An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
Questions tagged [elliptic-curves]
1505 questions
15
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6 answers
bad reduction for elliptic curves
Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?
flor.ian sprung
- 922
14
votes
4 answers
Supersingular elliptic curves
I've read that an elliptic curve is supersingular if and only if its endomorphism ring is an order in a quaternion algebra. Does anyone have a simple explanation of this (or a good reference)?
Jonathan Wise
- 7,754
12
votes
4 answers
addition-theorem polynomials
Suppose a function f(u) identically satisfies an equation of the form G{f(u+v),f(u),f(v)}=0 for all u and v and u+v in its domain. Here G(Z,X,Y) is a non vanishing polynomial in the three variables with constant coefficients. Then one says that f…
Mark B Villarino
- 201
11
votes
0 answers
What are the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)$ is an integer $>2$?
The problem of finding the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)=4$ turns out to be surprisingly difficult, and has made the rounds on the internet and social media, and Andrew Bremner and Allan Macleod have written a paper on…
Christopher D. Long
- 690
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10
votes
4 answers
Specific Elliptic Curves: Rank
Here's a challenge for elliptic curve descenders/programmers. It seems no public software or public tables can determine if the rank is zero for the following curves (over rational x,y):
y^2 = x^3 - 9122*x + 106889
y^2 = x^3 - x^2 - 42144*x +…
bobuhito
- 1,537
10
votes
2 answers
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 elliptic curve over $\mathbb{Q}$
having each of the…
Allan MacLeod
- 101
10
votes
3 answers
Division by 3 on elliptic curve
There is a classical theorem (cf. Theorem 4.1 on Husemoller "Elliptic Curve" book) that states conditions on the coordinates of a point P in an elliptic curve to be twice another point. The result is the following:
Theorem: Let $E$ be an elliptic…
9
votes
1 answer
Neron models of elliptic curves with level N structure?
In the Deligne-Rapoport paper entitled "Les schemas de modules de courbes elliptiques" the following is written (I translated in english):
Let $E$ be an elliptic curve with $\Gamma(N)$-level structure defined over
$\mathbb{C}((T))$. Let $E'$ be the…
Hugo Chapdelaine
- 7,521
9
votes
2 answers
Is it possible for the repeated doubling of a non torsion point of an elliptic curve stays bounded in the affine plane?
Let $P=(x_1,y_1)$ be a non torsion point on an elliptic curve $y^2=x^3+Ax+B$.
Let $(x_n,y_n)=P^{2^n}. x_n,y_n$ are rationals with heights growing rapidly. Can ${x_n} {y_n}$ stay bounded?
defgh
- 111
9
votes
3 answers
Is this naive test to tell whether a complex elliptic curve has complex multiplication effective?
I have a question about a naive test to tell whether a complex elliptic curve $E$ has complex multiplication.
Recall that the endomorphism ring $End(E)$ of $E$ is isomorphic to either $\mathbb{Z}$ or an order in an imaginary quadratic field $K$. In…
user1073
9
votes
6 answers
Two questions on isomorphic elliptic curves
Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.
But $E_1$ and $E_2$ could also be isomorphic over a subfield of…
wood
- 2,714
7
votes
0 answers
On discriminants of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ division fields $\mathbb{Q}(E[m_1]) \, \cap \,…
Jackson Morrow
- 933
6
votes
1 answer
How to prove there are exactly $8$ integer points on the elliptic curve $y^2 = x^3 + 17$
Consider the elliptic curve $y^2 = x^3 + 17$. I know that there are exactly $8$ integer points $(x,y)$ with $y>0$. But how do I prove it? Is there any specific approach to it or any proof for it?
Cody K.
- 69
- 1
6
votes
2 answers
j-invariant of a supersingular elliptic curve
Let E be a supersingular curve over a finite field. Why is the j-invariant always in F_p^2?
Josh
- 71
6
votes
3 answers
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 point, then $C$ can be given the structure of an…
352506
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