All Questions
663 questions with no upvoted or accepted answers
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
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
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 ...
- 6,006
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
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
2
votes
0
answers
264
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 ...
- 6,006
2
votes
0
answers
60
views
Local property of residual representations attached to elliptic curves over rational numbers
I found the following claim - without reference - in the (famous) book ''Modular Forms and Fermat’s Last Theorem'':
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. Let $\Delta_E$ be the ...
- 1,428
2
votes
0
answers
270
views
Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'
Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group.
Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence.
...
- 1,541
2
votes
0
answers
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
2
votes
0
answers
136
views
Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
- 913
2
votes
0
answers
65
views
Average rank of elliptic curves over one-parameter family
Let $E_t:y^2=x^3+f(t)x+g(t)$ be an one parameter family of elliptic curves with $f,g\in \mathbb{Z}[t]$. I found one Silverman's result https://www.degruyter.com/document/doi/10.1515/crll.1998.109/pdf ...
- 521
2
votes
0
answers
124
views
On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k
2
votes
0
answers
105
views
Torsion of an elliptic curve injects under reduction - question
Let $E/K$ be an elliptic curve over a number field. I am interested in the folowing statement: the map $E(K)[m]\rightarrow \tilde E_v(\tilde k_v)$ is injective for any place of $K$ provided there is ...
- 89
2
votes
0
answers
179
views
Is the Weil restriction of an elliptic curve self-dual?
$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$
be a prime split in $K$. Assume that
$$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
- 390
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
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 $T_p(E)$,
$$
\rho_E: G_{K} \rightarrow \mathrm{...
- 639
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
2
votes
0
answers
116
views
Pollard's rho algorithm for ECDLP using supersingular elliptic curves over a field with characteristic equal to a Mersenne prime
I have been playing with Pollard's rho algorithm for elliptic curves over finite fields. I have noticed after some experimenting, that the algorithm almost always 'fails' for supersingular elliptic ...
- 121
2
votes
0
answers
134
views
Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
- 21
2
votes
0
answers
78
views
Question on a certain reduced isogeny of CM elliptic curves
My question has to do with some hypotheses showing up in a Lemma of Joseph Silverman's Advanced Topics book. Here is some of the set up:
Let $K$ be an imaginary quadratic field and $E/H$ an elliptic ...
- 707
2
votes
0
answers
153
views
Order $4$ element of Tate-Shafarevich group
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows.
$$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
- 1,541
2
votes
0
answers
142
views
$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group
This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one.
To calculate the Selmer group of given elliptic curve, we ...
- 1,541
2
votes
0
answers
241
views
Inertia group representation from $p^{n}$-torsion of ordinary elliptic curve
Let $K$ be a complete local field. Suppose that $K$ is an unramified extension of $\mathbb{Q}_{p}$ and let $E$ be an elliptic curve over $K$ with good ordinary reduction. Let $G_{K}=\text{Gal}(\...
- 305
2
votes
0
answers
137
views
Tangential basepoint of a log singular local system
Consider the Legendre family $f: X\longrightarrow Y = \mathbb{P}^1\setminus\{0,1,\infty\}$, defined over $K = \mathbb{Q}/\mathbb{Q}_p$.
having fibre above $t\in Y$, the elliptic curve $E_t := y^2 = x(...
- 2,907
2
votes
0
answers
254
views
When is a prime considered to be ramified, split or inert in a non-maximal order of an imaginary quadratic number field?
I am reading this paper on "Averages of Elliptic curve constants" here and in section 2.2 page no. 693 the formula for the conjectural constant in the asymptotics of the Lang-Trotter ...
- 309
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
2
votes
0
answers
119
views
Finding a Hodge theoretic condition to measure the rank of isogeny of product abelian surfaces
Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$
to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain
...
- 91
2
votes
0
answers
132
views
How to compute torsion subgroup $E[24]$ over $\overline{\mathbb{Q}}$
If I have an elliptic curve $E: y^2=x^3-15x+22$ over $\mathbb{Q}$ with CM from the imaginary quadratic field $\mathbb{Q}(\sqrt{-3})$ then how do I compute the $24$-torsion subgroup $E[24]$ over $\...
- 309
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
2
votes
0
answers
142
views
Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields
Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...
- 565
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
2
votes
0
answers
145
views
Lattice relations and isogenous elliptic curves
Consider two (primitive) elements $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...
- 71
2
votes
0
answers
103
views
Selmer ranks unbounded?
Is it known if the Selmer ranks of quadratic twist families are unbounded?
Suppose that $E/K$ is an elliptic curve defined over a number field. For each quadratic extension $F/K$ I can form the twist $...
- 3,730
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 333
2
votes
0
answers
190
views
Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to prove
$\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$.
$ \hat{E}[p]$ denotes $p$ ...
- 1,541
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 12.9k
2
votes
0
answers
161
views
Embedding of a genus 1 hyperbolic curve
Let $E$ be an elliptic curve over a number field $k$. We define the affine curve $C := E \backslash \{p_1,...,p_n\}$ by removing a finite number of points from $E$. Here, I would like to declare that ...
- 895
2
votes
0
answers
97
views
Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
2
votes
0
answers
191
views
Picard and Rosati for elliptic curves
I would like to ask for confirmation whether the following argument is correct.
We work over an algebraically closed field $k$ of characteristic $0$. For an elliptic curve $E$, the Picard variety, or ...
- 533
2
votes
0
answers
127
views
Semi-stable elliptic curves and Szpiro ratios
This is a continuation of the following question: Szpiro ratios of elliptic curves over $\mathbb{Q}$
In that question I asked whether Szpiro ratio
$$\displaystyle \beta_E = \frac{\log |\Delta_{\min}(E)...
- 27k
2
votes
0
answers
171
views
Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
2
votes
0
answers
467
views
Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
- 1,805
2
votes
0
answers
2k
views
What's the best reference for Abelian varieties?
I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves.
I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
- 101
2
votes
0
answers
287
views
Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 183
2
votes
0
answers
117
views
Splitting of prime and order of reduction of point of infinite order in an abelian variety
I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here.
Let $A$ be an abelian variety defined over a number field $K$, $P \...
- 443
2
votes
0
answers
230
views
Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
In Siksek's notes The modular approach to Diophantine equations he uses the following result:
Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
- 131
2
votes
0
answers
85
views
Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?
Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
- 1,833
2
votes
0
answers
187
views
Factoring integers of the form $n=p q^2$ using elliptic curves
We got argument and strong experimental support
that integers of the form $n=p q^2$ can
be factored using elliptic curves easier than general integers
Q1 Is this known?
Added This is known since at ...
- 25.4k