All Questions
1,978 questions
2
votes
1
answer
222
views
Construction refuting the existence of nonisotrivial elliptic curve over $\mathbb{G}_m$
I have some troubles to understand the construction in detail presented here by Daniel Litt used to show that there cannot exist an elliptic curve over $\mathbb{G}_m/k$, $k$ of characteristic $p >3$...
- 5,986
1
vote
0
answers
102
views
Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist.
It is widely known that for all $E/\Bbb{Q}$: elliptic ...
- 1,541
0
votes
1
answer
353
views
Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$
$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$.
Let $\sigma$ be a generator of $\Gal(L/K)$.
Let $E/K$ be an elliptic curve defined ...
- 1,541
2
votes
0
answers
107
views
elliptic curves on general 3-folds of degree 7
Do there exist elliptic curves on a general 3-fold hypersurface $X_7 \subset \mathbb{P}^4$ of degree $7$?
Clemens proved that for $d \ge 8$ there are no elliptic curves on the general hypersurface $...
- 3,730
10
votes
1
answer
444
views
How fast can elliptic curve rank grow in towers of number fields?
Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
- 2,665
7
votes
1
answer
630
views
Cubic twist of elliptic curves and its rank
Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$).
Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$.
$E$ and $E_D$ are isomorphic over $\...
- 1,541
1
vote
0
answers
133
views
Select random point on elliptic curve
If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
- 11
4
votes
0
answers
112
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 5,230
2
votes
1
answer
344
views
On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension
Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$,
$$
\rho_E: G_{K} \rightarrow \mathrm{...
- 639
8
votes
0
answers
178
views
Do there exist Calabi-Yau 3-folds that contain a finite number of elliptic curves?
The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$...
- 3,730
0
votes
0
answers
235
views
Reference book on the relation between modular forms and elliptic curves
What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
- 43
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
5
votes
2
answers
543
views
When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
- 89
2
votes
2
answers
415
views
Upper bound on number of integral solutions of elliptic curves
I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating ...
- 61
6
votes
2
answers
1k
views
Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
- 171
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
6
votes
0
answers
166
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
user516086
0
votes
1
answer
272
views
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
- 1,833
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 23
3
votes
0
answers
171
views
Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants
$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
- 1,541
8
votes
1
answer
315
views
Is there an elliptic curve over a number field with a point of order 64 and Mordell-Weil rank zero?
It seems to me that there ought to be elliptic curves over number fields with arbitrarily large torsion subgroups but Mordell-Weil rank zero. But I'll settle for a point of order 64. Does anyone ...
- 315
1
vote
1
answer
212
views
The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism
Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...
- 1,833
2
votes
1
answer
292
views
One unexpected observation related to algebraic curves and their Jacobians
Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
- 1,833
9
votes
1
answer
426
views
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 $D \in \Bbb{Z}$?
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$.
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/...
- 1,541
2
votes
1
answer
402
views
Irreducibility of Tate module (as a Galois representation) of elliptic curves with good reduction
This question is following the previous question.
Definitions:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote ...
- 785
2
votes
0
answers
116
views
Pollard's rho algorithm for ECDLP using supersingular elliptic curves over a field with characteristic equal to a Mersenne prime
I have been playing with Pollard's rho algorithm for elliptic curves over finite fields. I have noticed after some experimenting, that the algorithm almost always 'fails' for supersingular elliptic ...
- 121
4
votes
0
answers
276
views
Special case of Eichler–Shimura
I'm reading ‘Rational Points on Elliptic Curves’ by Silverman and Tate, and the exercise 4.6 is about the following special case of the Eichler–Shimura theorem. Let $C$ be the elliptic curve given by ...
- 41
8
votes
1
answer
326
views
3-divisibility of Manin constant for elliptic curves with 3-torsion
Let $E/\mathbb{Q}$ be an elliptic curve with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$ (not necessarily $\Gamma_0$-optimal). Does $3$ necessarily divide one of: the Manin constant (not necessarily $...
- 91
3
votes
0
answers
330
views
Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,183
2
votes
0
answers
134
views
Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
- 21
13
votes
2
answers
1k
views
On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 12.6k
1
vote
0
answers
90
views
Isogenous elliptic curves in characteristic zero and in characteristic $p$
Assume two elliptic curves (with CM), $E_{1}$ and $E_{2}$, are isogenous over a field $K$ of characteristic zero. Are the following two statements true?
(a) Their $V_{p}$ modules are $G_{K}$-...
- 71
3
votes
0
answers
177
views
For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
- 5,422
1
vote
0
answers
120
views
Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$
Let $p$ be a prime number and $n$ be positive integer.
Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.
LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$.
This is the biggest size of $Sha(...
- 1,541
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
2
votes
0
answers
78
views
Question on a certain reduced isogeny of CM elliptic curves
My question has to do with some hypotheses showing up in a Lemma of Joseph Silverman's Advanced Topics book. Here is some of the set up:
Let $K$ be an imaginary quadratic field and $E/H$ an elliptic ...
- 707
0
votes
0
answers
179
views
Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
- 193
3
votes
0
answers
227
views
Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
- 371
0
votes
1
answer
289
views
Elliptic curve points with the same $y$-coordinate
Let $nP = (x,y)$ be a point on elliptic curve $E: x^3+ax+b=y^2$ modulo a prime $p$. If I want to create all points with the same $x$ coordinate the process is simple, $nP, -nP$ is the full list.
If I ...
- 591
18
votes
2
answers
2k
views
What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 7,182
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
0
votes
0
answers
281
views
Néron–Severi group of Abelian surfaces
Suppose that an Abelian surface $A$ is isogenous to the product of two elliptic curves $E \times E'$. When can we say that the Néron–Severi group is generated by the classes of these two elliptic ...
- 91
1
vote
0
answers
59
views
Density of integer representations of a a two variable polynomial
I was testing some data on the positive integer representations of $f(x, y) = x^3 - y^2$ such that $(x, y) \in [1, M] \times [1, M]$. I tested it for $M = 1000, 10000$. It turns out that the density ...
- 577
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
1
vote
1
answer
276
views
Family parametrizing elliptic curves with a rational 5-isogeny
I am looking for a reference for the following. I believe the set of elliptic curves $E/\mathbb{Q}$ admitting a rational 5-isogeny can be parametrized as
$$E: y^2 = x^3 + f(t)x + g(t), t \in \mathbb{Q}...
- 27k
8
votes
2
answers
424
views
$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
Let $E/\mathbb{Q} = E_{a,b}$
$$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$
be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K_n$ be ...
- 27k
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
4
votes
1
answer
183
views
Primes of bad reductions for quotients of elliptic curves
Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of ...
- 1,428
3
votes
0
answers
170
views
Explicit relationship between Gross--Zagier's On Singular Moduli, and Heegner Points and Derivatives of L-series
In various places in the literature surrounding the Gross--Zagier formula, the results in Heegner points and the derivatives of $L$-series (hereafter, Heegner points) are referred to as a ...
- 185
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630