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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162 views

Characterization of an Abelian surface

I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E_{1,t}$ and $E_{2,t}$, (I don't know what either $t$ runs through!) such that (1), for any i={1,2}, the closed ...
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111 views

Elliptic curves and localizations at various primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime at which $E$ has good reduction. Let $D=D_{E,p}$ be the $p$-torsion in the cokernel of the map $E(\mathbb{Q})\otimes\mathbb{Z}_p\...
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135 views

Magic hourglass of squares hyperelliptic equation

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$ $ $ $ $ $ $ $ $ $ $ $d^2$ $e^2$ $f^2$ $g^2$ ...
5
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2answers
574 views

Embedding torsors of elliptic curves into projective space

Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...
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98 views

What do congruences between modular forms tell us about $\mu$-invariants of elliptic curves?

This question is based off these notes by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $...
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128 views

Representations of elliptic curves over arbitrary fields

An elliptic curve over a field $k$ is a commutative algebraic group, so we can ask what its algebraic representations are, in particular what its characters (1-dimensional representations) are. Have ...
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1answer
408 views

The real part of the period of an elliptic curve

Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain: $$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \...
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134 views

Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...
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1answer
454 views

How do you compute modular symbols?

In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way. Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The ...
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197 views

To justify the intuition about #$E(\Bbb Q_p)$=$∞$

Let $E$ be an elliptic curve on $\Bbb Q_p$. $E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points. How to prove #$E(\Bbb Q_p)$=$∞$ directly ? According to Silverman's book 'the ...
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392 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
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1answer
244 views

Questions on the $j$-invariant

The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a ...
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334 views

Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?

In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich), he applies the theorem "When $K$ is complete with respect to it's discrete value, ...
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127 views

Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ? My ...
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1answer
123 views

Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?

Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$. Suppose the elliptic curve $E^D$ is a quadratic twist of $E$. I understand that ...
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127 views

Clarification of argument in "Elliptic curves over $\mathbb{Q}_{\infty}$ are modular"

In https://arxiv.org/abs/1505.04769 in the proof of Theorem 5 it is asserted that since $\rho_{E, l}:G_\mathbb{Q}\to\mathrm{GL}_2(\mathbb{Z}_l)$ is surjective then $E_{\mathbb{Q}_\infty}[l^\infty]=0$. ...
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84 views

The existence of two $p$-isogenies implies the existence of one $p^2$-cyclic isogeny

Let $E$ be an elliptic curve over $\mathbb{Q}$. (or over a number field $K$.) If $E$ has two $p$-isogenies over $\mathbb{Q}$, then $E$ has $p^2$ cyclic isogeny over $\mathbb{Q}$. I want to show it ...
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1answer
169 views

A closed subgroup $G$ of $\operatorname{GL}_2 \mathbb{Z}_\ell$ which surjects onto $\operatorname{GL}_2 \mathbb{F}_\ell$

Let $\ell \ge 5$ be a prime and $G$ be a closed subgroup of $\operatorname{GL}_2 \mathbb{Z}_\ell$ whose image in $\operatorname{GL}_2 \mathbb{F}_\ell$ is $\operatorname{GL}_2 \mathbb{F}_\ell$. Then $G ...
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101 views

Splitting of prime and order of reduction of point of infinite order in an abelian variety

I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here. Let $A$ be an abelian variety defined over a number field $K$, $P \...
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115 views

Brauer-Manin obstruction and affine curves

I'm looking for references that can justify to what extent is the following statement true: Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
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1answer
223 views

Coefficients of elliptic curves over function fields

Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$. Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...
5
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1answer
321 views

Weak Mordell-Weil for EC using Chevalley-Weil theorem

I am reading the book Applications of Diophantine Approximation to Integral Points and Transcendence by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they ...
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395 views

Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$

I'm looking at chapter 4 of Waterhouse's "Abelian varieties over finite fields"; and Theorems 4.1 and 4.2 seem to use the following fact: Suppose that $E/\mathbb{F}_q$ is an elliptic curve ...
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91 views

A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
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2answers
293 views

Sum of Legendre Symbol when $p\equiv 1,3\mod{4}$

Let $\ p\ $ be a prime. Prove that if $\ p\equiv 3\pmod{4}\ $ then the sum $$ S=\sum_{k=0}^{p-1}\left(\frac{k^3+6k^2+k}{p}\right)=0 $$ What is the value of the sum $\ S\ $ when $\ p\equiv 1\pmod{4}\,?\...
3
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1answer
259 views

An example of Serre on the cohomology of some CM elliptic curves

Let $E$ be the elliptic curve over $\mathbf{Q}_3$ with Weierstrass equation $y^2 = x^3-x$. It has complex multiplication by $\mathbf{Z}[\sqrt{-1}]$, with $\sqrt{-1}$ acting as $(x,y)\mapsto(-x,iy)$. ...
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1answer
122 views

Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$

Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$, define $E_a : x^3+a x z^2=y^2 z$ Let $B= \lfloor 2 \sqrt{p}\rfloor$ Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
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142 views

Showing that two families of elliptic curves are diffeomorphic

Consider a family of elliptic curves over the open unit disc $D\subset \mathbb{C}$ which degenerates to the nodal elliptic curve over the point $0$. I'd like to show that such a family is ...
3
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136 views

Making Virasoro uniformization explicit for elliptic curves

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
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0answers
120 views

Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$

In Siksek's notes The modular approach to Diophantine equations he uses the following result: Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
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54 views

Elliptic fibrations on some Kummer surface in characteristic $2$

In the question I ask about one elliptic fibration on the surface $$ K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2. $$ over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...
3
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0answers
88 views

A uniform version of Bashmakov's theorem for elliptic curves

Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
2
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2answers
228 views

Algorithm for finding integral points $P,n P$ on an elliptic curve

We found and implemented algorithm which finds integral points of infinite order $P=(X_1,Y_1)$ and $nP=(X_2,Y_2),n>1$ on an elliptic curve $E : y^2=x^3+a_4 x + a_6$. Let $X(x)/Z(x)$ be the $X$ ...
3
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1answer
244 views

Geometric line bundles on the Tate curve

Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$. ...
2
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1answer
105 views

Better way to compute elliptic curves over finite fields?

I've been using modular polynomials to compute isogeny vulcanoes with prime degree $l$ over finite fields $\mathbb{F}_p$, excluding cases containing the $j$-invariants $0$ and $1728$ or $j$-invariants ...
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141 views

Simultaneous reductions of elliptic curves: same number of points $|E(\Bbb F_p)| = |E'(\Bbb F_p)|$ for some prime $p$?

$ \newcommand{\End}{\mathrm{End}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\kb}{\overline{k}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\Q}{\mathbb{Q}} $ Let $E,E'$ be ...
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0answers
73 views

Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?

Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
5
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204 views

2-descent on elliptic curves, and units modulo squares of units

Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
1
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1answer
221 views

To integrate elliptic integral, we glue two Riemann surface to make torus

To deal with elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this in order to avoid indeterminacy of integral, in other word, to avoid the condition ...
3
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1answer
147 views

Mirror partners of some Calabi-Yau threefolds

I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance. Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. ...
1
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2answers
352 views

On the equation $x^3 + y^3 =cz^3$

What are the characteristics of the values of $c$ for which the equation $x^3 + y^3 = cz^3$ has pairwise coprime non-zero integral solutions where $x \neq \pm y$ ? For instance, it is known that $c$ ...
2
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1answer
234 views

Where does this clever choice of differential come from? (calculating $\mathrm{H}^1_{\mathrm{dR}}(E/k))$

In these notes of Kedlaya, he calculates the de Rham cohomology of an affine part $X$ of an elliptic curve $E$ over a field $K$, given by $y^2 = P(x) = x^3 + ax + b$. He uses these relations: $0 = y^...
2
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0answers
150 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
273 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 $E$)?
1
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1answer
152 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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0answers
128 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
68 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
239 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
135 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
178 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 ...

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