1
vote
0answers
8 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
-1
votes
0answers
24 views

Find point of maximal order $\operatorname{lcm}(a,b)$ on a curve

In Elliptic curve okamoto uchiyama there is a condition for picking base point $G$ such that $G$ belongs to $E_n$ of maximal order $\operatorname{lcm}(|E_{p2}|,|E_q|)$. I am not getting these ...
4
votes
1answer
36 views

Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
0
votes
0answers
33 views

Constant in an inequality for a height of an elliptic curve

I am trying to find explicitly a constant $\kappa$ in an inequality for the height of an elliptic curve. Suppose the curve $E$ is defined by $y^2 = x^3 - kx$ with $k \neq 0$, the curve is defined over ...
0
votes
1answer
79 views

How is this an isomorphism?

$\newcommand\O{\mathcal O}$I was reading Silverman and Tate's Rational Points on Elliptical Curves. In page 21 of the same book it was written We also want to mention that there is nothing ...
2
votes
1answer
36 views

Coefficients of an elliptic curve for which the torsion group is trivial

Consider an elliptic curve in the short Weierstrass form $$ y^2 = x^3 + bx + c, $$ defined over rational numbers ($b,c$ are integers). My goal is to provide an example of congruence relations on $b$ ...
2
votes
1answer
30 views

Height function constants

Consider an elliptic curve defined over the field of rational numbers and given by $$\mathcal{E}_n: y^2=x^3-kx,\ k \ne 0$$ Let $B = \left(\dfrac{r}{s^2},\dfrac{t}{s^3}\right)$ with $r,s,t$ coprime. ...
4
votes
1answer
81 views

Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
3
votes
1answer
54 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
3
votes
1answer
51 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
6
votes
2answers
69 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
0
votes
1answer
42 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
1
vote
1answer
32 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...
3
votes
1answer
79 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
3
votes
3answers
55 views

Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup

Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup. We've been told that for this problem, we are not allowed to use Mazur's Theorem. ...
2
votes
1answer
45 views

Show that the curve $2Y^2 = X^4 - 17$ has no points in $\mathbb{Q}$

There is a hint to show that if there were points in $\mathbb{Q}$, then there would exist $r,s, t \in \mathbb{Z}$ with $\gcd(r, t) = 1$ such that $2s^2 = t^4 - 17r^4$ , and then show that any prime ...
4
votes
1answer
65 views

Discriminants and Weierstrass form of elliptic curves

I'm confused by what appears to be contradictory information. In this post, the claim is made that "Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^{2}= x^{3}+ax+b$ where ...
3
votes
0answers
39 views

Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
2
votes
1answer
45 views

$L$-functions of elliptic curves over $\mathbb{Q}$

How to find out the $P_{v}(E/\mathbb{Q},X)$ theoretically given below in the definition of $L$-functions for elliptic curves over $\mathbb{Q}$ $?$ Please cite some references for the same. For an ...
1
vote
2answers
64 views

From $y^2=x^3+Ax^2+Bx$ to $y^2+(1-c)xy-by=x^3-bx^2$

I have two question How can I transfer with a change of coordinates from $$y^2=x^3+Ax^2+Bx$$ to $$y^2+(1-c)xy-by=x^3-bx^2?$$ In a note of Prof. Lozano "Elliptic Curves, Modular Forms and their ...
2
votes
0answers
64 views

On the rank of $y^2=x^3+k^2x$ [closed]

Which value of k leads the curve $y^2=x^3+k^2x$ to have the rank equal to zero? Can we find a family of this such curves with rank zero?
0
votes
1answer
31 views

On the rank of $y^2=x^3+a^2x^2-a^4x$

How can I prove that the rank of $y^2=x^3+a^2x^2-a^4x$ is zero where $a$ is rational and positive?
0
votes
1answer
108 views

Are $10^{10}$-digit-numbers too big for Lenstra's elliptic curve method (ECM)?

I would like to search prime factors of the numbers $$10^{10^{10}}-113$$ and $$10^{10^{10}}+13$$ Both numbers have no prime factor below $10^9$. Are these numbers still too big for ECM ? I also ...
1
vote
1answer
70 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
0
votes
3answers
180 views

$ax^3+by^3+cz^3=0$ and Elliptic curves

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
25
votes
0answers
265 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
2
votes
1answer
54 views

Projective coordinates for elliptic curves

If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)? I'm ...
2
votes
1answer
34 views

Uniqueness of points in Elliptic Curve addition

When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!
0
votes
0answers
46 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
3
votes
1answer
86 views

The group of $\mathbb{K}$--rational points for isomorphic elliptic curves

The Springer text by Tom Apostol on Dirichlet series and modular forms, which I have, defines modular functions and modular forms on page 34 and on page 114 respectively, not to mention the Springer ...
0
votes
1answer
89 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
0
votes
1answer
63 views

Order of subgroup on elliptic curve over $Z_p$

I should determine the order of subgroup on elliptic curve over $\mathbb{Z}_p$ where $p$ is prime, and point $X$ is generator of some subgroup. While generating the subgroup by points addition I found ...
6
votes
0answers
93 views

Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be ...
2
votes
1answer
72 views

Number of points on $Y^2 = X^3 + A$ over $\mathbb{F}_p$

Let $p\equiv 2\pmod{3}$ be prime and let $A\in\mathbb{F}^{∗}_p$ . Show that the number of points (including the point at infinity) on the curve $Y^2 = X^ 3 + A$ over $\mathbb{F}_ p$ is exactly $p + 1$ ...
7
votes
1answer
124 views

Challenge from Fermat

Fermat challenged Frenicle with finding a pythagorean triple (a,b,c) where $(a-b)^2-2b^2$ is itself a square. By making the substitution $a=m^2-n^2$, $b=2mn$, and $c=m^2+n^2$ into $(a-b)^2-2b^2=d^2$ ...
10
votes
0answers
211 views

Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
1
vote
2answers
85 views

How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
9
votes
1answer
641 views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
3
votes
0answers
161 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
5
votes
1answer
211 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
9
votes
2answers
192 views

Making an elliptic curve out of a cubic polynomial made a cube, or $ax^3+bx^2+cx+d = y^3$

What is the transformation such that a general cubic polynomial to be made a cube, $$ax^3+bx^2+cx+d = y^3\tag{1}$$ can be transformed to Weierstrass form, $$x^3+Ax+B = t^2\tag{2}$$ (The special ...
0
votes
0answers
99 views

number points and twist of elliptic curve

Let elliptic curve $y^2 = x^3 + 1$ has ($q^2 + 2q + 1$) $\mathbb{F}_{q^2}$-rational points. Let $\zeta$ is generator of $\mathbb{F}_{q^2}^*/\mathbb{F}_{q^2}^{*6}$. Curve $E'$ with equation $y^2 = x^3 ...
1
vote
0answers
103 views

Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
6
votes
1answer
130 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
4
votes
2answers
65 views

Other names for $E_{p+1}$ $\pmod{p}$?

If I want to know properties of $E_{p+1}$ modulo $p$, do you know a name for this modular form, so that it is easier to search via the internet? So far, what I know is that $E_{p-1}$ is the Hasse ...
0
votes
0answers
123 views

Analogy between Picard group and Ideal class group

Can you give a reference where the conformity between Picard group and Ideal class group is explained? What is analogy of Picard group of elliptic curve over finite field?
3
votes
1answer
78 views

How do infinite series contain “local” information?

I would like to know why we consider infinite series (Dirichlet series, zeta function, elliptic curve $L$-series) or their Euler product. How is the local information "stored/contained" in the ...
5
votes
2answers
190 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
2
votes
1answer
86 views

Elliptic curve, number of elements of finite order

I've been looking at this problem for some time now and just can't seem to get the right idea. The problem is: Consider the elliptic curve $C:y^2=x^3+bx$ defined over the rational numbers with $b$ a ...
21
votes
3answers
604 views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies ...