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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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finite $p$ extensions on adjoining $p$-torsion points of an elliptic curve

Let $K$ be a fixed number field and $E$ be any elliptic curve over $K$. When we adjoin to $K$ the $p$-torsion points $E[p]$, we obtain an extension whose Galois group can be embedded in $GL(2, \mathbb{...
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108 views

Field of Definition of Quotient of Elliptic Curve

In Silverman's Arithmetic of Elliptic Curves, Chapter III, Proposition 4.12, we have the statement that if $E/F$ is an elliptic curve and $\Phi$ is a $\mathrm{Gal}(\bar{F}/F)$-invariant subgroup then ...
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95 views

Structure of the End group scheme of an abelian scheme over ring of integers

Let $O$ be the integer ring of a p-adic field $K$ (finite extension of $\mathbb Q_p$), $\mathscr{A}$ be an abelian scheme over $S=\operatorname{Spec O}$, consider the group endohomorphism scheme of $\...
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1answer
260 views

Is there a substitution that relates every Fermat curve to an elliptic curve?

I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level. A Fermat Curve of degree $n$ is the set of solutions to $x^...
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67 views

Algebraic theta-functions of level $2$ on an elliptic curve

Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...
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What do we know about the ramification of the modularity map $X_0(N)\to E$?

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $N$ be its conductor. By the modularity of elliptic curves over $\mathbb{Q}$, there exists a surjective map $f:X_0(N)\to E$, where $X_0(N)$ is ...
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1answer
153 views

Kato's Euler System for Isogenous Elliptic Curves

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?
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1answer
113 views

Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?

TL;DR. Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $\mathbf{Q}(i)$ seem to have value in $\mu_4$. But apparently our computation via Rohrlich's ...
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130 views

trivial solutions for Diophantine equations

Let $K$ be an odd degree number field. Consider the Diophantine equation: $$ X^4 + bY^4 =Z^2 $$ where $b\neq 0$. Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
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1answer
135 views

Deuring's result on elliptic curves. Any proof reference

I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\...
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1answer
142 views

Distribution of Elliptic Curve Generators over Q

Considering elliptic curves over Q with positive discriminant and rank>0, are there any results or proposed heuristics regarding the fraction of generators that are located on the identity vs non-...
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165 views

A website which explains Mazur's torsion point theorem

I'm about to read Mazur's paper "Modular curves and the Eisenstein ideal". It's so long and difficult for me, but I found a website which shows the Mazur's theorem. This is very short and very very ...
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1answer
163 views

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{...
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1answer
127 views

How likely is it for Selmer groups to have mu invariant 0?

Given a number field $K$, how likely is it that we'll find at least one elliptic curve $E/K$ such that the $\mu$-invariant of its Selmer group is 0 (in a cyclotomic extension)?
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161 views

What are the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)$ is an integer $>2$?

The problem of finding the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)=4$ turns out to be surprisingly difficult, and has made the rounds on the internet and social media, and Andrew ...
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lcalc and the Analytic Rank of $y^2 = x^3 + 432764797 x^2 + 332896 x$

I'm looking at elliptic curves associated with $a/(b+c) + b/(a+c) + c/(a+b) = N$. For the case $N=10400$, Michael Rubinstein's lcalc gives the analytic rank of the associated elliptic curve $y^2 = x^3 ...
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259 views

Record analytic rank for an elliptic curve?

What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$? The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot ...
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1answer
158 views

When is this localization map injective, if at all?

Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map $$ E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \...
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112 views

Notation for endomorphism algebra of Elliptic Curves

$\newcommand{\End}{\operatorname{End}}$For an elliptic curve $E$, I understand that the notation $\End(E)$ denotes the ring of endomorphisms of $E$. Since $\End(E)$ is torsion free, it's possible to ...
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1answer
210 views

Do we need the Weber function to generate ray class fields of imaginary quadratic fields of class number one?

I'm a bit confused by the role of the Weber function in generating ray class fields of imaginary quadratic fields of class number one. More specifically, let $K$ be such a field and $E$ an elliptic ...
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1answer
268 views

Elliptic curves and $GL(2)$ Iwasawa theory

Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
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1answer
446 views

The valuation of j-functions vs number of isomorphisms for an elliptic curve

Gross and Zagier prove the following fantastic result in their paper "Singular Moduli": Let $R$ be a discrete valuation ring over $\mathbb Z_p$ with uniformizer $\pi$ such that $k = R/\pi$ is ...
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1answer
150 views

Sage: Evaluation precision for elliptic curves over p-adic fields

Consider the elliptic curve $E: y^2 = x^3 + 23x+11$ over p-adic fields. In Sage I use: k = GF(257) E = EllipticCurve(k,[23,11]) kp = Qp(257,5) # 257-adic Field with capped relative ...
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1answer
138 views

Raynaud's universal Tate elliptic curves

In the end of Section 9.2 of Bosch's book Lectures on Formal and Rigid Geometry, a rigid $S$-space $E_Q$ is constructed, for a variable $Q$ replacing the classical parameter $q\in k$. (Here $k$ is a ...
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1answer
183 views

Existence of newforms which are non-ordinary at a given prime

Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (...
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1answer
193 views

Existence of certain endomorphism of supersingular elliptic curve

Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
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1answer
408 views

BSD conjecture for rank 1 elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and ...
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On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$

For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations: $$x^4+16z^n=y^2 \qquad (1)$$ and $$x^4+z^n=y^2 \qquad (2)$$. (2) is special case of Fermat Catalan and is solved. For ...
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Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
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1answer
282 views

An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...
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3answers
686 views

Points of elliptic curves over cyclotomic extensions

Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
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Smooth morphisms to the moduli stack of elliptic curves

Fix a prime $p$, and let $\overline{M}$ be the Deligne-Mumford compactification of the moduli stack $M$ of elliptic curves (which, concretely, is the open substack of the stack of cubic curves which ...
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1answer
203 views

Is there infinitely many prime $p$ such that the normalized trace of Frobenius $\frac{a_p(E)}{2\sqrt{p}}$ is arbitrarily small (but not zero)?

I just encountered the following problem and failed to find proper reference, could anyone kindly help to give me some illustration? For an elliptic curve $E$ without complex multiplication (just ...
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301 views

Torsion subgroup of the group of points of an elliptic curve over local field

Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is ...
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Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?

In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article ...
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1answer
214 views

p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...
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1answer
289 views

projective plane cubics with exactly 9 real points

It is not hard to construct such curves explicitly, e.g. my favourite example is a curve $U$ singular at $(3:4:5i)$ and also passing through $(1:0:0)$, $(0:1:0)$, $(0:0:1)$, $(1:1:0)$, $(1:0:1)$, $(0:...
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1answer
306 views

Largest rank assumed by infinitely many elliptic curves

One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one ...
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2answers
181 views

Singular abelian surfaces that can be defined over $\mathbb Q$

An abelian surface $A$ is called singular if it has maximal Picard number $\rho(A) = 4$. By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E_1 \times E_2$ of two ...
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1answer
102 views

Confusion on supersingular reduction of elliptic curves with complex multiplication

Let $A/L$ be an elliptic curve, with complex multiplication by a quadratic imaginary field $K$. A theorem by Deuring ([13, paragraph 4], Theorem 12 on page 182 of Elliptic Functions by Serge Lang) ...
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Tangent Bundle of reducible genus one curves

I need to know what can be said in general about the tangent bundle of reducible curves over complex numbers with arithmetic genus one, say $I_N$. As far as I know for any Simpson semistable torsion ...
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1answer
87 views

Isogenies of degree 3 of elliptic curves with j-invariant 0

Let $E/\mathbb{Q}$ be an elliptic curve with $j(E)=0$. I.e., $E$ has Weierstrass equations $$ y^2 = x^3+ B$$ for some $B\in \mathbb{Q}$. $E$ has complex multiplication by $\mathcal{O}:= \mbox{ ring ...
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154 views

Growth of Selmer Groups

If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the Selmer or Tate-Shafarevich groups become large? I will ...
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99 views

The specific elliptic fibration on the Kummer surface of the superspecial abelian surface

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \...
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53 views

The quotient of a superspecial abelian surface by the involution

Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution $$ i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
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119 views

Elliptic fibrations on the Fermat quartic surface

Consider the Fermat quartic surface $$ x^4 + y^4 + z^4 + t^4 = 0 $$ over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$). Is there the full ...
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1answer
174 views

Modular parametrization of a curve of Heegner and Weber

The curve $$(X-16)^3=XY\tag{1}\label{1}$$ is essential to Heegner's approach to the class number one problem for imaginary quadratic fields. We have the following “modular” parametrization \begin{...
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1k views

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 ...
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1answer
91 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 ...
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1answer
102 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_{...