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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Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?

I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3. The authors define a $...
Aphelli's user avatar
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2 votes
1 answer
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Why do we get a connected 2-regular graph?

In reading "PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ ...
Shean's user avatar
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8 votes
0 answers
175 views

Elkies' family of elliptic curves of rank 19

There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list at [email protected]&...
Stepan Nesterov's user avatar
0 votes
0 answers
47 views

Reduction of elliptic curves over local fields

Let $E$ be an elliptic curve defined over a p-adic local field $K$, with $j$-invarient $j(E)\in K$. Let $\mathscr{O}_K$ be the ring of integer of $K$. If $j(E)$ does not belong to $\mathscr{O}_K$, ...
zzp's user avatar
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3 votes
1 answer
286 views

Counting points on elliptic curves

Consider the Legendre family of elliptic curves $$E_a: y^2=x(x-1)(x-a).$$ Let $p$ be an odd prime. QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$ over the ...
T. Amdeberhan's user avatar
8 votes
1 answer
260 views

A real-valued analogue of the Weierstrass $\wp$ Function

I am interested in the following function: $$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$ This function is analogous to the Weierstrass $\wp$ function, the only ...
Aobara's user avatar
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1 vote
0 answers
85 views

Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$. ...
stupid_question_bot's user avatar
8 votes
0 answers
121 views

Finding a rational point of large height on an elliptic curve knowing a real approximation

Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial rational point $(r,s)$...
Henri Cohen's user avatar
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2 votes
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Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group. Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence. ...
Duality's user avatar
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7 votes
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$

Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that $\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
Jeremy Rouse's user avatar
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0 answers
71 views

Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1. Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
Vik78's user avatar
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6 votes
2 answers
448 views

Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system, $$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$ $$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$ $$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
Tito Piezas III's user avatar
2 votes
0 answers
127 views

Prime splitting in the division field of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
Jeff H's user avatar
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1 vote
0 answers
72 views

Ramification of mod $\ell$ representation of elliptic curves [closed]

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers. Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
ZZP's user avatar
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7 votes
1 answer
376 views

A parametric elliptic curve for $x^4+y^4+z^4 = 1$?

Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
Tito Piezas III's user avatar
1 vote
1 answer
141 views

cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction. Let $\...
foivos's user avatar
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2 votes
1 answer
349 views

Can an abelian surface be bielliptic

Is an abelian surface containing an elliptic curve a bielliptic surface? Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then $A \to A/E$ is an ...
Stormblessed's user avatar
3 votes
0 answers
179 views

Galois image of CM elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with CM, with the endomorphism ring $R=\mathrm{End}_{\overline{\mathbb{Q}}}(E)$. Then for any integer $m$, we have the mod-$m$ Galois representation $\rho_m:\...
dragoboy's user avatar
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0 answers
110 views

Similar to a $d$-twist but over a cubic field

This question could be related to my old and Duality's newer questions. I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$: $$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$ For $...
Maksym Voznyy's user avatar
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0 answers
77 views

Elliptic curves and images of decompositions group exceptional?

Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...
did's user avatar
  • 595
2 votes
0 answers
61 views

Average rank of elliptic curves over one-parameter family

Let $E_t:y^2=x^3+f(t)x+g(t)$ be an one parameter family of elliptic curves with $f,g\in \mathbb{Z}[t]$. I found one Silverman's result https://www.degruyter.com/document/doi/10.1515/crll.1998.109/pdf ...
dragoboy's user avatar
  • 521
7 votes
1 answer
364 views

Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves

Let $K$ be a complete extension of $\mathbb{Q}_{p}$ with valuation $v$ over $p$, valuation ring $R$, maximal ideal $\mathfrak{m}$ and residue field $k$. It is well known that if $E/K$ is an elliptic ...
David Hubbard's user avatar
2 votes
0 answers
121 views

On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?

From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation, $$a^3+b^3+c^3 = (c+m)^3$$ if we solve the elliptic curve, $$E:=X^3+6d^2X-7d^3 = Y^2$$ More details can be found in this MSE ...
Tito Piezas III's user avatar
3 votes
2 answers
248 views

When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e [closed]

When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for $e$. The equation of the curve is: $y^2 = x^3 + ax + b \...
user520875's user avatar
0 votes
0 answers
80 views

Using the trace map to find rational points on elliptic curves

Let $K$ be a number field and let $E/K$ be an elliptic curve. Let $L/K$ be a finite extension. Consider the trace map $$ \operatorname{Tr}_{L/K}:E(L)\longrightarrow E(K),\qquad \operatorname{Tr}_{L/K}(...
わくわく's user avatar
6 votes
0 answers
195 views

Ranks of elliptic curves over cubic fields

We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations. D. Jeon,...
Maksym Voznyy's user avatar
7 votes
1 answer
180 views

Explicit equations for the universal vector extension of an elliptic curve

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
Vik78's user avatar
  • 487
2 votes
0 answers
88 views

Torsion of an elliptic curve injects under reduction - question

Let $E/K$ be an elliptic curve over a number field. I am interested in the folowing statement: the map $E(K)[m]\rightarrow \tilde E_v(\tilde k_v)$ is injective for any place of $K$ provided there is ...
わくわく's user avatar
1 vote
0 answers
80 views

Finiteness of elliptic curves with trivial conductor over function fields

Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
MightyGuy's user avatar
  • 121
1 vote
0 answers
136 views

Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$

Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
user267839's user avatar
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1 vote
0 answers
99 views

Criterion for an etale cover $E[\ell]\to \mathbb{G}_m$ to be tamely ramified in $0, \infty$

Suppose $E\to \mathbb{G}_m/k$ is an elliptic curve with $k$ field of characteristic $p>0$ and $E[m]$ it $m$-torsion group with $(m,p)=1$. Consider the induced finite etale cover $E[\ell]\to \mathbb{...
user267839's user avatar
  • 5,948
11 votes
2 answers
612 views

Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?

Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere: Let $E$ be an elliptic curve, and $f(q)$ its associated modular ...
KStarGamer's user avatar
2 votes
0 answers
166 views

Is the Weil restriction of an elliptic curve self-dual?

$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$ be a prime split in $K$. Assume that $$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
yhb's user avatar
  • 338
0 votes
0 answers
118 views

Trivializing covers of $\ell$- torsion of elliptic curve

Let $E \to \mathbb{G}_m/k$ an elliptic curve over $ \mathbb{G}_m$ ($k$ field of char $p>0$) and $E[\ell]$ for $(\ell,p)=1$ the $\ell$-torsion group. Let $f:T \to \mathbb{G}_m$ an finite etale ...
user267839's user avatar
  • 5,948
2 votes
1 answer
198 views

Construction refuting the existence of nonisotrivial elliptic curve over $\mathbb{G}_m$

I have some troubles to understand the construction in detail presented here by Daniel Litt used to show that there cannot exist an elliptic curve over $\mathbb{G}_m/k$, $k$ of characteristic $p >3$...
user267839's user avatar
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1 vote
0 answers
64 views

Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist. It is widely known that for all $E/\Bbb{Q}$: elliptic ...
Duality's user avatar
  • 1,406
0 votes
1 answer
285 views

Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $\Gal(L/K)$. Let $E/K$ be an elliptic curve defined ...
Duality's user avatar
  • 1,406
2 votes
0 answers
101 views

elliptic curves on general 3-folds of degree 7

Do there exist elliptic curves on a general 3-fold hypersurface $X_7 \subset \mathbb{P}^4$ of degree $7$? Clemens proved that for $d \ge 8$ there are no elliptic curves on the general hypersurface $...
Ben C's user avatar
  • 3,281
9 votes
1 answer
399 views

How fast can elliptic curve rank grow in towers of number fields?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
David Lampert's user avatar
7 votes
1 answer
407 views

Cubic twist of elliptic curves and its rank

Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$). Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$. $E$ and $E_D$ are isomorphic over $\...
Duality's user avatar
  • 1,406
1 vote
0 answers
90 views

Select random point on elliptic curve

If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
R Artur's user avatar
  • 11
4 votes
0 answers
86 views

Elliptic integral as quantity associated with Riemann surface?

There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]): $$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
Student's user avatar
  • 5,008
2 votes
1 answer
270 views

On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension

Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$, $$ \rho_E: G_{K} \rightarrow \mathrm{...
Hetong Xu's user avatar
  • 579
8 votes
0 answers
167 views

Do there exist Calabi-Yau 3-folds that contain a finite number of elliptic curves?

The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$...
Ben C's user avatar
  • 3,281
0 votes
0 answers
218 views

Reference book on the relation between modular forms and elliptic curves

What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
Cosimo's user avatar
  • 43
3 votes
1 answer
256 views

Global duality theorem for 2-part

$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field. Let $E/K$ be an elliptic curve over $K$. Suppose finiteness of $\Sha(E/K)$. According to Global duality ...
Duality's user avatar
  • 1,406
5 votes
2 answers
436 views

When are two elliptic curves with zero j invariant isogenous?

Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
わくわく's user avatar
0 votes
0 answers
69 views

Is there an $\mathbb{F}_{\!q}$-curve of geometric genus 3 and $\mathbb{F}_{\!q^3}$-cover to an elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant 0?

Let $E$ be an elliptic curve $y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $3 \mid (q-1)$. Is there an absolutely irreducible $\mathbb{F}_{\!q}$-curve $C$ of ...
Dimitri Koshelev's user avatar
2 votes
2 answers
318 views

Upper bound on number of integral solutions of elliptic curves

I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves" And came across a very fascinating ...
Navvye's user avatar
  • 21
6 votes
2 answers
1k views

Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

I am posting this question on MO since I haven't received any answers on MSE. Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
user108580's user avatar

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