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 in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

learn more… | top users | synonyms

0
votes
0answers
60 views

Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?

Suppose that $G=\langle g\rangle$ is a general group of order $p$. Maurer has introduced an algorithm to reduce the discrete log problem to the Diffie-Hellman problem under a conjecture about smooth ...
4
votes
2answers
169 views

Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
6
votes
0answers
59 views

Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
5
votes
1answer
140 views

On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...
13
votes
2answers
332 views

Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$. Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
4
votes
0answers
169 views

Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1. One can show that for a general ...
3
votes
2answers
229 views

n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
-4
votes
1answer
93 views

Elliptic Curve Multiplication [closed]

What would happen if I performed Elliptic Curve multiplication on some random point within the FiniteField that wasn't actually on the curve? I assume that I would get a point in return but would that ...
10
votes
0answers
408 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
13
votes
1answer
506 views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
7
votes
1answer
217 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
4
votes
0answers
138 views

Models for the moduli space $\overline{M}_{1,n}$

Let $\overline{M}_{1,n}$ denote the coarse moduli space of $n$-pointed elliptic curves. Is there an explicit description of these spaces (a la Kapranov's construction) for low $n$? Apparently this ...
3
votes
1answer
123 views

Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve

Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...
0
votes
1answer
122 views

homological invariant of the “universal elliptic curve” over the punctured $j$-line

My question considers the curve $E$ over the affine $j$-line $S$ given by $$Y^2 - (j-1728)XY = X^3 - 36(j-1728)^3X - (j-1728)^5$$ This curve has the property that it's $j$-invariant is $j$ (see ...
3
votes
1answer
204 views

Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$

Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
0
votes
1answer
57 views

The group of real points on quadratic twist of elliptic curve has one connected component

I am trying to understand the proof of assertion (i) in Proposition 3.10 (page 14) of this paper http://arxiv.org/pdf/1312.3884v3.pdf $M$ stands for a square free integer which is prime to $7$, $A$ ...
0
votes
0answers
99 views

Ward's formula for elliptic divisibility sequences

M. Ward in his Memoir on elliptic divisibility sequences proved that the sequence $\{a_n\}$ defined by recurrence $$a_{n+2}a_{n-2}=a_2^2a_{n+1}a_{n-1}-a_1a_3a_n^2$$ and initial conditions ...
2
votes
1answer
128 views

Frey's Formula and utilisation of the Hasse Invariant in “Links between Stable elliptic curves and Diophantine equations.”

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then ...
8
votes
2answers
580 views

Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does $$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$ have nontrivial solutions ...
2
votes
0answers
247 views

When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$. When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...
8
votes
1answer
309 views

A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves

Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
1
vote
0answers
48 views

If $E(K)=E(L)$ for an elliptic curve $E$ and an algebraic extension $L/K$, what can we say about $Sel(E/K), Sel(E/L), L/K$?

More generally, when $E(K_m)$ is stable as $m$ increases for an extension equence $K_0<K_1<K_2<\cdots<K_m<\cdots$ ? In the case, is $\mathrm{Sel}(E/K_m)$ stable as $m\rightarrow ...
1
vote
1answer
104 views

Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C is a square

Trying to understand the proof of Corollary 2.3 in the following paper, http://arxiv.org/pdf/1312.3884.pdf I would like to be able to justify that the root number of the quadratic twist ...
0
votes
1answer
170 views

Hyperelliptic curve of genus 2 over R

I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve ...
2
votes
1answer
130 views

How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero?

Let $E \subset \mathbb{P}^3_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization $$ \chi: (\mathbb{R}/\mathbb{Z})\times ...
3
votes
1answer
173 views

Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
0
votes
0answers
41 views

Equivalent of Lauricella $F_D$ on an elliptic curve?

Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. ...
4
votes
1answer
570 views

How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?

Related to this question where degree $2$ algebraic curve is good approximation to vanishing of the real part of expression involving zeta. Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...
2
votes
1answer
181 views

An explicit formula for Weil pairing on a complex torus

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers. Let ...
0
votes
0answers
68 views

Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?

Let $E$ be elliptic curve over the rationals and $P=(X_P,Y_P)$ point on $E$. $\psi_n$ are the division polynomials. Define $a_n=\psi_n(X_P,Y_P)$. Is it possible $a_n$ to be perfect power ...
4
votes
2answers
533 views

Unable to find any information regarding this fact (Frey, elliptic curves)

Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following "The most important fact about elliptic curves with reduction of muItipIicative type is due to ...
0
votes
0answers
17 views

Hardness assumptions on composite order bilinear groups

I am not at all knowledgeable in elliptic curve cryptography. So, here lies a couple of questions that I failed to find answers for to my satisfaction. Is there any known Type-III bilinear pairing ...
2
votes
0answers
109 views

Galois descent for a non-Galois extension

Suppose that $k$ is an algebraically closed field of characteristic $p > 0$ and $E/k$ is a supersingular elliptic curve equipped with a full level $N$ structure $\phi$ for some $N \ge 3$ that is ...
1
vote
1answer
186 views

Heegner points on elliptic curves

I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?
1
vote
0answers
161 views

Fourier expansions at the cusps of $\Gamma_0(N)$

My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form ...
4
votes
1answer
188 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
2
votes
0answers
147 views

A morphism of elliptic schemes that preserves the identity is a homomorphism

I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group ...
2
votes
1answer
172 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
2
votes
1answer
162 views

rational point of a curve [closed]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
0
votes
0answers
28 views

Reading roots of the supersingular polynomial from an explicit formula

I wish to obtain an explicit expression for the supersingular polynomial such that the roots of the polynomial can be easily read from it. I am aware of the work of Brillhart and Morton -- ...
5
votes
1answer
530 views

The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. If there are $a,b,c,d,e,f$ such that, $$a+b+c = d+e+f\tag1$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$ $$3u^3-3uv+w=-def\tag3$$ where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then, $$(a + u)^k + (b + ...
1
vote
0answers
86 views

Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups

Let me start with the simplest version of the question since already there I don't know anything. For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb ...
0
votes
0answers
97 views

Asymptotic Expansion of Double integral

Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely. Define $$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} ...
5
votes
0answers
281 views

is the modular curve X(N) defined over Q?

In most sources, the field of definition of the modular curve $X(n)_\mathbb{C}$ (quotient of the upper half plane by the subgroup $\Gamma(n)$ of $SL_2(\mathbb{Z})$ congruent to $I\mod n$) is ...
2
votes
2answers
524 views

Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?

Let $f(x,y)=0$ be irreducible elliptic curve over the rationals. Are there $f$ for which: Both $x,y$ are arbitrary large powers infinitely often, i.e. infinitely many rational points $(u^k,v^m)$ ...
3
votes
0answers
228 views

On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation, $$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$ for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...
24
votes
1answer
986 views

$x^4+y^4$ powerful for relatively prime $x,y$

I asked this question on the NMBRTHRY mailing list on 17 February 2014, but it remains unsolved as far as I know. Recall that a "powerful number" is a positive integer whose prime factorizations $m = ...
5
votes
1answer
289 views

On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve, $$x(x+a^2)(x+b^2)=y^2\tag1$$ can be used to prove that, $$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$ has an infinite number of primitive ...
11
votes
1answer
790 views

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

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.) The equation discussed in a paper by Jacobi and Madden, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$ or equivalently, $$(p-2q + ...
3
votes
2answers
146 views

Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?

Consider an elliptic curve $E/\Bbb Q$ and let $\Phi:\Gamma_0(N)\backslash\overline{\mathfrak{H}}\rightarrow E(\Bbb C)$ be the analytical description of its modular parametrization. We know that this ...