Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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, ...

11
votes
0answers
236 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 ...
4
votes
1answer
142 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 \...
2
votes
0answers
105 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 ...
6
votes
1answer
198 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 ...
5
votes
1answer
200 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_\...
21
votes
1answer
429 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 ...
1
vote
1answer
145 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 ...
7
votes
1answer
136 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 ...
7
votes
1answer
178 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 (...
9
votes
1answer
185 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_{\...
13
votes
1answer
389 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 ...
1
vote
0answers
112 views

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 ...
6
votes
0answers
95 views

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 ...
3
votes
1answer
225 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 ...
19
votes
3answers
673 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 ...
6
votes
0answers
199 views

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 ...
5
votes
1answer
201 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 ...
5
votes
0answers
292 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 ...
3
votes
0answers
2k views

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 ...
6
votes
1answer
192 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, ...
8
votes
1answer
285 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:...
15
votes
1answer
295 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 ...
4
votes
2answers
179 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 ...
1
vote
1answer
99 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) ...
1
vote
0answers
86 views

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 ...
2
votes
1answer
85 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 ...
2
votes
0answers
150 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 ...
1
vote
0answers
95 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 \...
2
votes
0answers
52 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, ...
2
votes
0answers
103 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 ...
2
votes
1answer
167 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{...
14
votes
0answers
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 ...
2
votes
1answer
90 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 ...
1
vote
1answer
99 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_{...
7
votes
1answer
320 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 ...
6
votes
2answers
399 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 ...
1
vote
0answers
107 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 ...
10
votes
0answers
216 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 ...
11
votes
0answers
332 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 ...
28
votes
2answers
1k 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 ...
2
votes
0answers
127 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 ...
2
votes
1answer
211 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. ...
6
votes
0answers
108 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 ...
5
votes
1answer
250 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 ...
8
votes
1answer
176 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 ...
2
votes
2answers
317 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 ...
5
votes
2answers
182 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 \...
2
votes
1answer
115 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 = ...
16
votes
1answer
643 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 ...
1
vote
0answers
76 views

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{...