For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
5
votes
2answers
962 views
Local-Global Principle and the Cassels statement.
In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that
There is not merely a local-global principle for curves of genus-$0$, but
...
13
votes
2answers
1k views
Elliptic Curves and Points at Infinity
My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ...
3
votes
1answer
496 views
Intuition and Stumbling blocks in proving the finiteness of WC group
After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle,
and coming to its ...
2
votes
3answers
901 views
Group Law for an Elliptic curve
I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages.
The ...
3
votes
1answer
51 views
How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$
Let $C$ be an elliptic curve over $ℚ$. The group
$C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
1
vote
1answer
67 views
Violating assertion in Cohen's instructions for Weierstrass reduction
I am trying to follow case 2 of the procedure given in Cohen:
for the cubic $f(x,y,z) = x^3 + 3 y^3 - 11 z^3$ using the rational point $P_0 = (2 : 1 : 1)$. The tangent at this point is $y = - ...
1
vote
1answer
191 views
Modular functions and elliptic functions
Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions?
Any advice much appreciated.
...
0
votes
0answers
48 views
Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?
The motivation to this question can be found in
How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$
We know that there is an isomorphism ...
0
votes
0answers
39 views
Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?
Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ...
34
votes
3answers
395 views
The resemblance between Mordell's theorem and Dirichlet's unit theorem
The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group.
If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
6
votes
1answer
1k views
Reading the mind of Prof. John Coates (motive behind his statement)
To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as
...
4
votes
3answers
450 views
Integral points on an elliptic curve
Let's start with an elliptic curve in the form
$$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$
I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
3
votes
3answers
136 views
Reference: Elliptic curves as complex tori
I'm looking for books which contain a more or less self-contained description of how elliptic curves over $\mathbb{C}$ - that is, nonsingular plane cubic curves - can be realized as a quotient of the ...
5
votes
1answer
310 views
How are the Tate-Shafarevich group and class group supposed to be cognates?
How can one consider the Tate-Shafarevich group and class group of a field to be analogues?
I have heard many authors and even many expository papers saying so, class group as far as I know is ...
1
vote
2answers
138 views
$N^2=2M^4-2p^2e^4$ has no integer solution
If $\gcd(M,e)=\gcd(N,e)=1$ and $p$ is prime and $p\equiv 5 \mod(16)$ then how I can show that
$N^2=2M^4-2p^2e^4$ has no integer solution.
4
votes
1answer
111 views
UPDATE: How to find the order of elliptic curve over finite field extension
I want to find the order of elliptic curve over the finite field extension $\mathbb{F}_{p^2}$, where $E(\mathbb{F}_{p^2}):y^2=x^3+ax+b $
I am using the method illustrated by John J. McGee in his ...
2
votes
1answer
99 views
writing down the minimal discriminant of an elliptic curve
Let $j$ be an integer.
Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way?
For example, can ...
2
votes
1answer
244 views
How to find all rational points on the elliptic curves like $y^2=x^3-2$
Reading the book by Diophantus, one may be led to consider the curves like:
$y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$,
the first two of which are easy (after calculating some eight curves to be solved ...
2
votes
2answers
1k views
How elliptic arc can be represented by cubic Bézier curve?
If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
2
votes
2answers
218 views
Trying to piece together an integral addition theorem
If we have a curve $C:\{ P(x,y) = 0 \}$ and define $\omega=\frac{\mathrm{d}x}{y}$ then is
$$\int_0^A \omega + \int_0^B \omega = \int_0^{A \oplus B} \omega$$
(with $\oplus$ being addition on a group ...
1
vote
0answers
34 views
Frobenius endomorphism on supersingular elliptic curve
Let we have supersingular curve $E(\bar{\mathbb{F}_q})$.
Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
1
vote
1answer
227 views
Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group
This question is an interesting one,not like my previous one.
Can we judge the size of a Quotient Group by seeing the size of its constituents ?
To add something ,Suppose consider a group ...
0
votes
2answers
303 views
Nontrivial Rational solutions to $y^2=4 x^n + 1$
Are there any nontrivial rational solutions to the following equation:
$$y^2=4 x^n + 1,$$
where $n>2$?
0
votes
1answer
360 views
An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”
This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$
which are generally called as Pell-conics, so all in this question $K$ refers to ...
-9
votes
1answer
628 views
A Hunt for a Mathematical Machine That Gives Points
The central question is :
Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ?
Explanation:
...
