**0**

votes

**0**answers

58 views

### Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...

**4**

votes

**0**answers

188 views

### The Modularity Theorem and Serre's/Faltings's Isogeny Theorem

Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. ...

**1**

vote

**1**answer

123 views

### Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...

**2**

votes

**0**answers

88 views

### padic BSD vs. BSD for algorithm to compute rank

Just to be specific, I deal only with the elliptic curves $E$ over $\mathbb{Q}$, and most of the explaination here are obtained from the paper: Algorithms for the Arithmetic of Elliptic Curves using ...

**1**

vote

**1**answer

107 views

### $p$-adic Regulators

Is there some relationship between the $p$-adic regulators of isogenous curves over $\mathbb{Q}$? I've done some computations and their ratio seems to be related (equivalent in all calculations so ...

**4**

votes

**1**answer

155 views

### Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...

**8**

votes

**1**answer

420 views

### Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...

**5**

votes

**1**answer

135 views

### Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$.
...

**10**

votes

**2**answers

323 views

### Existence of a family of elliptic curves with large torsion subgroup

Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html
gives example of the (current) largest known rank of an ...

**10**

votes

**1**answer

273 views

### K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...

**1**

vote

**2**answers

177 views

### BSD leading-term coefficient in terms of places without distinction

After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely
$$
a_0 \stackrel{?}{=} ...

**0**

votes

**0**answers

53 views

### If the $L$-series does not vanish

I refer to this paper http://wstein.org/papers/shark/shark.pdf
At the top of page 24, we are dealing with the issue where the $L$-series does not vanish for the case where $p$ is good and ordinary. ...

**2**

votes

**1**answer

204 views

### Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than
$x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...

**0**

votes

**0**answers

61 views

### Can we deduce that all the real zeros of those $k^{th}$ derivatives are also simple?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...

**2**

votes

**1**answer

69 views

### How the equality in the first case is equivalent to the inequality in the last case?

The motivation to this question can be found in:
About equivalent statements of the Birch and Swinnerton-Dyer Conjecture
My question is about the last equivalences:
$\mathrm{ord}_{s=1} L(E/K,s) = ...

**0**

votes

**0**answers

36 views

### Can we deduce something about the nature of those solutions?

To describe the problem, we note that we can find an affine model for any elliptic curve $C$ over $ℚ$ in Weierstrass form
$$C:y^2=x^3+ax+b$$
with $a,b∈ℤ$. The full L-series of $C$ is given by
...

**4**

votes

**0**answers

80 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

**1**

vote

**2**answers

139 views

### On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...

**2**

votes

**0**answers

257 views

### Proportion of rational elliptic curves of a given rank

This morning appeared on Arxiv the following article by Manjul Bhargava et al: http://arxiv.org/pdf/1407.1826.pdf, in which the authors give a lower bound for th proportion of rational elliptic curves ...

**2**

votes

**2**answers

353 views

### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
...

**6**

votes

**1**answer

160 views

### Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?

Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to ...

**6**

votes

**1**answer

348 views

### Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...

**1**

vote

**1**answer

163 views

### Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...

**8**

votes

**2**answers

411 views

### Elliptic Curves with equal trace of Frobenius Values

Suppose we have two elliptic curves over $\mathbb{Q}$ with trivial rational torsion. Is there some density $\delta$ such that if the trace of Frobenius values of the two elliptic curves are equal on a ...

**3**

votes

**0**answers

84 views

### Lang's height conjecture over $\mathbb{F}_q(T)$?

Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or ...

**1**

vote

**1**answer

106 views

### Can we use this formula to construct rational points on the curve $C$?

One of the techniques used to quantifying the size of a point on an elliptic curve is the so called canonical height defined as follow: Let $R=(x,y)∈C(ℚ)$ where $x=(p/d),p,d∈ℤ$. Define the naive or ...

**0**

votes

**1**answer

184 views

### Is there is a known relation or expression containing the algebraic rank $r$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...

**6**

votes

**3**answers

324 views

### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group ...

**3**

votes

**1**answer

156 views

### Galois representation attached to $3$-torsion points of an elliptic curve

Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points ...

**4**

votes

**1**answer

211 views

### motives of elliptic curves, modular forms, Hecke characters

Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives:
(a) The Chow motive $h^1(E)$ ...

**14**

votes

**2**answers

189 views

### S integral points of an elliptic curve, with S of positive density

Let E be an elliptic curve over Q of non-zero rank. Let S be the union of the primes of bad reduction of E with a Chebotarev set [1]. Suppose additionally that S has density strictly less than one.
...

**7**

votes

**1**answer

170 views

### How is the propagator computed on an elliptic curve?

I've been struggling for a while now understanding why the propagator for the action
$$
S(\varphi) = \int_E \partial \varphi \bar\partial\varphi + \frac{\lambda}{6}(\partial\varphi)^3
$$
on an ...

**3**

votes

**2**answers

136 views

### Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...

**2**

votes

**0**answers

61 views

### Elliptic surfaces with different Kodaira symbols

Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre?
Can ...

**4**

votes

**1**answer

178 views

### Confusion regarding the definition of semistable reduction of an elliptic curve at a prime $p$

I am consulting the recent paper ''On the Integrality of Modular Symbols and Kato's Euler system for Elliptic Curves'' by Chris Wuthrich. But I am confused regarding the definition of semistable ...

**1**

vote

**0**answers

69 views

### Find all points on the elliptic curve using irreducible polynomials [closed]

Hello I am new to the subject of Cryptography. Following is the question I have to solve.
Topic is Elliptic Curve Cryptography (ECC)..
In the elliptic curve E (g2, g7) over the GF(2^4) field: ...

**1**

vote

**1**answer

140 views

### Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...

**4**

votes

**2**answers

213 views

### Argument for unboundedness of integral points of elliptic curves over number fields

Probably this is well known to those who know it.
Got an argument and numerical support that over
number fields elliptic curves in minimal models
might have unbounded number of integral points,
the ...

**21**

votes

**1**answer

927 views

### Possible counterexample to a theorem assuming Lang's conjecture

Looks like I found a counterexample to a theorem assuming Lang's conjecture,
but not sure it is correct.
Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture
p. 2
...

**1**

vote

**1**answer

220 views

### Embedding of an elliptic curve into $\mathbb{P}^2 \times \mathbb{P}^2$

Let $E$ be a smooth elliptic curve over a field $k$. Let
$$
i : E \to \mathbb{P}^2 \times \mathbb{P}^2,
$$
be an embedding. How one can find an explicit canonical forms of equations cutting $E$ in ...

**11**

votes

**1**answer

333 views

### Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$

Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} ...

**0**

votes

**0**answers

115 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 ...

**2**

votes

**0**answers

170 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 ...

**4**

votes

**2**answers

359 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

**4**

votes

**3**answers

178 views

### Quadratic twist of an elliptic curve given by non-Weierstrass model

Suppose $f(x)$ is a polynomial of degree 4 with integer coefficients and nonzero discriminant. Let $C$ be the hyperelliptic curve of genus 1 defined by $y^2=f(x)$. If we assume that $C$ has a rational ...

**1**

vote

**1**answer

112 views

### Elliptic curves with square conductor

Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?

**0**

votes

**1**answer

175 views

### Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...

**1**

vote

**2**answers

138 views

### Anomalous elliptic curves over finite rings

I was wondering if it is possible to solve the discrete logarithm on an Elliptic Curve E(Z/nZ) (defined over the ring of integers modulo a composite n) with #E(Z/nZ)=n by applying a method analogous ...

**1**

vote

**1**answer

210 views

### On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c_4,c_6$.
Via elliptic curves it might have ...

**6**

votes

**1**answer

660 views

### Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...