**2**

votes

**1**answer

121 views

### isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...

**0**

votes

**0**answers

126 views

### Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$
splits? I assume $K $ to be a number field and ...

**1**

vote

**0**answers

91 views

### References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...

**-1**

votes

**0**answers

41 views

### Real Lie groups and elliptic curves [migrated]

Let $f:A\to A'$ be a morphism of elliptic curves over the real numbers $\mathbb R$. It canonically induces a morphism $f(\mathbb R): A(\mathbb R)\to A'(\mathbb R)$ between the sets of real points, ...

**0**

votes

**2**answers

131 views

### Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.
Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...

**1**

vote

**1**answer

124 views

### From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...

**1**

vote

**0**answers

52 views

### What is the complexity of finding a generator for the cyclic elliptic curves?

Let $E$ be an elliptic curve which is defined over a finite field $\mathbb{F}_p$, where $p$ is a prime number. If we know that $E(\mathbb{F}_p)$ is cycyclic, is there an algorithm to find its ...

**11**

votes

**2**answers

889 views

### Serre's Open Image Theorem Without Shafarevich's Theorem

In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ associated to an elliptic ...

**4**

votes

**2**answers

199 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 ...

**0**

votes

**0**answers

70 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 ...

**6**

votes

**0**answers

65 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

**1**answer

152 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

**2**answers

361 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

**0**answers

183 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

**2**answers

255 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 ...

**2**

votes

**1**answer

353 views

### Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$.
So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...

**13**

votes

**1**answer

521 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 ...

**12**

votes

**3**answers

567 views

### Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?

**-4**

votes

**1**answer

107 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 ...

**11**

votes

**0**answers

427 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 ...

**5**

votes

**2**answers

576 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

**7**

votes

**1**answer

232 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 ...

**3**

votes

**1**answer

127 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 ...

**4**

votes

**0**answers

141 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

**3**answers

2k views

### Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers

I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 ...

**3**

votes

**0**answers

332 views

### Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...

**0**

votes

**1**answer

130 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 ...

**14**

votes

**2**answers

770 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

208 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

550 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

104 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

58 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

135 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

589 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

313 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

251 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

49 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

105 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

174 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

139 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 ...

**12**

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

178 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, ...

**39**

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

575 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

189 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

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

**2**answers

543 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

18 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 ...

**3**

votes

**0**answers

116 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 ...