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.
1,504
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Embedding of a genus 1 hyperbolic curve
Let $E$ be an elliptic curve over a number field $k$. We define the affine curve $C := E \backslash \{p_1,...,p_n\}$ by removing a finite number of points from $E$. Here, I would like to declare that ...
3
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0
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178
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Extending the analogy between cyclotomic units and elliptic units
There is a nice analogy between cyclotomic units and elliptic units given as follows:
Cyclotomic units are related to special values of the Riemann Zeta function. This is because the logarithmic ...
5
votes
1
answer
273
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Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?
For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field.
For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...
0
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0
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86
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Determine the CM type of a CM elliptic curve
I have something don't understand about the CM types of CM elliptic curves. I want to determine the CM type of certain elliptic curves. Let $K=\mathbb{Q}(\sqrt{-3})$ be the CM field and $\omega=\frac{-...
4
votes
1
answer
834
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What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?
Let $E_1$ and $E_2$ be Elliptic curves over the field $K$ and $\ell\neq\mathrm{char}(K)$ be a prime number. Let $T_\ell(E_i)$ is the Tate module of $E_i$, $i=1,2$ and $\mathrm{Hom}_K(T_\ell(E_1),T_\...
7
votes
1
answer
525
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A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
4
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0
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166
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Primes of supersingular reduction for non-CM elliptic curves
When $E/\mathbb{Q}$ is a non-CM elliptic curve, Serre had shown that there are density 0 primes of supersingular reduction. His proof can be generalized to elliptic curves over arbitrary number fields....
4
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1
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164
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Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve
Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
Lange and Ruppert - Complete ...
3
votes
1
answer
307
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Lang's proof concerning ray class fields of imaginary quadratic number fields
Crosspost from Math.SE as I did not receive an answer there:
In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...
3
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0
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222
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Birationally equivalent elliptic curves and singularities
I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3
\alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma
-\beta ^2$ for known ...
1
vote
0
answers
80
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The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions
Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map
\begin{equation}
\mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
-1
votes
1
answer
76
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Characterization of tori/elliptic curve isogenies
I am reading Chapter 11 of Dale Husemöller's Elliptic Curves Springer book and I got stuck on Theorem (1.4) (c.f., image below).
Notation and definitions: Let $L$ and $L'$ be two complex lattices ...
2
votes
1
answer
139
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Mordell-Weil rank growth in Iwasawa tower
This is more of a reference request in case anyone can direct me to the right literature. I asked originally on MathStack, but I was suggested to better post it here.
If you have an elliptic curve $E/\...
5
votes
2
answers
238
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Model of an elliptic curve with p-torsion
Suppose I have an elliptic curve $E$ defined over a number field $K$.
I know that if it has
a $2$ $K$-torsion, it has a model of the form:
$E: Y^2=X^3+aX^2+bX$
a $3$ $K$-torsion, it has a model of ...
3
votes
0
answers
160
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Rank and Taylor coefficient in Birch and Swinnerton–Dyer
I am trying to get a better understanding of the Birch and Swinnerton–Dyer conjecture. I have two questions
Why might one expect that the analytic rank of $L(E,s)$ is equal to the rank of $E(\mathbb{...
1
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0
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245
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4-distance problem and elliptic curves
The 4-distance problem is an open question(as far as I know it is still open) that asks if there exists a point P on the Euclidean plane such that its distances to all four points of a unit square are ...
1
vote
1
answer
304
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Primes of bad reduction for CM elliptic curves
$\DeclareMathOperator\Norm{Norm}$Suppose $E/\mathbb{Q}(j(E))$ is a CM elliptic curve and $d$ is a non-square. Let $E_d$ denote the twist of $E$ by $\mathbb{Q}(j(E))(\sqrt{d})$. I know if $d$ is ...
1
vote
1
answer
297
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Cohomology of the dual Abelian variety
I am interested in the (degree $1$) Betti cohomology of the dual $A^\vee$ of an Abelian variety $A$ (say, over $\overline{\mathbb{Q}}$). We can even assume $A$ to be an elliptic curve, if this makes ...
2
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0
answers
177
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Picard and Rosati for elliptic curves
I would like to ask for confirmation whether the following argument is correct.
We work over an algebraically closed field $k$ of characteristic $0$. For an elliptic curve $E$, the Picard variety, or ...
2
votes
2
answers
326
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Mordell-Weil rank of some algebraic surface
Consider an elliptic curve $S:y^2 = x^3 + t^2x + (t^3 + 1)$ over $k(t)$, where char($k$) is 0.
How can I calculate the Mordell-Weil rank of the surface, or how to get its Picard number $\rho(S)$ of ...
2
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0
answers
113
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Semi-stable elliptic curves and Szpiro ratios
This is a continuation of the following question: Szpiro ratios of elliptic curves over $\mathbb{Q}$
In that question I asked whether Szpiro ratio
$$\displaystyle \beta_E = \frac{\log |\Delta_{\min}(E)...
4
votes
1
answer
290
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Szpiro ratios of elliptic curves over $\mathbb{Q}$
For an elliptic curve $E/\mathbb{Q}$, let us denote by $\Delta_{\min}(E)$ the minimal discriminant of $E$ and $N(E)$ the conductor of $E$. Then it is well-known that $N(E) | \Delta_\min(E)$.
The ...
2
votes
1
answer
482
views
Galois invariants and tensor products
Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group
$$E(L)\otimes L^{\times}.$$
Every element $g$ in $\text{Gal}(...
6
votes
1
answer
346
views
Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
4
votes
1
answer
387
views
Tate-Shafarevich groups under finite Galois field extensions
Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$.
Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{...
2
votes
0
answers
467
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Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
4
votes
1
answer
209
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An analogy of product formula for homogeneous space?
$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)...
2
votes
1
answer
142
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Does $\mu=0$ for an imaginary quadratic field $K$ imply $\mu=0$ for $\mathbf{Q}$?
Suppose that $E/\mathbf{Q}$ is an elliptic curve and $K$ is an imaginary quadratic field. Let $\mathbf{Q}_{\infty}$ denote the cyclotomic $\mathbf{Z}_p$ extension of $\mathbf{Q}$, and let $K_{\infty}$ ...
2
votes
1
answer
307
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$2$-isogenous to a curve in the Tate normal form
It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a $2$-isogenous curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in
A. Dujella, ...
8
votes
0
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251
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Simultaneous rank jumping of elliptic curves over number fields
Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
7
votes
0
answers
212
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Counting elliptic curves over finite fields with a prescribed number of points
Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, and let $\mathcal{O}$ be an order in $K$ of discriminant $D$ and class number $h(\mathcal{O})$. Then the Hurwitz-...
4
votes
1
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222
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Is Galois representation induced by semistable elliptic curve semistable?
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l]) $ be mod $\ell$ ...
3
votes
1
answer
226
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Existence of congruences between modular forms / elliptic curves
I'd like to ask two questions about congruences: one about modular forms and one about elliptic curves.
Suppose we are given a cusp form $f$ of weight $2$ and level $\Gamma_0(N)$. Given a good ...
4
votes
2
answers
214
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Bounds on $p$-primary Selmer groups when $E[p]$ is irreducible
My question is: if $E$ is an elliptic curve over $\mathbf{Q}$, and $p$ is a prime number such that $E[p]$ is irreducible as a Galois module, how does one go about bounding the $p$-primary Selmer group ...
7
votes
2
answers
568
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ℤ/18ℤ elliptic curves over cubic fields
I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
11
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0
answers
499
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A curious observation on the elliptic curve $y^2=x^3+1$
Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end).
Take a point of $y^2=x^3+1$ and ...
3
votes
1
answer
251
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Rationalizing and minimizing elliptic curve coefficients
I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E_1=[0,a,0,b,0]$ following the formulas on p. 13 of
L. ...
2
votes
1
answer
132
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$p$-adic valuation of $L$ values for elliptic curves
I'm wondering if the following conjecture is true:
Let $\mathcal{A}$ be an isogeny class of elliptic curves over $\mathbf{Q}$. Fix an odd prime $p$ of good reduction. Then there is a curve $E \in \...
1
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0
answers
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What's the best reference for Abelian varieties?
I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves.
I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
0
votes
1
answer
350
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Systems of equations for elliptic curves without $3$-torsion
In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank ...
0
votes
1
answer
160
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Lower bound related to derivative of $j$-invariant
Recall the $j$-invariant function, namely,
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{2\pi i \tau}$ and the coefficients $(c_k)_k$ are in the OEIS sequence A000521.
By using some ...
0
votes
1
answer
163
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How to glue a section of $T^*\mathbb{P}^1$ to create an elliptic curve
Consider a meromorphic section of the cotangent bundle $T^*\mathbb{P}^1$. Such a section has two poles, say at $0$ and $\infty$ with residues $a,-a$ for some $a\in\mathbb{C}$. I'd like to take this ...
1
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0
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58
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A lower bound for a sum related to the $j$-invariant function
There are some days that I am thinking in the following problem.
For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \...
3
votes
3
answers
444
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Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
1
vote
0
answers
116
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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)/\...
4
votes
1
answer
229
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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 \...
4
votes
0
answers
158
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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 ...
2
votes
1
answer
257
views
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 ...
5
votes
0
answers
259
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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 ...
1
vote
1
answer
218
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 ...