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 ...
user124294's user avatar
2 votes
1 answer
136 views

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 ...
Licheng Wang's user avatar
1 vote
1 answer
137 views

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_{...
Licheng Wang's user avatar
7 votes
1 answer
638 views

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 ...
Nico A's user avatar
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6 votes
2 answers
716 views

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 ...
TeaFor2's user avatar
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1 vote
0 answers
117 views

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 ...
giulio bullsaver's user avatar
11 votes
1 answer
471 views

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 ...
François Brunault's user avatar
13 votes
0 answers
537 views

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 ...
Ming Hao Quek's user avatar
28 votes
2 answers
2k views

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 ...
Kimball's user avatar
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2 votes
0 answers
227 views

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 ...
Wei Pin Wong's user avatar
2 votes
1 answer
294 views

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. ...
Zlatan P.'s user avatar
6 votes
0 answers
153 views

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 ...
Eduardo R. Duarte's user avatar
5 votes
1 answer
419 views

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 ...
user avatar
8 votes
1 answer
204 views

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 ...
Wolfgang's user avatar
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2 votes
2 answers
487 views

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 ...
maddels's user avatar
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5 votes
2 answers
241 views

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 \...
Vlad's user avatar
  • 51
3 votes
1 answer
182 views

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 = ...
Lynn's user avatar
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18 votes
1 answer
1k views

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 ...
Nico A's user avatar
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1 vote
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{...
debanjana's user avatar
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4 votes
0 answers
298 views

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 ...
Sam Streeter's user avatar
5 votes
1 answer
249 views

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/...
user avatar
3 votes
0 answers
325 views

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 $K$, ...
Johnny T.'s user avatar
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2 votes
0 answers
142 views

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 ...
Johnny T.'s user avatar
  • 3,547
1 vote
0 answers
99 views

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 ...
dspyz's user avatar
  • 263
18 votes
0 answers
711 views

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 $S$...
R. van Dobben de Bruyn's user avatar
6 votes
0 answers
350 views

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 ...
Johnny T.'s user avatar
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5 votes
0 answers
312 views

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}...
Alexey Ustinov's user avatar
7 votes
1 answer
922 views

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,...
Rdrr's user avatar
  • 881
0 votes
1 answer
144 views

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 ...
Tito Piezas III's user avatar
13 votes
2 answers
781 views

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 ...
FusRoDah's user avatar
  • 3,680
4 votes
2 answers
338 views

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-...
user230394's user avatar
4 votes
0 answers
344 views

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. ...
Marion's user avatar
  • 577
5 votes
0 answers
319 views

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 $...
The Thin Whistler's user avatar
4 votes
2 answers
548 views

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., $a_1=1$ ...
ZolaElliptic's user avatar
4 votes
0 answers
118 views

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{...
François Brunault's user avatar
4 votes
1 answer
196 views

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) \...
Eins Null's user avatar
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8 votes
2 answers
392 views

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]\...
Rdrr's user avatar
  • 881
2 votes
1 answer
262 views

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?
Ahmed Matar's user avatar
1 vote
1 answer
218 views

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 ...
Hanno's user avatar
  • 2,736
0 votes
1 answer
298 views

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 ...
cehptr's user avatar
  • 21
3 votes
0 answers
167 views

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 ...
Eduardo R. Duarte's user avatar
10 votes
1 answer
507 views

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 ...
naf's user avatar
  • 10.5k
1 vote
0 answers
91 views

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 \...
cehptr's user avatar
  • 21
6 votes
1 answer
219 views

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:...
M.O.'s user avatar
  • 125
4 votes
1 answer
583 views

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 $...
User0829's user avatar
  • 1,378
2 votes
1 answer
436 views

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 (...
wkf's user avatar
  • 637
6 votes
0 answers
140 views

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. ...
Desiderius Severus's user avatar
3 votes
1 answer
338 views

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 ...
stupid_question_bot's user avatar
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?
Youmbai's user avatar
  • 31
4 votes
1 answer
175 views

$\#E(\mathbb{F}_p) \in \{p,p+2\}$ 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 $27a^2+27a+7$ ...
joro's user avatar
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