**4**

votes

**3**answers

337 views

### Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and
$$G=\{P,2P,3P,4P,\cdots\}.$$
Is there an integral point $Q\in G?$

**4**

votes

**1**answer

277 views

### What is the complexity of finding an integral point on an elliptic curve?

Let $E$ be an elliptic curve over rational numbers. We know that the set of integral points on $E$ is finite. What is the complexity of finding a point $P\in E(\mathbb{Z})$?
Indeed I'm trying to find ...

**5**

votes

**1**answer

327 views

### Are the Szpiro ratios of 37b1 over certain number fields {33,39,42,48,51,66}?

Related to this question.
According to Hindry p.7 Conj 3.1
and Stein's notes Szpiro's conjecture over number fields states that the Szpiro ratio is:
$$ ...

**4**

votes

**1**answer

342 views

### Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field?

According to Hindry p.7 Conj 3.1
and Stein Szpiro's conjecture states that the Szpiro ratio is:
$$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$
Given $ \varepsilon ...

**3**

votes

**2**answers

214 views

### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...

**3**

votes

**1**answer

276 views

### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

**1**

vote

**0**answers

166 views

### which sections of elliptic curves are conjugate?

Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...

**8**

votes

**2**answers

424 views

### How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...

**6**

votes

**0**answers

227 views

### Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...

**1**

vote

**0**answers

308 views

### How to find generators to Mordell weil groups of elliptic curves?

I am new to branch of elliptic curve and algebraic number theory .I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$ and class number of $\mathbb ...

**9**

votes

**2**answers

398 views

### Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...

**2**

votes

**2**answers

314 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**0**

votes

**0**answers

242 views

### Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...

**8**

votes

**1**answer

360 views

### Separation of lattice points on the Mordell elliptic curve

Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) ...

**0**

votes

**0**answers

78 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

271 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

142 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

115 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

135 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

170 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

543 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

161 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

401 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

360 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 ...

**2**

votes

**2**answers

213 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

58 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

287 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: $x=3,y=15$, $x=5,y=55$, ...

**2**

votes

**1**answer

96 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) = ...

**4**

votes

**0**answers

98 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

**1**answer

200 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

283 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

450 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
...

**7**

votes

**1**answer

200 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

429 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

217 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

486 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

95 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

125 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

195 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

369 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

187 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

293 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

215 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.
...

**8**

votes

**1**answer

195 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

165 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

74 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

253 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

106 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

147 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

250 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 ...