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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Relation between $L$-values of elliptic curves and Manin constants
Given an elliptic curve $E$ over $\mathbf{Q}$, we can attach two numbers two it.
the so-called Manin constant $c_E$. (Defined below the fold.)
the "algebraic $L$-value" given by $L(E,1)/\...
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integral points on elliptic curves in terms of discriminant
I am curios where in the literature was the first time written the following conjecture.
Say we have we have an elliptic curve $E$ given by the Weierstrass equation $y^2=x^3+AX+B$ with $A,B\in \...
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Derivative of dual isogeny is pullback on $H^1$
Apologies if this is not quite at the level of MathOverflow, but I have already asked at MSE and received no answer after two+ weeks and a bounty.
Let $X$ and $Y$ be elliptic curves (over an ...
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Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$
Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. My question is: how do you compute the local cohomology group $H^1(K_v, \, E[p^{\infty}])$?
As to why I'm asking this, it ...
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Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
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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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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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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:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
...
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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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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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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:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \...
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Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?
For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...
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How do you compute modular symbols?
In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way.
Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The ...
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To justify the intuition about #$E(\Bbb Q_p)$＝$∞$
Let $E$ be an elliptic curve on $\Bbb Q_p$.
$E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points.
How to prove #$E(\Bbb Q_p)$＝$∞$ directly ?
According to Silverman's book 'the ...
- 1,405
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Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?
Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
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Questions on the $j$-invariant
The j-invariant as a modular function is typically defined
$$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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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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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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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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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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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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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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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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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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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{\...
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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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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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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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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 $$
What is the value of the sum $\ S\ $ when $\ p\equiv 1\pmod{4}\,?\...
user317834
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An example of Serre on the cohomology of some CM elliptic curves
Let $E$ be the elliptic curve over $\mathbf{Q}_3$ with Weierstrass equation $y^2 = x^3-x$.
It has complex multiplication by $\mathbf{Z}[\sqrt{-1}]$, with $\sqrt{-1}$ acting as $(x,y)\mapsto(-x,iy)$.
...
user178246
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Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$
Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$,
define
$E_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
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Showing that two families of elliptic curves are diffeomorphic
Consider a family of elliptic curves over the open unit disc $D\subset \mathbb{C}$ which degenerates to the nodal elliptic curve over the point $0$. I'd like to show that such a family is ...
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Making Virasoro uniformization explicit for elliptic curves
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
user108998
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Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
In Siksek's notes The modular approach to Diophantine equations he uses the following result:
Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
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Elliptic fibrations on some Kummer surface in characteristic $2$
In the question I ask about one elliptic fibration on the surface
$$
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 ...
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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 ...
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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$ ...
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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$.
...
user178246
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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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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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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{...
- 1,801
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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))$, ...
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457
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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 ...
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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$. ...
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413
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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$ ...
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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^...
- 1,102
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0
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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 ...
- 24.2k
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299
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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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198
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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 ...
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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 ...
user176661