16
votes
What is the taxicab number for rational fourth powers?
It has been known since Euler that the quartic surface defined by
$$\displaystyle x_1^4 + x_2^4 = x_3^4 + x_4^4$$
contains a rational curve defined by
\begin{gather*}
\displaystyle x_1(t) = t^7 + t^5 -...
15
votes
Accepted
Upper bound for Hall's conjecture on separation of squares and cubes
The best $\theta$ is $0$. It is known that there are infinitely many
solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain
"Pell equations"; indeed one such family attains $...
13
votes
Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?
Let $Z$ be the number of zeroes of $y\mapsto f(iy)$ for $0<y<\infty$ and let $r$ be the analytic rank of $E$. Then $Z\geqslant r$ and $Z\equiv r \pmod{2}$ as I will explain below. I don't know ...
12
votes
Accepted
With 6 inverted, is the ring of Weierstrass curves a quotient of the Lazard ring by a regular sequence?
I'll refer to my notes on formal groups at https://strickland1.org/courses/formalgroups/fg.pdf. There are results about the formal group law of an elliptic curve in Section 19. That is written in ...
12
votes
Accepted
Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
Here is a proof using a bit of commutative algebra and algebraic geometry, elaborating on comments by @abx and @ChrisWuthrich. The ring
$$R=\mathbb{C}[x,y]/(x^3+y^3-1)$$
is a Dedekind domain, because ...
11
votes
Accepted
Cubic twist of elliptic curves and its rank
There is a formula but it involves both cubic twists. Let $E: y^2 = x^3+B$ be an elliptic curve over $\mathbb{Q}$ with $j=0$ as the one in the question. Let $D$ be a cubefree integer. Set $E_1: y^2=x^...
10
votes
Accepted
Definition of modular curve associated to $\Gamma(N)$
This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).
Your $S(N)$ is naturally a scheme over $\mathbb{...
10
votes
Accepted
Integral points near elliptic curves
You can take $\theta = 0$, even $\theta = -1/6$ works.
Fix an integer $A \ne 0$ and an integer $B$. If $r$ is an integer, the elliptic curve $E : y^{2} = x^{3} + Ax + r^{2} A^{2}$ has the obvious ...
10
votes
Accepted
Discrepancy in Magma's calculation and Sage's of elliptic curve?
Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the ...
10
votes
Accepted
$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
No. The modular curve $X_E(n)$, which paramaterizes pairs $(E',\iota)$, where $E'$ is another elliptic curve and $\iota \colon E'[n] \to E[n]$ is an isomorphism (of group schemes, or equivalently, of ...
10
votes
Accepted
On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
This is a partial answer.
Using the group law of elliptic curves, we have,
$b_{10}=\frac{\left(n^{102}+133370 n^{100}+235431945 n^{98}-53960558412 n^{96}+\,\dots\,+607383986505 n^{6}+2125016730 n^{4}-...
10
votes
Accepted
When are two elliptic curves with zero j invariant isogenous?
The curves $E_B$ and $E_C$ are isogenous over $\mathbb{Q}$ if and only if $C=u^6B$ or $C=-27u^6B$ for some $u\in\mathbb{Q}^\times$. In other words, up to isomorphism there are exactly two curves ...
10
votes
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e
Adding on a bit to David Speyer's answer, the elliptic curve discrete logarithm problem (ECDLP) has been studied intensively since the 1980s, and in addition to commercial applications, there is a ...
10
votes
Accepted
A real-valued analogue of the Weierstrass $\wp$ Function
Sums of this kind are called Epstein Zeta function.
More generally, they are studied under the name of Lattice Sums. There is the book Lattice sums then and now devoted to this subject. There one can ...
10
votes
Accepted
Counting points on elliptic curves
Corresponding sums of Legendre symbols are know as Jacobsthal sums $$J(u)=\sum_{x\mod p}\left(\frac {x^3+ux}p\right).$$
They are equal to $0$ for prime $p\equiv 3\pmod4$. If $p\equiv 1\pmod4
$, $\...
9
votes
Accepted
Family parametrizing elliptic curves with a rational 5-isogeny
Let $\phi:E\to E'$ be an isogeny of degree $5$, and $j, j'$ be the $j$-invariants of $E$ and $E'$, respectively. Let $r$ be the rational function $$r(z)=\frac{(z^2+10z+5)^3}{z}.$$ Then there is $\tau\...
9
votes
Accepted
Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
For question 1.
The condition $p$ is represented by $(1,9,81)$ or equivalently by $(1,1,61)$ is equivalent to the condition that $p\equiv 1\mod 3$ and $3 \mod p$ is not a cube (this is exercise 9.10 ...
8
votes
Accepted
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded by some constant $M$, for all integers $D \in \Bbb{Z}$?
Perhaps a few remarks too long to fit in a comment will help clarify the issue.
The question of whether there exist elliptic curves of arbitrarily large rank has a surprisingly long and interesting ...
8
votes
Accepted
Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?
The elliptic curve $E = 990.e2$ (Cremona label $990b1$) has rank $0$, and the number of zeros of the associated newform $f$ on the imaginary axis seems to be 2 (numerically). Here is some PARI/GP code ...
8
votes
Accepted
Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves
There is a similar story: One can write $E$ as a $y^2 +a_1(\lambda-1) xy + a_3(\lambda-1) y = x^3+ a_2(\lambda-1)x^2 + a_4(\lambda-1) x + a_6 (\lambda-1)$ where $a_1,a_2,a_3,a_4,a_6$ are power series ...
7
votes
Accepted
Ker of corestriction of Galois cohomology
(Not sure why this question comes up naturally. The more interesting question, and the one analogue to the kernel of restriction, is to ask what is the cokernel of corestriction. That turns up a lot. ...
7
votes
Accepted
Primes of bad reductions for quotients of elliptic curves
As has already been remarked, but more generally, if $K$ is a number field and $A/K$ and $B/K$ are abelian varieties that are isogenous over $K$, then the criterion of Neron-Ogg-Shafarevich has as a ...
7
votes
Accepted
How fast can elliptic curve rank grow in towers of number fields?
I think it is a good idea to compare the growth of the rank to the degree. I would say that we have excessive growth in an extension $F/K$ if $$\DeclareMathOperator{\rank}{rank}\rank E(F) - \rank E(K) ...
7
votes
Accepted
Explicit equations for the universal vector extension of an elliptic curve
(Not a complete answer but too long for a comment)
My guess is that such a description is not known, probably because there isn't an easy one. There is a very nice complex analytic description as $\...
6
votes
Upper bound on number of integral solutions of elliptic curves
First, the bound that you cite has been significantly improved by Alpöge and Ho. Here is Theorem 1.1 therein. Let $A,B\in\mathbb{Z}$ satisfy $\Delta_{A,B}:=-16(4A^3+16B^2)\neq 0$. If $\mathscr{E}_{...
6
votes
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e
This is called the "elliptic curve discrete logarithm problem". It is believed to be hard, and several commercial cryptographic schemes are based on this belief.
6
votes
Accepted
Can an abelian surface be bielliptic
By definition https://en.wikipedia.org/wiki/Hyperelliptic_surface the Albanese morphism of a bielliptic surface has 1-dimensional fibers, while the Albanese morphism of an abelian surface is the ...
5
votes
Accepted
Twist of the Tate Curve
Chris provided a reference, but for those who don't have a copy of the book:
$$
E(K) \cong \bigl\{ u\in L^*/q^{\mathbb Z} : \operatorname{\textsf{Norm}_{L/K}}(u) \in q^{\mathbb Z}/q^{2\mathbb Z} \bigr\...
5
votes
Accepted
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
I don't see any contradiction: the Selmer group also has a contribution of rational points. Indeed, the group of 2-torsion rational points on this elliptic curve is isomorphic to $\mathbb Z/2\mathbb Z$...
5
votes
Accepted
Galois cohomology of Tate modules
Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T_pE$ and $G_S$ for the Galois group of the maximal extension ...
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