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An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.

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I have been working on this diophantine problem: $Q^2 - ((6n+3)^2 + t) \cdot Q + 2(6n+3)(36n^3 + 54n^2 + 27n - 4) = 0$ For arbitrary values of $n$ from $n = 1,2,3,4,5,6,7,8,9,10,11,12,13,14$, the ...
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The question of bounding the number of integer points on an elliptic curve $E/\mathbb{Q}$, shown to always be finite by Siegel, is an old question. There are various aspects to this problem. The ...
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I have been working on this elliptic curve equation parameterized by $n \in \mathbb{Z}$: $E_n : y^2 = x^3 + (-102n - 51) \cdot x - (432n^6 + 1296n^5 + 1620n^4 + 468n^3 - 513n^2 - 378n - 142)$ This ...
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$$y^{2} = x^{6} + 4n^{2}x^{2}$$ here $x,y$ have infinitely many rational solutions if and only if 'n' is a congruent number because this elliptic curve is a quadratic twist of the elliptic curve 32a1 ...
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Let $p$ be an odd prime. Suppose an elliptic curve $E/\mathbb{Q}$ has reduction of type $I_n$ at $p$. Let $\mathcal{E}$ be its Néron model, and assume that the component group $\tilde{\mathcal{E}}/\...
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I am considering an elliptic curve of positive rank given by the equation $$y^2 = x^3 + a x + b $$ where $a ≠ 0 $ Define $$ z = y^2 - x^3 $$ so we have $$z = a x + b$$ We know that the elliptic curve ...
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Kummer theory gives a unifying perspective on several 'discriminant factorization' phenomena. In the setting of a number field $K$, there is the short exact sequence of Galois modules $$ 1 \...
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Elliptic curves have an additive group law. The Elliptic Curve Discrete Logorithm Problem (ECDLP) is defined as, given $G,P\in E(\mathbb F_p)$, finding $k$ from $P=kG$. Let $q=\#E(\mathbb F_q)$. One ...
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Do you know an (explicit) example of a superelliptic curve $C\!: y^{p} = f(x)$ over a field of zero characteristic for which $p = 11$ and there is a cover $\phi\!: C \to E$ onto an elliptic curve $E$? ...
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Are there coprime integers $x,y$ and integers $A>1,D>6$ such that $xy(x+y)=A^D$? abc implies finitely many solutions for all $A>1,D>3$. For fixed $A$ the problem is elliptic curve and sage ...
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Let $E/\mathbb{Q}$ be a fixed elliptic curve. For a nonzero square-free integer $u$, let $E^{(u)}$ denote the quadratic twist of $E$ by $u$. What are the best known upper bounds for $\mathrm{rank}(E^{(...
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Let $p$ be a prime that splits as $p\mathcal{O}_F=\mathfrak{p}\bar{\mathfrak{p}}$ where $F$ is an imaginary quadratic field. Let $H_F$ denote the Hilbert class field of $F$. Finally let $E/H_F$ be an ...
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Let $E$ be supersingular elliptic curve defined over $\mathbb{F}_{p^2}$ and let $\ell$ be a prime. I'd like to know how one can check if E shares an edge, in the $\ell$-isogeny graph, with the set of ...
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We can view an elliptic curve $E$ as an anticanonical divisor $-K_{\mathbb{P}^2}$ in $\mathbb{P}^2$, and elliptic curves have their (self-dual) mirrors by Polischuk-Zaslow: https://arxiv.org/abs/math/...
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In cryptography, it seems to be a common choice to use the so-called Jacobian coordinates to represent a point of an elliptic curve (see e.g. Elliptic Curves: Number Theory and Cryptography, L. C. ...
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Let $C\subset\mathbb{P}^2$ be a plane quintic with an ordinary triple point in $[1:0:0]$, and two ordinary double points in $[0:1:0],[0:0:1]$, and otherwise general. Then $C$ has geometric genus $1$. ...
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I have been working on a problem involving magic squares where the equations below were developed: $x^2 = 2 n^2 \cdot (m^2 - n^2)^2 \cdot k^4 + [2 \cdot(m n)^2 - 4 \cdot m n \cdot (m^2 - n^2) + \frac{...
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Let $T_{g,b}$ be the Teichmüller space of $\mathbb{C}$-curves $\Sigma_{g,b}$ with genus $g$ with $b$ marked points. The end of Section 4 of Drinfeld's famous "On Quasitriangular Quasi-Hopf ...
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For prime $p$, let $$ H_p(t)=\sum_{i=0}^{(p-1)/2}\binom{\frac{p-1}{2}}{i}^2t^i$$ be the Deuring polynomial, whose roots correspond to the supersingular elliptic curves of the form $$E_t:\ y^2=x(x-1)(x-...
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Let $E$ be an elliptic curve of good reduction over a local field with residue field characteristic $p\ge 5$ and suppose $E$ has complex multiplication by $\mathcal{O}_d$ for some discriminant $d$ ...
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Surely this is well-known in the arithmetic geometry community, which I (unfortunately) do not regard as a member of, so I will include some basic exposition for the laymen (including myself). We deal ...
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Let $I$ be an infinite set of elliptic curves over $\bar{\mathbb{Q}}$ with a set $S$ of good primes of $\bar{\mathbb{Z}}$ for all elliptic curves in $I$. For $E\in I, v\in S$, let $\bar E_v$ be the ...
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Consider two Diophantine equations $$3n^2+p^2-q^3=1$$ and $$3n^2+r^2-s^3=-1.$$ Here, $n,p,q,r,s$ are positive integers. Is there any way to find an integer solution pair for $(p,q)$ or $(r,s)$ for ...
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In Silverman's book Advanced Topics in the Arithmetics of Elliptic Curves, the table in the section on Tate's algorithm lists reduction types of elliptic curves. For reduction types $IV$ and $IV^*$, $...
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I came across the abstract of the lectures given by Chantal David in the Winter School and Workshop on Frobenius Distributions on Curves (2014) here, the following point caught my attention: "In ...
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First, remember bn curves is a class of elliptic curves defined over curve $y^2=x^3+3$ with embedding degree 12 and $\mathbb G_2$ points lying over the curve twist $\frac {Y^2 = X^3 + 3}{i+9}$ defined ...
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Let it be a finite field $FF$ with 2 finite field elements having their discrete logarithm in a large prime subgroup $s$ of $FF$… Will the only way to map the discrete logarithm of $FF$ always be to ...
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Let $j$ be the j-invariant defined by $$j(\tau)=\frac{E_4(\tau)^3}{\Delta(\tau)},\quad \Im(\tau)>0$$ where $$E_4(\tau)=1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n},\quad q=e^{2\pi i\tau},$$ $$\...
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Let $E$ be an elliptic curve defined over $\bar{Q}$, for example a plane cubic curve. We know that the first de-Rham cohomology is generated by $\frac{dx}{y}$ and $\frac{xdx}{y}$. Denote by $E_{\...
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Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM) by an order $\mathcal{O}$ in an imaginary quadratic field $K$. Let $p$ be an odd prime of good reduction for $E$, and let $\...
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Let $E/K$ be an elliptic curve over a local field. I understand that the Kodaira type of $E/K$ refers to the isomorphism class of the special fiber of the Néron minimal model of $E/K$ as a scheme over ...
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A cyclotomic equation is the equation obtained by setting the minimal polynomial, over $\mathbb{Q}$, of $\mathrm{e}^{2\pi i/n}$ equal to zero. Gauss wrote about solving cyclotomic equations by ...
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Let $E/\mathbb{Q}$ be an elliptic curve without CM. Fix a finite set of primes $S$ containing all primes of bad reduction for $E$, and let $p\notin S$ be a prime of good ordinary reduction. Fix $T = S\...
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Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. Elkies famously showed there are infinitely many primes $p$ at which $E$ has supersingular reduction. One may reinterpret this as follows: there ...
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I am working on the elliptic curve $$ E_k: y^2 = x^3 - a_k^2 $$ This elliptic curve is deeply connected with the Pell’s equation: $$ a_k^2 - 3b_k^2 = 1 $$ We know that $$ a_k + \sqrt{3} b_k = (2 + \...
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There are many NP-complete problems, e.g. SAT, CVP, SIS, graph colouring, Minesweeper etc. By definition there are polynomial time reductions from one to another of these, at least in their decision ...
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I am trying to understand the behaviour of reduction types of elliptic curves at $p=2$ over $K$. I know that if I begin with curve that has type $I_n^*$ at a prime $p \ge 3$, then it becomes of the ...
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Let K be an imaginary quadratic field $Q(\sqrt{-3})$. Let $p$ be a prime which split in $K$ and Let $\varpi$ be one of the factors of $p$ in $K$. Let $E$ be an elliptic curve $y^2=x^3+\varpi^2/4$ with ...
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Fix an elliptic curve $E/\mathbb Q$. What is known about the proportion of the quadratic twists $E_D$ of $E$ over $\mathbb Q$ with rank $\ge 2$? Specifically, (1) Gouvea-Mazur show that $\gg_\...
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Given an elliptic curve over the rationals Mazur proved that the order of the torsion subgroup of $E(\mathbb{Q})$ is bounded by 16, and specified which groups can occur. If one only needs a bound of ...
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I am studying elliptic curves and came across an interesting pattern. For the elliptic curve: $$ y^2 = x^3 - 219x + 1654. $$ In this elliptic curve there are only $8$ solution couples $(x,y)$, the ...
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I am studying elliptic curves of the form: $$y^2 = x^3+ ax + b$$ Suppose this curve has positive rank, meaning it has infinitely many rational points $(x,y)$. Now, I consider the case when $y=0$. ...
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I would like to propose and ask about a conjecture involving a new family of elliptic curves that may be connected to the classical congruent number problem. We know that a positive integer '$n$' is a ...
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A friend recently nerd-sniped me with a (seemingly elementary) geometry question: Let $\triangle ABC$ (with corresponding side lengths $a$, $b$, and $c$) be an obtuse triangle with $\angle C > 90^\...
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We got some unexpected to us results. Let $E$ be an elliptic curve over a finite field of large characteristic. For positive integer $k$, let $D=2^k$ and assume the order of $E$ is $\rho=D t$ with $t$ ...
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Let $C$ be a singular projective curve over an imperfect field $K$ given by a Weierstrass equation. What is the structure of the group of non-singular rational points $C_\text{ns}(K)$? Here is a ...
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During my research, I came across this question. Let $p>11$ a prime number with $a=\text{card}\{(x,y) \in \mathbb Z/ p \mathbb Z: y^2=x^3+1\}$ and $b=\dfrac 1 {((p-1)/2)! \times ((p-1)/3)! \times ((...
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Let $E$ be an elliptic curve over a number field $K$ (I’m more generally interested in global fields). Cremona and Mazur in section 3 of these notes explain that any element $C$ of the Tate-...
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I'm trying to understand the theorem predicting the existence of a normalized newform $f \in S_2(\Gamma_0(N))$ associated to an elliptic curve $E/\mathbb{Q}$, such that there is a modular ...
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Apologies for the confused title, I wanted to appeal to two communities but have probably just alienated one and intimidated the other. This is really a very elementary question. In a fantastic talk ...
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