Questions tagged [elliptic-curves]
For questions about elliptic curves.
3,525 questions
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From sum of three cubes to Desboves/Selmer elliptic curve
Let given $n \in \mathbb{Z}^+$ and equation $x^3 + y^3 + z^3 = n$ over $\mathbb{Q}$.
Let $x = -\dfrac{4 a^3-4 nb^3+1}{3 b},y = \dfrac{a^3-nb^3+1}{3 b},z = \dfrac{a}{b}$, where unknowns $a,b\in \mathbb{...
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Why elliptic curves instead of Edwards curves?
I have been wondering this for a while; I get how elliptic curves of the Weierstrass form $y^2=4x^3-g_2x-g_3$ have the lowest degree and are the simplest way to study a genus $1$ curve. But why aren’t ...
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Existence of rational elliptic curves with trivial torsion over $\mathbb{Q}$ acquiring 5-torsion over a cubic field
I am investigating the torsion growth of rational elliptic curves upon base change to cubic fields.
Specifically, I am looking for a rational elliptic curve $E/\mathbb{Q}$ with trivial rational ...
- 1,259
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Rational points and sections on a family of genus-3 hyperelliptic curves
I am interested in the proof or disproof of some conjectures about rational points and sections over $\mathbb{Q}$ for the following family of genus-3 hyperelliptic curves:
$$
C_t: f(x,a)\, g(x,a)\, h(...
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Computing large division fields of elliptic curves with CM
Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by $\mathbb{Q}(i)$, e.g. any curve of the form $E:y^2 = x^3 + Ax$ with $A \in \mathbb{Q}^\times$.
I would like to compute generators ...
- 335
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Rational points on a hyperelliptic curve defined by a product of two quartics
I am interested in finding all rational points on the hyperelliptic curve
$$
C: f(x)\, g(x) = y^2,
$$
where
$$
f(x) = \left(625x^4 + 3100x^3 - 11344x^2 + 6200x + 2500\right), \quad
g(x) = \left(961x^4 ...
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Sturm bound and the modular forms of elliptic curves
Let $p$ be a prime, and let $E/\mathbb{Q}$ and $F/\mathbb{Q}$ be elliptic curves. Suppose the modular forms $f$ attached to $E$ and $g$ attached to $F$ have $q$-expansions that agree modulo $p$ in ...
- 6,877
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If modular form satisfies $a_n(f_E) \equiv a_n(f_F) \pmod{p}$ for all $n \geq 1$, are the semisimplification isomorphic as $G_{\mathbb{Q}}$-modules?
Let $p$ be a prime. Let $E$ and $F$ be elliptic curves over $\mathbb{Q}$, and let $f_E$ and $f_F$ denote the associated modular forms. Suppose that all Fourier coefficients are congruent modulo $p$, i....
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Two questions on Deuring's Lifting Theorem
I have two questions about Deuring's Lifting Theorem:
Let $E/\mathbb F_q$ be an elliptic curve over a finite field and let $\phi \in End(E)$ be nonzero. There exists an elliptic curve $E^*$ over a ...
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Inequality for the height of the sum and difference of two points on an elliptic curve
(This is exercise 3.2(b) of Rational Points on Elliptic Curves)
Let $C$ be a non-singular curve $y^2 = x^3 + a x^2 + b x + c$, where $a$, $b$ and $c$ are integers.
I want to prove that there is a ...
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Cancelling factors in isogenies between elliptic curves
I am currently working with isogenies of supersingular elliptic curves over finite fields and I do not really understand when and how I am allowed to cancel single factors.
For exapmle, let $\phi,\psi:...
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Understanding modular curves as "moduli of Hodge structures"
It is well-known that modular curves parametrize elliptic curves with level structures. For the purpose of this question, I will work complex-analytically and describe analytically the moduli space $$\...
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Is there any relation between group law of smooth projective Elliptic curve and divisor class group of completion of coordinate ring?
Let $f$ be a homogeneous polynomial in $\mathbb C[x,y,z]$ defining an Elliptic curve in $\mathbb P^2_{\mathbb C}$. In general, is there a relation between the Elliptic group law of this Elliptic curve ...
- 1,901
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Simple/Elegant proof of a weakened form of the Cannonball Problem?
I am wondering if anyone has a simple and elegant proof of the following fact:
The diophantine equation $\frac{n(n+1)(2n+1)}{6}=m^2$ has finitely many integral solutions.
Of course, this is a very ...
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Isogeny classes of elliptic curves with period in given number field
Let $F\subseteq\mathbb{C}$ be the cyclotomic field generated by a primitive $5$th root of unity. Is anything known about the number of isogeny classes of elliptic curves $E_\tau:=\mathbb{C}/\langle1,\...
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