**13**

votes

**2**answers

736 views

### Simple proofs for the existence of elliptic curves having a given number of points

Yesterday, after he gave a nice talk, Dick Gross and I were chatting and he brought up the following annoying problem: suppose that for $p$ a prime that $H_p$ is the "Hasse interval" $[p+1- 2 ...

**3**

votes

**1**answer

192 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?

**6**

votes

**1**answer

522 views

### Can elliptic integral singular values generate cubic polynomials with integer coefficients?

For the elliptic integral of first kind, $K(m)=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-m^2sin^2\theta}} $, it is well-known that $K(m)$ can be expressed in what Chowla and Selberg call "finite terms" ...

**0**

votes

**0**answers

87 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 ...

**0**

votes

**1**answer

49 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 $7$, $A$ ...

**2**

votes

**1**answer

116 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

**2**answers

560 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 ...

**8**

votes

**1**answer

299 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 ...

**2**

votes

**0**answers

243 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 ...

**1**

vote

**0**answers

37 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

**1**answer

95 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

**1**answer

159 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

**1**answer

113 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 ...

**11**

votes

**5**answers

1k views

### Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...

**3**

votes

**1**answer

166 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, ...

**38**

votes

**0**answers

2k views

### Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1

Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...

**4**

votes

**1**answer

562 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 ...

**0**

votes

**0**answers

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. ...

**2**

votes

**1**answer

169 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

**0**answers

64 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

**2**answers

519 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

**0**answers

14 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

**0**answers

101 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

**1**answer

175 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

**0**answers

144 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

**1**answer

179 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

**0**answers

143 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

**1**answer

171 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 ...

**3**

votes

**1**answer

634 views

### Does the following Diophantine equation have nontrivial rational solutions?

Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am ...

**2**

votes

**1**answer

158 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

**0**answers

24 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 -- ...

**35**

votes

**0**answers

725 views

### The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# ...

**5**

votes

**1**answer

517 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

**0**answers

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 ...

**19**

votes

**2**answers

863 views

### State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
...

**24**

votes

**1**answer

929 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 = ...

**0**

votes

**0**answers

84 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} ...

**2**

votes

**2**answers

520 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 $(u^k,v^m)$ ...

**5**

votes

**0**answers

261 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 ...

**3**

votes

**0**answers

282 views

### Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...

**3**

votes

**0**answers

220 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 ...

**5**

votes

**1**answer

284 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

**1**answer

721 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

**2**answers

142 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 ...

**2**

votes

**1**answer

93 views

### Simple Isogeny Question

I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally.
One can define the degree of an isogeny as the degree of extension fields of the ...

**7**

votes

**2**answers

336 views

### Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?

**11**

votes

**2**answers

503 views

### Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?

Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?

**1**

vote

**1**answer

154 views

### Canonical form of cubic curves over general fields

Given a field of characteristic not 2 or 3 containing a primitive third root of unity, is it true that every nonsingular cubic curve, i.e. a curve defined by one homogeneous form of degree 3 in 3 ...

**3**

votes

**1**answer

289 views

### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

**5**

votes

**1**answer

333 views

### Many integral points on quartic models of elliptic curve via differences of squares

Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...