All Questions
1,978 questions
75
votes
9
answers
17k
views
Why is an elliptic curve a group?
Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the ...
- 1,446
7
votes
1
answer
464
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 ...
- 1,986
4
votes
1
answer
450
views
Other elliptic curves for $x^4+y^4+z^4 = 1$
Given,
$$a^4+b^4+c^4 = d^4\tag{0}$$
we have the identity,
$$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$
where,
$$591800025 + 20030510 u + 1671327 u^2 +...
- 12.6k
1
vote
1
answer
179
views
cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$
Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.
Let $\...
2
votes
1
answer
373
views
Can an abelian surface be bielliptic
Is an abelian surface containing an elliptic curve a bielliptic surface?
Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then
$A \to A/E$ is an ...
- 39.3k
16
votes
1
answer
2k
views
Reference to a Don Zagier result and the congruent number problem
I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...
- 4,713
3
votes
0
answers
218
views
Galois image of CM elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with CM, with the endomorphism ring $R=\mathrm{End}_{\overline{\mathbb{Q}}}(E)$. Then for any integer $m$, we have the mod-$m$ Galois representation $\rho_m:\...
- 28.2k
2
votes
0
answers
136
views
Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
- 913
0
votes
0
answers
89
views
Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...
- 63.9k
2
votes
0
answers
65
views
Average rank of elliptic curves over one-parameter family
Let $E_t:y^2=x^3+f(t)x+g(t)$ be an one parameter family of elliptic curves with $f,g\in \mathbb{Z}[t]$. I found one Silverman's result https://www.degruyter.com/document/doi/10.1515/crll.1998.109/pdf ...
- 521
7
votes
1
answer
492
views
Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves
Let $K$ be a complete extension of $\mathbb{Q}_{p}$ with valuation $v$ over $p$, valuation ring $R$, maximal ideal $\mathfrak{m}$ and residue field $k$. It is well known that if $E/K$ is an elliptic ...
- 63.9k
2
votes
0
answers
124
views
On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k
3
votes
2
answers
287
views
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e [closed]
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for $e$.
The equation of the curve is: $y^2 = x^3 + ax + b \...
- 63.9k
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 63.9k
0
votes
0
answers
100
views
Using the trace map to find rational points on elliptic curves
Let $K$ be a number field and let $E/K$ be an elliptic curve. Let $L/K$ be a finite extension. Consider the trace map
$$
\operatorname{Tr}_{L/K}:E(L)\longrightarrow E(K),\qquad \operatorname{Tr}_{L/K}(...
- 89
4
votes
0
answers
506
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
CommunityBot
- 1
6
votes
0
answers
219
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,...
- 913
7
votes
1
answer
303
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 ...
- 30.5k
2
votes
0
answers
179
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
$$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
- 47.4k
8
votes
1
answer
945
views
Where do nonstandard elliptic curve angles come from?
This is a question which has bounced around my head over the past few years. At the same time, I am answering Riemann hypothesis for zeta function of algebraic curves over fields of infinite ...
- 63.9k
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
2
votes
0
answers
105
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 ...
- 89
1
vote
0
answers
90
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 ...
- 121
1
vote
0
answers
169
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 ...
- 12.9k
1
vote
0
answers
103
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{...
- 5,986
11
votes
2
answers
694
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 ...
- 20.8k
0
votes
0
answers
136
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 ...
- 5,986
0
votes
1
answer
353
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 ...
- 12.9k
2
votes
1
answer
222
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 $p >3$...
- 10.7k
1
vote
0
answers
102
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 ...
- 1,541
2
votes
0
answers
807
views
Why is the congruent number problem open?
I was reading up about the congruent number problem.
One of the theorems on the subject says how the two things are equivalent: a positive integer $n$ being a congruent number and elliptic curve $y^...
- 317
2
votes
0
answers
107
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 $...
- 3,730
10
votes
1
answer
444
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 ...
- 2,665
7
votes
1
answer
630
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 $\...
- 8,945
1
vote
0
answers
133
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 ...
- 11
4
votes
0
answers
112
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})}}...
- 5,230
8
votes
1
answer
327
views
3-divisibility of Manin constant for elliptic curves with 3-torsion
Let $E/\mathbb{Q}$ be an elliptic curve with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$ (not necessarily $\Gamma_0$-optimal). Does $3$ necessarily divide one of: the Manin constant (not necessarily $...
CommunityBot
- 1
8
votes
0
answers
178
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)$...
- 3,730
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
6
votes
1
answer
268
views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
- 63.9k
0
votes
0
answers
235
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?
- 12.9k
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 ...
- 4,985
5
votes
2
answers
543
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$ ...
- 1,403
2
votes
2
answers
415
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 ...
- 61
0
votes
1
answer
272
views
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
- 9,497
6
votes
0
answers
166
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
CommunityBot
- 1
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 8,945
3
votes
0
answers
171
views
Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants
$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
- 1,541
8
votes
1
answer
316
views
Is there an elliptic curve over a number field with a point of order 64 and Mordell-Weil rank zero?
It seems to me that there ought to be elliptic curves over number fields with arbitrarily large torsion subgroups but Mordell-Weil rank zero. But I'll settle for a point of order 64. Does anyone ...
- 8,945
1
vote
0
answers
320
views
Tate-Shafarevich group of Elkies curve
The Elkies curve
$$
E:y^2+xy+y=x^3−x^2−20067762415575526585033208209338542750930230312178956502x+34481611795030556467032985690390720374855944359319180361266008296291939448732243429
$$
conductor of ...
- 245