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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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 ...
oleout's user avatar
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3 votes
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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 ...
Adithya Chakravarthy's user avatar
5 votes
1 answer
273 views

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 ...
Stabilo's user avatar
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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{-...
yhb's user avatar
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4 votes
1 answer
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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_\...
user avatar
7 votes
1 answer
525 views

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 $x$, $y$, $z$ ...
Dimitri Koshelev's user avatar
4 votes
0 answers
166 views

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....
debanjana's user avatar
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4 votes
1 answer
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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 ...
Dimitri Koshelev's user avatar
3 votes
1 answer
307 views

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 $...
mxian's user avatar
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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 ...
DaveWasHere's user avatar
1 vote
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80 views

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 ...
A. Maarefparvar's user avatar
-1 votes
1 answer
76 views

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 ...
DaveWasHere's user avatar
2 votes
1 answer
139 views

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/\...
foivos's user avatar
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5 votes
2 answers
238 views

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 ...
did's user avatar
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3 votes
0 answers
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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{...
Rdrr's user avatar
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1 vote
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245 views

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 ...
Yuan Yang's user avatar
  • 537
1 vote
1 answer
304 views

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 ...
Rdrr's user avatar
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1 vote
1 answer
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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 ...
57Jimmy's user avatar
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2 votes
0 answers
177 views

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 ...
57Jimmy's user avatar
  • 533
2 votes
2 answers
326 views

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 ...
mathleaf's user avatar
2 votes
0 answers
113 views

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)...
Stanley Yao Xiao's user avatar
4 votes
1 answer
290 views

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 ...
Stanley Yao Xiao's user avatar
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}(...
user avatar
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 ...
user avatar
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{...
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2 votes
0 answers
467 views

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 ...
The Thin Whistler's user avatar
4 votes
1 answer
209 views

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)...
Shenxing Zhang's user avatar
2 votes
1 answer
142 views

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 $K_{\infty}$ ...
Adithya Chakravarthy's user avatar
2 votes
1 answer
307 views

$2$-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, ...
Maksym Voznyy's user avatar
8 votes
0 answers
251 views

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 ...
Ariyan Javanpeykar's user avatar
7 votes
0 answers
212 views

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-...
Tristan Phillips's user avatar
4 votes
1 answer
222 views

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 $\ell$ ...
user11333's user avatar
  • 343
3 votes
1 answer
226 views

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 ...
Adithya Chakravarthy's user avatar
4 votes
2 answers
214 views

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 ...
Adithya Chakravarthy's user avatar
7 votes
2 answers
568 views

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 ...
Maksym Voznyy's user avatar
11 votes
0 answers
499 views

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 ...
KhashF's user avatar
  • 2,588
3 votes
1 answer
251 views

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. ...
Maksym Voznyy's user avatar
2 votes
1 answer
132 views

$p$-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 \...
Adithya Chakravarthy's user avatar
1 vote
0 answers
1k views

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....
Tejas Rao's user avatar
0 votes
1 answer
350 views

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 ...
Maksym Voznyy's user avatar
0 votes
1 answer
160 views

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 ...
Jean's user avatar
  • 515
0 votes
1 answer
163 views

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 ...
EJAS's user avatar
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1 vote
0 answers
58 views

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 \...
Jean's user avatar
  • 515
3 votes
3 answers
444 views

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 ...
Jean's user avatar
  • 515
1 vote
0 answers
116 views

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)/\...
Adithya Chakravarthy's user avatar
4 votes
1 answer
229 views

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 \...
Vlad Matei's user avatar
4 votes
0 answers
158 views

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 ...
Hank Scorpio's user avatar
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 ...
Adithya Chakravarthy's user avatar
5 votes
0 answers
259 views

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 ...
cartesio's user avatar
  • 233
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 ...
Yuan Yang's user avatar
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