1
vote
0answers
111 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 ...
0
votes
1answer
156 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
0answers
113 views

Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...
1
vote
1answer
201 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 ...
5
votes
1answer
482 views

Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
20
votes
1answer
765 views

Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
10
votes
1answer
411 views

Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties - 1) $E_{1}, E_{2}$ have no rational torsion points. 2) $E_1[9] \cong E_2[9]$ ...
1
vote
1answer
107 views

A special curve with points of order 3(or 6)

Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2-x$ where $$s = -\frac{1}{432}\frac{(81k^8-2592k^4-6912)}{k^6}$$ and $k$ is rational?
20
votes
3answers
627 views

Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square? It is known that the ...
0
votes
1answer
154 views

Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$
1
vote
0answers
129 views

Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...
4
votes
1answer
370 views

Point of order 5 over an elliptic curve

For this curve $y^2=x^3+b^2x^2-a^2b^2x$ where $a \neq b$ and $a,b$ are rational. I can prove that if $b^2+4a^2$ is square then torsion group of curve is $\mathbb Z2 \times \mathbb Z2$, and when ...
7
votes
2answers
308 views

Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...
7
votes
1answer
216 views

Algebraic equations for modular parameterizations

I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way. One type of model for $X_0(N)$ is just given by a single algebraic ...
11
votes
1answer
315 views

$S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be $$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v ...
3
votes
1answer
333 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $ $ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $ with both good ...
3
votes
2answers
184 views

$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$ 2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ ...
3
votes
1answer
226 views

Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely: Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...
0
votes
1answer
110 views

Does the modified Szpiro conjecture require minimal model?

The modified Szpiro conjecture is described in Wikipedia and here and here. The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such ...
1
vote
1answer
204 views

Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity. My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...
2
votes
1answer
240 views

Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...
8
votes
0answers
255 views

divisibility of Tamagawa numbers

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization ...
1
vote
0answers
107 views

Cube-root of j-invariant [closed]

Cube-root of j-invariant is a modular function of level 3, does it have similar property as j-invariant? How about the its minimal polynomial? In particular, are it's coefficients much smaller like ...
1
vote
1answer
82 views

The relative sizes of coordinates of a point on projective genus 1 curve (second try)

Hopefully this is better than what I asked yesterday and Milton solved. Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with no restriction on the degree. Write a point $P = ...
2
votes
1answer
306 views

Invariants for isogenous elliptic curves

How to prove that for elliptic curves, the $\lambda$-invariant is always unchanged by an isogeny ?
5
votes
2answers
160 views

Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
1
vote
1answer
92 views

The relative sizes of coordinates of a point on projective genus 1 curve

Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with no restriction on the degree. Write a point $P = (X , Y , Z)$ with the smallest coprime integers $X,Y,Z$. Is it true that ...
27
votes
0answers
492 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 $$ E_d \colon y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# ...
1
vote
2answers
206 views

Isogeny classes and elliptic curves over finite fields

Fix a conductor and a prime $p$. Then 1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$ 2) Do the ...
12
votes
1answer
429 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 ...
4
votes
2answers
400 views

Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points: Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
2
votes
0answers
219 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 ...
3
votes
1answer
178 views

The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM? ...
3
votes
2answers
254 views

The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how ...
6
votes
1answer
268 views

Fundamental group of the moduli stack of ordinary generalized elliptic curves

Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme ...
4
votes
1answer
216 views

Isogeny classes and reduction types of elliptic curves at primes of bad reduction

Fix a conductor. Then 1) Do the elliptic curves in the same isogeny class have the same reduction type at a prime of bad reduction of the curve ? 2) Do the elliptic curves belonging to two ...
1
vote
0answers
208 views

in the preface of “A first course in modular forms”

In the preface of "A first course in modular forms", the author considered the quadratic equation $Q: x^2=d, \ \ d \in \mathbb{Z}, d \not=0$, and for each prime $p$ define an integer $a_p(Q)=\#\tilde ...
1
vote
0answers
189 views

Elliptic curve is to Tate module as genus one curve is to blank?

Let $E$ be an elliptic curve over $\mathbf{Q}$, and let $\rho:\mathrm{Gal}(\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{Z}_\ell)$ be its Tate module. Let $T$ be the set of isomorphism classes of ...
13
votes
3answers
441 views

Average rank of elliptic curves, excluding those of low rank

It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or ...
0
votes
0answers
134 views

Can one find elliptic curve and a point of known order over $\mathbb{Z}/n\mathbb{Z}$?

Let $n$ be composite with unknown factorization. Can one efficiently find elliptic curve $E$ over $\mathbb{Z}/n\mathbb{Z}$ and a point $P$ on $E$ of known order $m > 5$? $P$ should be nontorsion ...
0
votes
2answers
370 views

Rational points or a Weierstrass model for degree 8 elliptic curve

Related to rationally derived polynomials. Neither Maple nor Magma online couldn't solve it. Choose $s,h \in \mathbb{Q}$. I am looking for rational points (possibly of finite order) on this ...
3
votes
0answers
173 views

Other elliptic curves for $x^4+y^4+z^4 = 1$

Given, $$a^4+b^4+c^4 = d^4\tag{0}$$ we have the identity, $$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$ where, $$591800025 + 20030510 u + 1671327 u^2 ...
6
votes
0answers
211 views

Modular interpretation of Ramanujan theta operator?

I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta ...
3
votes
1answer
173 views

relation between Faltings height and periods

Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods $$ \int_{\gamma} ...
1
vote
1answer
280 views

Determining $\mu$-invariants of elliptic curves over $\mathbb{Q}$

From Pollack's table on his homepage, I have the values of mu invariant of elliptic curves 38B1 & 38B2 (labeled as in Cremona table). But I need to know the values of mu invariants of 38A1, 38A2, ...
7
votes
2answers
608 views

Supersingular elliptic curves over $\mathbb{Q}$

what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?
7
votes
2answers
538 views

BSD conjecture for X_0(17)

I use Magma to calculate the L-value, yields E:=EllipticCurve([1, -1, 1, -1, 0]); E; Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E); Elliptic Curve defined by y^2 + x*y + ...
1
vote
2answers
375 views

Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve 1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
4
votes
0answers
182 views

Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$

As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion ...
2
votes
1answer
344 views

Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$

In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$. In each case the x coordinates are ...