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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69 views

Pro-representability and the obstruction to deformations of “stable curve of genus one + a section”

I have 2 questions about the theorem III.1.2 of Deligne-Rapoport's "Les shemas de modules de courbes elliptiques". 1. Let $k$ be a field, $\Lambda$ a complete noetherian local ring with the ...
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105 views

Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?

Consider the Frey-Hellegouarch curve given $a,b$ natural numbers: $$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$ Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(...
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1answer
119 views

The number of perfect squares which can occur in an arithmetic progression of length n

This is a similar question to https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487 Let f(n) be the maximum ...
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566 views

Recent developments in the proof of Fermat's last theorem

I posted on Mathematics Stack Exchange, but was encouraged to post on MathOverFlow instead. It has been 20 years since Fermat's last theorem was proved by Andrew Wiles. Has there been any ...
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2answers
411 views

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$. Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
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2answers
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primes of multiplicative reduction for elliptic curves with rational $\ell$-torsion

Very recently, I made an observation from scanning lists of elliptic curves on the LMFDB that leads me to the following (unproven) statement: Fix $\ell \in \{5, 7\}$. Let $E$ be an elliptic curve ...
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84 views

Drinfeld basis of the universal formal deformation of a supersingular elliptic curve

Now I'm trying (5.3.2) of Katz-Mazur's Arithmetic moduli of elliptic curves. Let $k$ be an algebraically closed field of characteristic $p > 0$, $E_0$ a supersingular elliptic curve over $k$, ...
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3answers
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Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation $$ y^2 + y = x^3 - x^2. $$ My guess is that there is some problem ...
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1answer
241 views

Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$

I read an article where it is said: $E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$ where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$. The ...
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1answer
115 views

abc triples with a symmetric condition

Recently, I have asked a question about the balance of abc triples. Since then I have come up with a different idea of a new criterion that somewhat combines balance and magnitude and has two ...
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123 views

Is there a way to show that the $m$-torsion group of an elliptic curve is $(\mathbb{Z}/m\mathbb{Z})^2$ without using analytical methods?

All of the proofs that I have seen of this fact are by saying that it is true for the complex numbers (which is clear by looking at an elliptic curve over $\mathbb{C}$ as the quotient of $\mathbb{C}$ ...
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Is there an infinite order $\mathbb{F}_{p}$-section for a certain elliptic surface $\mathcal{E}_n$?

Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
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54 views

What conditions are sufficient for two points to be independent in the Mordell-Weil group?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t), $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
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69 views

Is there a way to calculate the Picard $\mathbb{F}_q$-number of an (rational or K3) elliptic surface?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6, $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way ...
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1answer
289 views

How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ in rational? [duplicate]

The motiviation of this question is to look if there is such solution in rational number to the identity which montioned here, I have done many attempts using wolfram alpha to find such pairs of ...
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53 views

properties of anti-cyclotomic extension

Let $K$ be an imaginary quadratic field, and $p$ be a primes number, there exists an unique $\mathbb{Z}_p$-extension of $K$, we denote it by $K_{p,\infty}$ such that the action of complex conjugate $c$...
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145 views

Hecke correspondence and the trace map of differential forms

Let $k$ be a field, $X$, $Y$, $Z$ smooth geometrically connected curves, and $f: Z \to X$, $g : Z \to Y$ finite morphisms. Suppose that $f$ is separable. Then we have $f_* \circ g^* : \Gamma(Y, \...
2
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2answers
106 views

Supersingular elliptic curves and their automorphisms

If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology). Do all ...
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2answers
276 views

$3$-ranks of elliptic curves and representations $p=ax^3+by^3$

Let $p$ be a prime with $p\equiv2\pmod3$ and $E_p$ the elliptic curve $y^2=x^3+9p^2$ which has a rational $3$-torsion point. Let $\alpha$ from $E_p(\mathbb Q)$ to $\mathbb Q^*/{\mathbb Q^*}^3$ be the $...
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96 views

FLT and integral points on elliptic curves

For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$. For $ n > 2$, Fermat's Last Theorem implies there are no integral solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are ...
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1answer
182 views

Degree of morphisms and isogenies

$\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$ I am reading this paper by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \...
5
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106 views

reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$

In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...
3
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1answer
297 views

Family of elliptic curves in $\mathbb P^3$

For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$ in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$. Let $$C=l_{12} \cup ...
5
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1answer
320 views

An explicit description of $X(3)$ and its universal generalized elliptic curve

I'm struggling with the proof of 2.21 of Saito's "Fermat's Last Theorem". Let $\omega$ be a primitive 3rd root of unity, $X(3) = \mathbb{P}^1_{\mathbb{Q}(\omega)}$, and $E = \{ X^3 + Y^3 + Z^3 - 3 \...
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1answer
88 views

isogenies between elliptic curves with multiplicative reduction

Let $ K $ be a $ p $-adic field. Suppose we have an isogeny of elliptic curves $ \phi : E \to E' $ defined over $ K $, where $ E $ and $ E' $ both have multiplicative reduction. 1) Is there anything ...
7
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1answer
195 views

Finding $Q(\sqrt{-2})$-rational points on $X_0(33)$

Let $K = Q(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(33)$? Recall that $X_0(33)$ is of genus $3$ and has the following affine model, $$y^2 +(-x^4-x^2-1)y = 2x^...
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0answers
296 views

Finding elliptic curve with $P=[m]R$

To avoid X-Y problem I am going to write my problem down in detail, so plz bear with me. The elliptic curve over $Q$ given by a Weierstrass equation is - $E := y^2 +a_1 xy +a_3 y = x^3 + a_2 x^2+...
3
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1answer
216 views

Rank 3 Lagrangian vector bundles on an elliptic curve

Let $k$ be an algebraically closed field of characteristic zero (feel free to assume $k= \mathbb{C}$) and $E$ an elliptic curve over $k$ with identity $P \in E(k)$. I am interested in certain ...
16
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2answers
660 views

How can I see the relation between shtukas and the Langlands conjecture?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about. Drinfeld ...
7
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1answer
192 views

The $S$-unit equation for functions on curves

Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
1
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1answer
109 views

Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve

From Silverman's AEC page 332: I need to understand why the determination of the following local kernel $$ ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big) $$ is straightforward. The ...
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0answers
116 views

The Picard scheme of an ordinary singular curve

Let $k$ be an algebraically closed field, $C$ a proper reduced connected scheme over $k$ of dimension 1, whose singularity is at worse ordinary, $\pi : \tilde{C} \to C$ the normalization of $C$ and $...
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0answers
157 views

Perfect square quadratic expression

For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square. I start with $y^2=(5cx+100)(5cx-64c+36)$ and ...
5
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1answer
170 views

Indecomposable vector bundles on elliptic curves

Let $X$ be a smooth complex projective curve of genus 3, $E$ an elliptic curve, and $f: X \to E$ a finite map of degree 2. Let $L$ be a line bundle on $X$, and $R^0f_*(L)$ its direct image. Question: ...
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0answers
59 views

Cubic extensions of number fields and their local nature

Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$. Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^...
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151 views

Elliptic curves and archimedean place

here is my question : Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place. We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such ...
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71 views

Sato-Tate for length zero intervals

Let $E$ be an elliptic curve (without CM) over a number field $K.$ Is it known that $a(\mathfrak{p})=N(\mathfrak{p})+1-|E(\mathbb{F}_{\mathfrak{p}})|$ is neither zero, not $2\sqrt{N(\mathfrak{p})}$ ...
1
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1answer
95 views

Elliptic curves and its Neron model

Let $E$ be an elliptic curve over $\mathbb{Q}$. For a prime $p$, let $\mathcal{E}_p$ denote its Neron model over $\mathbb{Z}_p$. Also, let $\Phi_p(E)$ denote the component group of $\mathcal{E}_p$. ...
2
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0answers
86 views

Some questions regarding computation of the Mordell-Weil group

I was reading the theory relevant to Selmer and Shafarevich-Tate groups from Silverman's AEC. And I have a lot of doubts related to these topics: First, I don't understand the reasoning behind the ...
4
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1answer
280 views

A generator needed for a Z/6 elliptic curve

We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in A. Dujella, J.C. Peral, P. Tadić, ...
7
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0answers
261 views

Stacky proof of no elliptic curves over Z

It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
3
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1answer
129 views

Wiman's method for bounding the rank of an elliptic curve

In 1945 Wiman [W] showed that certain elliptic curves $E$ over $\mathbf Q$ have rank* at least 4. (It seems this was the highest known rank of an elliptic curve over $\mathbf Q$ until 1974, when ...
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101 views

The minimal equation of the Frey curve

In the paper of G.Frey there is a link between stable elliptic curves and certain Diophantine equations. The Frey curve of the equation $A-B=C$ is $$E :\;y^2=x(x-A)(x-B)$$ where $A=a^p$, $B=b^p$, ...
9
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0answers
150 views

Good reduction of finite etale covers of abelian varieties

Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$. Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
8
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1answer
687 views

Hard: One more generator needed for a Z/6 elliptic curve

We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in A. Dujella, J.C. Peral, P. Tadić, Elliptic ...
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0answers
49 views

double-and-add algorithm with(z,w)-coordinates

At "The Arithmetic of Elliptic Curves" by Joseph H. Silverman p.390 VI Example 6.7 given $E:y^2 = x^3+19x+112$ over ${F}_{127}$ points $P= (106,72)∈E$(${F}_{127}$), $Q= (12,121)∈E$(${F}_{127}$) lifted ...
1
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1answer
68 views

Super-singular reduction at a given prime

Given a prime $p$ should there always exist an elliptic curve over $\mathbb{Q}$ having super-singular reduction at $p$ ? I know examples with $p \equiv 2 \pmod 3,$ or $\equiv 3 \pmod 4$. But I am ...
7
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2answers
316 views

Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
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0answers
188 views

Generate algorithmically an elliptic curve with its exact class group structure?

Is it possible to generate an elliptic curve $E$ (randomly), together with knowing its class group $\mathrm{Cl}(\mathcal{O})$ structure? where $\mathcal{O}$ is its endomorphism rings $\mathsf{End}(E)$ ...
1
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1answer
193 views

Resolved: Two more generators needed for a Z/6 elliptic curve

We are searching for the rank 8 elliptic curves with the torsion subgroup Z/6 using newly discovered families similar to Kihara's (Kihara's family is described in https://arxiv.org/pdf/1503.03667.pdf)....

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