Questions tagged [elliptic-curves]
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
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Explicit example of elliptic curve of the kind needed for IUTT
At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
2
votes
1
answer
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How to compute Weber polynomials efficiently?
Given $\tau\in H$ (up-half plane) and $q=e^{2\pi i \tau}$, Weber polynomail is defined as
$$f(\tau)=q^{-\frac{1}{48}}\prod_{i=0}^{\infty}(1+q^{i-\frac{1}{2}}).$$
My question is: How can I compute a ...
1
vote
1
answer
137
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How to compute the Müller modular polynomials?
According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as
$$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
7
votes
1
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How is the Eichler-Shimura congruence related to L-functions?
My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...
6
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2
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Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
1
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0
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117
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Differentials on tori realised as double of annuli
In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus.
In short, the torus is realised ...
11
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1
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471
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Non-vanishing modular forms
Prompted by this MO question, I have the following question about modular forms which do not vanish on the upper-half plane.
Q1. Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal ...
13
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Case D=4l in Elkies' paper on Supersingular Primes of an Elliptic Curve over $\mathbb{Q}$
My question is regarding Elkies' paper on "The existence of infinitely many supersingular primes for every elliptic curve over $\mathbb{Q}$".
In the section "Nuts and Bolts", Elkies has the ...
28
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2
answers
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When did people start thinking of elliptic curves as groups?
I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
2
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0
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Global minimal Weierstrass equation over function fields
Let $K$ be a field of characteristic zero, $V/K$ a smooth projective variety and $F=\overline{K}(V)$ be the function field of $V$ over $\overline{K}$. For any elliptic curve $E/F$, it has a Weiertrass ...
2
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1
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Explicit families of elliptic curves
I am interested in finding families $X\to C$ of elliptic curves over a projective curve $C$. The fibers are not all isomorphic, not all smooth, and the singular fibers should be nodal stable curves. ...
6
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Descent via an explicit isogeny (genus 2)
This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...
5
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1
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419
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Tate-Shafarevich group over number fields
Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.
Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is ...
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1
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Integral complete 4-partite graphs
For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K_{a,b,c,d }$.
Can such a graph be integral, i.e. have only integer eigenvalues?
It is easy to see that the ...
2
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2
answers
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Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
5
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2
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How to prove that a certain curve does not lie in a proper abelian subvariety of abelian variety
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation
$$E: y^2=x^3-Ax+B=:f(x).$$
Consider the abelian variety $E^3:=E \times E \times E \subset \mathbb{P}^2 \times \mathbb{P}^2 \...
3
votes
1
answer
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Trace of elliptic curve in CM method
I am trying to understand the CM method for elliptic curves. Suppose we fix a discriminant $D<0$ and a prime $p$. In the CM method, we look for integer solutions $(t,y)$ to the norm equation $4p = ...
18
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1
answer
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How does Saito's treatment of the conductor and discriminant reconcile with an elliptic curve?
Saito (1988) gives a proof that
$$\textrm{Art}(M/R) = \nu(\Delta)$$
Here, $M$ is the minimal regular projective model of a projective smooth and geometrically connected curve $C$ of positive genus ...
1
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0
answers
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Abelian group extensions
Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...
4
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Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
5
votes
1
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regulator of an elliptic curve rational/irrational/transcendental?
Let $K$ be a number field and $E/K$ an elliptic curve (or abelian variety) with $\mathrm{rk}\,E(K) > 0$. Can/will the elliptic (abelian) regulator $\mathrm{Reg}(E/K)$ be rational/irrational/...
3
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Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves
I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations.
Let $E$ be an elliptic curve over $K$, ...
2
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0
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Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
1
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0
answers
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What size factor-base for Lenstra Elliptic Curve factorization
I'm writing a program to factorize numbers using Lenstra Elliptic Curve Factorization.
According to the wikipedia article, I should pick some k with a lot of small factors and then take a random ...
18
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0
answers
711
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Infinite extensions such that every elliptic curve has finite rank
The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
6
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0
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What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?
Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
5
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0
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Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
7
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1
answer
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Fields of Definition of Elliptic Curves
I am currently studying the theory of complex multiplication and I find myself confused by the language in a lot of the literature.
In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves,...
0
votes
1
answer
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On the elliptic curves $u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y_1^2$ and $v(v - 1)(v - n^2) = y_2^2$
A certain problem in equal sums of like powers for $7$th powers entails the elliptic curve,
$$u(u+127^2)(u+129^2) = y^2$$
I was looking at the general case,
$$u \big(u + (n - 1)^2\big) \big(u + (n ...
13
votes
2
answers
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GRH and the rank of elliptic curves
I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up ...
4
votes
2
answers
338
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On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
4
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0
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Weierstrass model of an elliptic curve: a line bundle over the base
Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface.
...
5
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0
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Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $...
4
votes
2
answers
548
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What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?
Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...
4
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0
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118
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Finding short linear combinations in abelian groups
Let $M$ be a finitely generated abelian group. Assume we are given a presentation of $M$, that is
\begin{equation*}
M = \frac{\bigoplus_{i=1}^r \mathbf{Z}g_i}{\sum_{j=1}^s \mathbf{Z} r_j}
\end{...
4
votes
1
answer
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Example of two p-Ordinary Elliptic Curves congruent to each other
I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \...
8
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2
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Mazur's Question on Mod $N$ Galois representations
In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\...
2
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1
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Local-Global divisibility of points on elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve and $P \in E(\mathbb{Q})$. If $P$ is divisible by $p$ in $E(\mathbb{Q}_p)$ for every prime $p$, does it follow that $P$ is a torsion point?
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1
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Fields of definition for torsion points on elliptic curves over finite fields
This question was previously asked on Math.Stackexchange.
It is about a proposition from R. Schoof's article Nonsingular Plane Cubic Curves over Finite Fields:
Proposition 3.7: Let $E$ be an elliptic ...
0
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1
answer
298
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Weil Pairing Example fails
I've been reading about Weil pairing from Pairings for Beginners and in section 5.1 an example is given. I took a look on the magma code of that example (see here) and it works.
All I did was to ...
3
votes
0
answers
167
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Scalar multiplication via the Kummer surface of a genus $2$ curve by $\sqrt{5}$
I hope this is a good question.
Recently I worked with genus two curves $H$ that have multiplication by $[\zeta_5]\in \text{Aut}(H)$, that is, multiplication by $e^{2\pi i/5}$. This automorphism is ...
10
votes
1
answer
507
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Orders of reductions of rational points on elliptic curves
I am looking for references where the following (or similar questions) have been studied:
Let $K$ be a number field or a function field in one variable over a finite field and let $E$ be an elliptic ...
1
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0
answers
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Anti-trace applied on the trace zero subgroup and distortion maps
I was reading Pairings for Beginners by Craig Costello and he talks about the base field subgroup $\mathcal{G}_1 = E[r] \cap \ker(\pi - 1)$ and the trace zero subgroup $\mathcal{G}_2 = E[r] \cap \...
6
votes
1
answer
219
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rational points and a local perturbation of an elliptic curve
Let $E_{a,b}$ be an elliptic curve defined by the equation $y^{2}=x^3+ax+b$ where $a,b \in \mathbb{Q}$.
Suppose that for $a=a_{0}$ and $b=b_{0}$ the rank of $E_{a_{0},b_{0}}(\mathbb{Q})=1$.
question:...
4
votes
1
answer
583
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Irreducibility of residual Galois representations attached to an elliptic curve
Let $E$ be a given elliptic curve over a number field $F$. For each prime $p$, one has the Galois representation $\mathrm{Gal}(\bar{F}/F)\to GL(E[p])$ where $\bar{F}$ is a fixed algebraic closure of $...
2
votes
1
answer
436
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Modular parametrization in terms of the moduli of shtukas
The modular parametrization of an elliptic curve over $\mathbb{Q}$ (and maybe over a number field in general?) is well-known; also for an elliptic curve over global function field with some condition (...
6
votes
0
answers
140
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Modular forms for non-arithmetic subgroups
Modular forms are usually considered as complex-valued functions on the upper half-plane quite invariant by a discrete subgroup $\Gamma$ of isometries and satisfying smoothness and growth condition.
...
3
votes
1
answer
338
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When does the module of Katz modular forms contain a basis for the vector space of classical modular forms?
Let $\Gamma\le SL(2,\mathbb{Z})$ be a congruence subgroup of level $N$. Let $R$ be a $\mathbb{Z}[1/N]$-algebra.
Let $\mathcal{Y}(\Gamma)_R$ denote the moduli stack over $R$ of elliptic curves ...
3
votes
1
answer
191
views
The rank of the elliptic curve $E(m):y^2 + xy - my = x^3 - mx^2$
Let $E(m)$ denote the elliptic curve
$$
E(m):y^2 + xy - my = x^3 - mx^2.
$$
Are the smallest values of $m$ such that $E(m)(\mathbb Q)$ has rank 3 (and 4) known?
4
votes
1
answer
175
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$\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form $27a^2+27a+7$
Related to this question.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Numerical evidence up to $2 \cdot 10^5$ suggests:
Conjecture: $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form
$27a^2+27a+7$ ...