All Questions
1,978 questions
3
votes
1
answer
366
views
Descent for étale covers of proper regular models of elliptic curves
Let $K$ be a complete (but think Henselian suffice for purposes of this question) local field of characteristic $0$ with residue field $k$ of characteristic $p>0$ and ring of integers $R=\mathcal{O}...
- 5,986
5
votes
0
answers
172
views
"Genus theory" for elliptic curve $L$-functions
Let $d$ be a positive integer, so that $-d$ is a fundamental discriminant. This means that $d$ is square-free at odd primes and $\nu_2(d) \in \{0, 2, 3\}$. Further, if $\nu_2(d) > 0$ then $-d 2^{-\...
- 27k
1
vote
0
answers
85
views
Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
0
votes
0
answers
108
views
looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 195
2
votes
0
answers
96
views
On the root numbers of quadruples of quadratic twists of elliptic curves
We got strong numerical evidence for the root numbers and analytic ranks
of quadruples of elliptic curves over the rationals.
Related to this question.
Let $k,k_1,k_2$ be squarefree pairwise coprime ...
- 25.4k
2
votes
1
answer
195
views
Can we get a homomorphism between two elliptic curves if we know a homomorphism between the respective formal groups?
Let $E_1$ and $E_2$ be two elliptic curves over a field of characteristic $0$, and let $\hat{E_1}$ and $\hat{E_2}$ be two formal group laws associated with $E_1$ and $E_2$, respectively. It is known ...
- 195
7
votes
3
answers
349
views
The rank of elliptic curves and related quadratic twists
Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves
$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}...
- 27k
9
votes
3
answers
584
views
Do rational points on $X(1)$ correspond to elliptic curves up to rational isomorphism?
I'm not sure whether this is obvious or not. The curve $X(1)$ parametrices all elliptic curves up to isomorphism class over $\mathbb{
C}$, and a $K$-rational point corresponds to an elliptic curve ...
- 195
3
votes
0
answers
241
views
Generating algebraic points on elliptic curves
Let $E$ be an elliptic curve over $\mathbf{Q}$. One has the modular parameterisation
\begin{align*}
\mathbf{H} \to X_0(N)(\mathbf{C}) \to E(\mathbf{C})
\end{align*}
where $X_0(N)$ is the modular curve ...
- 265
3
votes
1
answer
103
views
special $g_3^1$ on elliptic curve
Let $C$ be a planar smooth cubic curve. We know that projection from a point $p$ that does not lie on $C$ produces a 3-to-1 map $\pi: C\to \mathbb P^1$, which has 6 ramification points in general.
Now,...
- 1,803
7
votes
0
answers
170
views
Nondeterminism in Magma software while computing generators of an elliptic curve
Computing generators of a Mordell curve
$$y^2 = x^3 - 44275089430000,$$
can be done in Magma by running the following code:
...
- 34.4k
1
vote
1
answer
143
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
0
votes
0
answers
49
views
Is there a correlation between the bifurcation points of dynamical system and the integral points of the elliptic curve $E_d$?
Motivated by an interest in the interplay between dynamical systems and elliptic curves also On a question of Mordell, I derived a dynamical system corresponding to the elliptic curve:
$
E_d: Y^2 = X^...
18
votes
2
answers
888
views
Serre’s comment on Hurwitz: connecting FLT to points of finite order on elliptic curves
In the paper Sur les représentations modulaire de degré 2 de Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$, Serre makes the following comment:
Remarque. La relation existant entre "solutions de l'...
- 6,734
0
votes
0
answers
81
views
Computing elliptic periods from modular form
How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
- 2,907
8
votes
0
answers
208
views
Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
0
votes
0
answers
100
views
Algebraic degrees of $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + \sqrt{-n}}{2}\right)$ and class numbers of $Q(\sqrt{-n})$
Let $n\in\mathbb N$ be squarefree. Denote by $h(n)$ the class number of $Q(\sqrt{-n})$ and by $d_1(n)$ and $d_2(n)$ the degrees of the algebraic numbers $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + ...
- 13.4k
2
votes
1
answer
345
views
Why this genus one curve over $\mathbb{F}_5$ appear to violate Hasse-Weil bound?
Working over $\mathbb{F}_5$, the affine curve $x^4+2=y^2$ has no points.
The projective curve $x^4+2y^4=z^2y^2$ has only one point $(0 : 0 : 1)$.
Both curves appear to violate Hasse-Weil bound of $4....
- 25.4k
0
votes
0
answers
108
views
Isogeny classes for elliptic curves over quadratic field
Question. Is it possible for an elliptic curve $E$ over quadratic field $K$ to have two separate (yet connected) isogeny classes?
There are two $\mathbb{Z}/14\mathbb{Z}$ elliptic curves, $E_1$ and $...
- 913
3
votes
1
answer
222
views
Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
- 25.4k
5
votes
1
answer
203
views
Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
- 81
2
votes
0
answers
52
views
Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
- 25.4k
1
vote
1
answer
235
views
Are there conditions for an elliptic curve to have a quadratic $\mathbb{F}_q$-cover of the line without ramification $\mathbb{F}_q$-points?
Consider an elliptic curve $E: y^2 = f(x) := x^3 + ax + b$ over a finite field $\mathbb{F}_q$ of characteristic $> 3$. Obviously, the projection to $x$ is a quadratic $\mathbb{F}_q$-cover of the ...
- 1,833
2
votes
0
answers
92
views
Tate curve and components of special fibre II
Let $K$ be complete local field of characteristic $0$ with ring of integers $R=\mathcal{O}_{K}$ and residue field $k=R/\mathfrak{m}_R$ of characteristic $p>0$. Let $E/K$ be an elliptic curve of ...
- 5,986
2
votes
1
answer
161
views
Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?
In this and this question we show that if $p=27a^2+27a+7$ is prime, then the order of the elliptic curve
$y^2=x^3+2$ modulo $p$ is either $p$ or $p+2$.
Q1 Can we unconditionally show that the order ...
- 25.4k
2
votes
0
answers
103
views
Existence of supersingular abelian variety over $\mathbb F_p$ with $\mathcal O_E$-action where $E/\mathbb Q$ is quadratic imaginary ramified at $p$
Let $E/\mathbb Q$ be a quadratic totally imaginary number field with ring of integers $\mathcal O_E$. Let $p > 2$ be an odd prime number which ramifies in $\mathcal O_E$. Let $\mathcal P$ be the ...
- 769
1
vote
0
answers
153
views
Tate curve and components of special fibre
Let $K$ be a complete local field of characteristic $0$ with ring of integers $R=\mathcal{O}_{K}$ and residue field $k=R/\mathfrak{m}_R$ of characteristic $p>0$. Let $E/K$ be an elliptic curve of ...
- 5,986
3
votes
1
answer
350
views
Elements of $\mathbb{F}_p$ represented by an irreducible polynomial $f(x) = x^3 +a_2 x^2 + a_1 x + a_0$, $f(x) \in \mathbb{F}_p[x]$
Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of ...
- 31
0
votes
0
answers
61
views
Is generating semirandom blake256 hashes until packed points is on the curve, a safe algorithm to avoid the discrete log between the generated points?
I know there are more robust methods, but I wanted to know about this specific one
For any distinct said randomly generated point : $P_i,P_j\in \{P_1,...,P_k\}$ it should be hard to find $s$ such that ...
- 87
15
votes
1
answer
413
views
Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
- 267
8
votes
0
answers
333
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 831
8
votes
1
answer
365
views
Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
- 83
0
votes
0
answers
80
views
Elliptic curve over global function field: poles of $j$-function & ramification of torsion fields [duplicate]
Let $E/ \Bbb C(t)$ be an elliptic curve over $ \Bbb C(t)$ with nonconstant $j$-invariant $j_E \in \Bbb C(t)-\Bbb C$ and $p>2$ some prime such that it is bigger than an order of a pole $v$ of $j_E$. ...
- 5,986
0
votes
0
answers
197
views
On the integer solutions of the equation $y^2 = x^3 + n$
Let $n$ be a nonzero integer. I am interested in the integer solutions $(x, y)$ to the equation $y^2 = x^3 + n$.
Let $S$ be the set of all integer solutions $(x, y)$ to this equation.
I am wondering ...
- 95
1
vote
0
answers
115
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
1
vote
1
answer
591
views
Argument for non-existence of elliptic curve over $\mathbb{C}[t, t^{-1}]$
I have a couple of questions about this answer by Noam D. Elkies showing that there exist no elliptic curve $E$ over $\mathbb{C}[t, t^{-1}]$ having nonconstant $j_E$-invariant.
The strategy is to ...
- 5,986
1
vote
0
answers
157
views
Does this subset of elliptic curves over $\mathbb{Q}$ have positive proportion?
Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times ...
- 61
2
votes
0
answers
234
views
Elliptic curve with rank at least $6$
I was going through a research paper which proves the existence of an infinite family of rank $6$ elliptic curves over $\mathbb{Q}$ with invariant equal to $0$.
Let $k$ be a field of characteristic ...
- 127
4
votes
0
answers
166
views
Semistability of Frey curves: why no additive reduction?
Let $(a,b,c)$ be a hypothetical nontrivial integer solution to the Fermat equation $x^p + y^p + z^p = 0$, where $p \geq 5$ is prime, and assume $a, b, c$ are (pairwise) coprime. From this solution, we ...
- 283
2
votes
0
answers
263
views
Abelian extensions of number fields generated by torsion points of elliptic curve (as analogy to Lubin-Tate theory)
According to a remark from wikipedia the motivation of Lubin-Tate theory arose from the analogy to the way in which elliptic curves $E/K$ over a number field $K$ with extra endomorphisms (ie those ...
- 5,986
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
5
votes
2
answers
319
views
Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
First I shall begin by laying out some notation (I shall be using the conventions that are used by both DLMF and Mathematica which occasionally differ from the standard literature):
Let $\Lambda:=\...
- 533
5
votes
1
answer
253
views
Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$
Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$.
Here is my ...
- 622
4
votes
1
answer
525
views
Normalisation of models of elliptic curves in finite extensions and reducedness of fibres
Let $K$ be a complete local field of characteristic 0 with residue field $k$ of characteristic $p>0$ and ring of integers $R=\mathcal{O}_{K}$. Suppose we have an isogeny of elliptic curves:
$$f:F\...
- 305
4
votes
1
answer
458
views
Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$
Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$.
My question is, Does there exist a finite set $S\subset M_K$ such that
$\forall C$: $E/K$-torsor, $\...
- 1,541
3
votes
1
answer
187
views
Difficulties in the proof of finiteness of n-Selmer group using cohomology
I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.
1st ...
- 127
4
votes
1
answer
215
views
Atkin-Lehner involution on the modular abelian varieties
Let $f$ be a CM modular forms with coefficients in the imaginary qudaratic field $K=Q(\sqrt{-3}))$ which correspond to an elliptic curve $E$ defined over $K$. Then Shimura constructed an abelian ...
- 390
7
votes
1
answer
514
views
Field of definition of elliptic curves
Let $a,b$ be positive integers, $F=\mathbb{Q}(a^{1/3},b^{1/2})$. Let $E$ be the elliptic curve defined over $F$ by the cubic equation $$y^2=x^3+3a^{1/3}x+2b^{1/2}.$$
Then the $j$-invariant $j(E) = \...
- 622
0
votes
1
answer
208
views
Finiteness of Selmer group
I was reading the proof of finiteness of $S^n(E/\mathbb{Q})$ but I am unable to understand from the following lemma how it follows that $S^n(E/L)$ finite.
LEMMA 3.13 For any finte subset $T$ of $\...
- 127