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 ...

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,...

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 $...

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 ...

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 ...

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.
...

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 $...

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 ...

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 ...

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^{...

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_{\...

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 ...

Community wiki

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\...

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 ...

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_{\...

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 ...

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