Questions tagged [elliptic-curves]

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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2
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0answers
114 views

Factoring integers of the form $n=p q^2$ using elliptic curves

We got argument and strong experimental support that integers of the form $n=p q^2$ can be factored using elliptic curves easier than general integers Q1 Is this known? Added This is known since at ...
4
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1answer
218 views

Torsion points on $E/\mathbb{Q}$ with large coordinates

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points. What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
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1answer
107 views

Common prime of the finite number of order of imaginary quadratic field

This is from Silverman's 'the arithmetic of elliptic curves', exercise 5.5. Let $K$ be an imaginary quadratic field, and let $R_1...R_n$ be orders in $K$. I would like to prove that there are more ...
3
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103 views

Trace map on rational points of elliptic curves

Let $L/K$ be finite Galois ext. of number fields and $E/K$ an elliptic curve. Define trace $$Tr : E(L) \rightarrow E(K), \;\; P \mapsto \sum_{\sigma \in G_{L/K}} P^{\sigma}$$ When is this map ...
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0answers
56 views

An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
3
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1answer
185 views

Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer): ''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
3
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1answer
119 views

Combinations of rank and torsion attainable by $E/\mathbb{Q}$

Suppose $r\geq 0$ is a rank attainable by infinitely many elliptic curves over $\mathbb{Q}$. Let $T$ be one of the fifteen finite abelian groups in Mazur's theorem. Is there an elliptic curve $E/\...
2
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1answer
160 views

Finite quotients of the absolute Galois group of $\mathbb{Q}$ via torsion of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime. Then there is an action of the absolute Galois group of $\mathbb{Q}$ on $E[p]$ that factors through a finite quotient. Does any finite ...
5
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0answers
152 views

Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...
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144 views

Z2xZ6 elliptic curves with missing generators

By implementing the techniques described in and similar to A. Dujella, J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z, arXiv, Number Theory [math.NT] (2013), arXiv:1306.0027v1 A....
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1answer
228 views

What's the average order of the reduction of a section of an elliptic curve

Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(...
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1answer
268 views

How to prove there are exactly $8$ integer points on the elliptic curve $y^2 = x^3 + 17$ [duplicate]

Consider the elliptic curve $y^2 = x^3 + 17$. I know that there are exactly $8$ integer points $(x,y)$ with $y>0$. But how do I prove it? Is there any specific approach to it or any proof for it?
4
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1answer
150 views

Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$. Question. What are some examples of ...
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1answer
89 views

Analytic function with q- difference equation involving theta

Consider the analytic space $\mathbb{C}^{*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}...
3
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1answer
168 views

Universal bundles over algebraic stacks

$\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ ...
3
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1answer
193 views

Frobenius actions on de Rham cohomology of ordinary elliptic curves

In appendix 2 of Katz's "p-Adic properties of modular schemes and modular forms", he describes a certain "Frobenius" endomorphism on the de Rham cohomology of ordinary elliptic ...
3
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1answer
203 views

Mordell–Weil rank of some elliptic $K3$ surface

Consider a finite field $\mathbb{F}_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}_q^*)^2 \setminus (\...
4
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1answer
235 views

Elliptic curves of high rank over Quadratic extensions

Are there examples of elliptic curves which has rank 0 over $\mathbb{Q}$, but acquires a high rank ( $\geq 2$) over some quadratic extension? More generally, are there known bounds for a given ...
7
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1answer
346 views

Splitting of small primes in number fields generated by the torsion of elliptic curves

Suppose $E/\mathbb Q$ is a non CM elliptic curve and we look at the number field $K_d$ generated by the $d$-torsion of $E$. What is known about the (complete) splitting of small primes in $K_d$? More ...
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0answers
71 views

A subgroup of the $n$-Selmer group

Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$. Let $E$ be an elliptic curve over a number field $F$. The $n$-Selmer group, denoted by $S_n(...
4
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1answer
242 views

3-, 6-, 12-descent for Z2xZ6 elliptic curves

We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
1
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1answer
167 views

Find basis for the set of torsion points E[m]

In paper "On the Cost of Computing Isogenies Between Supersingular Elliptic Curves" (source) reads Let ${P, Q}$ be a basis for $E[2^{e/2}]$. Let $R_0 = [2^{e/2}−1]P , R_1 = [2^{e/2}−1]Q, ...
4
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1answer
101 views

The name of the equianharmonic curve

I have found several references where the elliptic curve $y^2=x^3-1$ is called the equianharmonic curve, and, more often, where the half-period of this curve $$ \omega_1 = \frac{\Gamma(1/3)^3}{4\pi} $$...
4
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1answer
255 views

An explicit equation for $X_1(13)$ and a computation using MAGMA

By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$. And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(...
2
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0answers
99 views

Moduli interpretation and Ogg's notation for the cusps on modular curves

In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps, that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
2
votes
1answer
176 views

Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?

Help me please. Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
8
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0answers
390 views

Elkies' theorem on supersingular primes and inertness

Suppose $E_{/\mathbb{Q}}$ is an elliptic curve over $\mathbb{Q}$ without CM. By Elkies' theorem, there exist infinitely many primes $p$ for which $E$ has supersingular reduction at $p$. Question. Is ...
8
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4answers
1k views

Status of $x^3+y^3+z^3=6xyz$

In Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML the author has studied the Diophantine equation \begin{equation} x^3+y^...
9
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2answers
409 views

Z/8Z elliptic curve with a missing generator

We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in A. J. MacLeod, A Simple Method for ...
2
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0answers
148 views

Finding rational points via birational map

Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$ and let $\overline{C}$ denote the projective closure of $C$. For ...
3
votes
1answer
155 views

What upper bounds on $\sum_{n<x}a_n$ are known where $L(E,s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is an Elliptic L Function?

What are the current best known upper bounds on $$\sum_{n<x}a_n$$ where $a_n$ are defined implicitly by $L(E,s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ where $L(E,s)$ is an Elliptic L Function and $E/\...
1
vote
3answers
337 views

What heuristic suggest for the number of solutions of $x^n+y^n=A$?

For integers $x,y,n,A$ with $n>1$ and $A>0$ we are interested how many solutions $x^n+y^n=A$ has for fixed $n$ and infinitely many $A$. What is unconditionally known $n=2$ or $n=3$ the number of ...
1
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0answers
55 views

Monodromy Representation on $H_1$ of Elliptic Curve

I'm reading this post by Charles Siegel on Monodromy Representations and there is a construction in example a not unterstand. We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic ...
4
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2answers
158 views

Are there upper bounds on the L function $|L(E,s)|$ for $|s|<C$?

Given some absolute constant $C$ (In my case, $C=4$ would suffice) and an elliptic curve $E/\mathbb{Q}$, are there upper bounds on $|L(E,s)|$ that are uniform for $|s|<C$? Using the functional ...
2
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1answer
528 views

Summation form of the Hasse-Weil zeta function?

The only way I know to write the Hasse-Weil zeta function of an elliptic curve is as a product over the local zeta factors which are rational functions. To me, this appears like an Euler product. Is ...
3
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1answer
145 views

Understanding the implementation of the $p$-adic(?) sigma function in SageMath

I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this: <...
4
votes
1answer
225 views

What effect do Torsion points have on an Elliptic Curve's L function?

Given an elliptic curve $E/\mathbb{Q}$, is it possible to determine whether or not $E$ has torsion points just by looking at it's Hasse-Weil L function $L(E,s)$? In general, what effects does an ...
2
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0answers
200 views

What is the minimal model of $E:y^2=x^3-x-n$?

Does it hold that $E:y^2=x^3-x-n$ is a minimal model for any choice $n$? Using the Sage programming language we can check that $E$ is indeed minimal for every $n\leq20,000$.
5
votes
1answer
272 views

Higher order inflection points

Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\...
3
votes
1answer
168 views

Distribution of the rank of $y^2=x^4+x+b^2$

For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$. $C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$ and one more point from the reciprocal of the polynomial y=0 ...
3
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0answers
76 views

Number of integer solutions to a system of Diophantine inequalities

Let $N\in\mathbb{N}$ and $a,b\in\mathbb{N}$ be such that $a+b\in(N/2,2N)$ (then of course $\max\{a,b\}\simeq N$). I'm interested in getting an upper bound (in terms of $N$) for the number of positive ...
9
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0answers
360 views

Kihara-like Z/6Z elliptic curve families

Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
3
votes
1answer
123 views

Ramification index and additive reduction of elliptic curves

Let $N \ge 5$ be a prime number and $E/ \mathbb{Q}_N$ be an elliptic curve with additive reduction. Then it is easy to see that there exists a finite extension $K$ over which $E$ has stable reduction. ...
2
votes
1answer
140 views

Is every sufficiently general monic quartic rational square infinitely often?

Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$. Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$. $C$ is birationally equivalent to $f(x)=y^2$. The constant coefficient of $g(x)$ is $1$ since $f$ is monic and $(...
0
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0answers
85 views

how often can a fixed prime be anomalous?

Let $p$ be a fixed prime. Say for simplicity $p>5$. As we vary over all elliptic curves $E/\mathbb{Q}$ of height $< X$, can one (expect to) say anything about what proportion of elliptic curves ...
2
votes
2answers
507 views

A new simple formula is needed

The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$. The SageMath/Python code below produces a list of small fractions $a$ for ...
5
votes
0answers
169 views

Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?

Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
5
votes
1answer
273 views

Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$

I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$ Now clearly this is very difficult, ...
4
votes
2answers
156 views

Rank of jacobians of twists of hyperelliptic curves of genus one

For $a,b \in \mathbb{Z}$ we define the binary quartic form $$\displaystyle F_{a,b}(u,v) = a(u^2 - v^2)^2 + 4bu^2 v^2.$$ We shall assume throughout that the discriminant $$\Delta(F_{a,b}) = 4096a^2 b^2 ...
4
votes
0answers
174 views

Computing homology class of curve in product of elliptic curves

I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...

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