.
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Weierstrass Form of degree 4 equation
Take the equation $$y^2 = x^4 - 2x^3 - 2x - 1$$
I found that this is a genus 1 curve, because it is well known that for $y^2 = f(x)$ where $f$ is of even degree, the genus is $\frac{\deg{f} - 2}{2}$, ...
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An action of $\Gamma_0(N)$ having finitely many orbits
For positive integers $m$ and $N$ let
\begin{align}
M_{2, m, N}(\mathbf{Z}) = \bigg\{\gamma \in M_2(\mathbf{Z}) \; \bigg\vert \; \det(\gamma) = m, \gamma \equiv \bigg(\begin{matrix} \ast & \ast \\ ...
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Modular parametrization of elliptic curves expressed as rational function of $j(\tau)$
I have learned that one can parametrize an elliptic curve using meromorphic modular functions on group $\Gamma_0(N)$ for certain level $N$. Using the example on wikipedia, the parametrization of $y^2-...
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Why does the continuous homomorphism group equal the homomorphism group?
I have the following question on Silverman's book The Arithmetic of Elliptic Curves, 2nd edition.
First, let $K$ be a perfect field, $G_{\overline{K}/K}$ be the absolute Galois group of $K$, and $M$ ...
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Order of Picard groups of non-hyperelliptic algebraic curves
Let $q$ be prime. When $E/\overline{\mathbb{F}_q}$ is an elliptic curve, it is well-known that the group of $\mathbb{F}_q$-points of $E$ is isomorphic to the Picard group of degree $0$ divisors on $E$ ...
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Uniformization Theorem and Non Existence of Family of Elliptic Curves over Riemann Sphere
A question concerning a statement from these notes introducing/motivating period map. On first page, left column, last sentence states:
Since $\Bbb{H}$ (=complex upper half plane) is biholomorphic to ...
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Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?
I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
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Example II.3.5 in Arithmetic of Elliptic curves
The example is
Let $C$ be a smooth curve, let $f \in \overline{K}(C)$ be a nonconstant function, and let $f:C\rightarrow \mathbb{P}^1$ be the corresponding map (II.2.2). Then directly from the ...
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Twist of elliptic curve by degree $n$ extension
Let $E/K : y^2=x^3+ax+b$ be an elliptic curve over number field $K$.
For quadratic extension $L=K(\sqrt{D})/K$, $E_D/K : Dy^2=x^3+aX+b$ is called a twist of $E/K$ by $D$.
This curve $E_D$ has ...
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Determining rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$
I am trying to find all rational solutions to $y^2 = 2(x^4 - 2x^3 - 2x - 1)$
So far, I have tried to show that rational solutions to the above form an isomorphism with rational solutions to an ...
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Relating discriminants of hyperelliptic curves to discriminants of their defining polynomials
Let $C$ be a hyperelliptic curve defined by an equation of the form
$$
C: y^2=f(x)
$$
where $f$ is a polynomial of prime degree $p\geq3$, over a complete field $K$ of residue characteristic $p$. ...
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Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."
I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here.
That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
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Elliptic curves with same $a_p$ for every $p$ are isogenous
If two curves E,F satisfy $\#E(\mathbb{F}_p) = \#F(\mathbb{F}_p)$ for each large prime then E and F are isogenous (conversely, two isogenous curves must have the same values of $\#E(F_p)$ for every $p$...
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Computing degree of $x$ map for elliptic curve given by Weierstrass equation
Suppose $E$ is an elliptic curve given by the Weierstrass equation
$$
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6
$$
I want to calculate the degree of the map
$$
\varphi\colon E\to\mathbb{P}^1\qquad\quad[x,y,1]...
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Proof of Grothendieck's Semistability Criterion for Elliptic Curves
We have the following result:
Let $E/\mathbb Q$ be an elliptic curve and let $p$ be a prime. Then $E$ is semistable at $p$ if and only if $\rho_{E,\ell}|_{I_p}$ is unipotent for some (all) prime(s) $\...
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