15
votes
Background for the Elkies-Klagsbrun curve of rank 29
With regard to question 3:
In earlier work, specifically the paper "New rank records for elliptic curves having rational torsion", the same authors, Elkies and Klagsbrun, attained new ...
- 143k
10
votes
Accepted
Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
One more or less systematic method which you might use roughly goes as follows. Let $X_1\cong\mathbb{P}^1$ denote the $j$-line.
Describe the branched covering $f : X_G\rightarrow X_1$ combinatorially,...
- 10.4k
9
votes
Accepted
Equations for dual cubic curves
This is the discriminant of $x^3 + ax + b - (ux+v)^2$ in the variable $x$ as a function of $u,v$, since the line $y=ux+v$ is tangent to $C$ if and only if the cubic polynomial you get by substituting $...
- 143k
7
votes
Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
Another approach is to directly access differentials on $X_{G}$ via modular forms. David Zywina has implemented this (see Sections 4 and 5 of his preprint here). Roughly speaking and when the genus of ...
- 20.1k
4
votes
Accepted
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$
I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$.
In full generality, I believe you are right that none of the properties you state are known for all elliptic ...
- 10.7k
4
votes
What does the Serre functor of equivariant category of fractional CY category look like?
I don't think you can understand $\kappa$ abstractly. In some sense, the computation of $\kappa$ is a more precise version (taking into account the group action) of computation of the Serre functor.
...
- 38.4k
4
votes
The rank of elliptic curves and related quadratic twists
You are looking at the biquadratic extension $K:=\mathbb Q(\sqrt{k_1},\sqrt{k_2})$ of $\mathbb Q$ and want to relate properties of $E(F)$ where $F$ is $\mathbb Q$ or one of the quadratic subfields $...
- 47k
4
votes
Rank and generators of elliptic curve
Further update: gp now has a command ellrank that finds
the rank quickly:
allocatemem(2^26)
E = ellinit([0,1,0,-15662264585,746984342506759]);
ellrank(E)
takes ...
- 79.1k
3
votes
On Mordell equation $y^2=x^3+k$
It is not clear what you mean by "completely solved". There are algorithms whose input is $k$ and output is the set of integer solutions. See for example Mordell's equation: a classical ...
- 103k
3
votes
The rank of elliptic curves and related quadratic twists
Let $E, E^{(k)}$ be defined over a number field $K$ and let $k \in K^\times \setminus K^2$. Then it is well-known that
$$\operatorname{rk}(E/K(\sqrt d)) = \operatorname{rk}(E/K) + \operatorname{rk}(E^{...
- 363
2
votes
The rank of elliptic curves and related quadratic twists
Here is some experimental data.
For positive integer $k$ let $E_k: y^2=x^3+k x $ and $k_1=2,k_2=3$.
According to computations with sage, for $0 < k < 2000$:
At least one of $\displaystyle r_{\...
- 25.2k
2
votes
Accepted
Flexes and projective equivalence of smooth cubics
Let me unravel the kind suggestion by Sasha in the comments section.
The point $F=[0:1:0]$ is a flex of the cubic $Y^2Z=X(X-Z)(X-\lambda Z)$.
Let us call $t_0$ the tangent line through $F$, which is ...
2
votes
Background for the Elkies-Klagsbrun curve of rank 29
The announcement on the NMBRTHRY listserv seems by Noam Elkies seems worth reproducing here.
The elliptic curve
$$E29 :y^2 + xy = x^3 - 27006183241630922218434652145297453784768054621836357954737385 ...
2
votes
Background for the Elkies-Klagsbrun curve of rank 29
Not sure if this is surprising, but the discriminant factors as $-2^{19}\cdot 3^7\cdot 5^7\cdot 7^4\cdot 11^5\cdot 13^3\cdot 17^4\cdot 31^3\cdot 41^2\cdot 43^2\cdot 61^2\cdot 233\cdot 241^2\cdot 4139\...
- 774
1
vote
Accepted
A correspondence between pairs of isogenies and representation numbers
This follows by unwinding the definitions of the quantities involved and using that $\mathrm{deg}$ is a positive definite quadratic form on isogenies.
We can take $R = \mathbb{Z}$, and the quadratic ...
- 46
1
vote
Accepted
Descent for étale covers of proper regular models of elliptic curves
Okay let me try my hand at an answer. Specifically, I'll address the final question, which is whether there exists a unique descent of a geometrically connected finite etale connected cover $D_{\...
- 10.4k
1
vote
Can we get a homomorphism between two elliptic curves if we know a homomorphism between the respective formal groups?
This is a temporary answer, please let me know if any argument is incorrect:
SI am answering the third question only. With the assumption $\hat{E_1} =\hat{E_2}$, my answer is yes, provided the other ...
- 83
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