For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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29
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532 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
14
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376 views

More elliptic curves for $x^4+y^4+z^4 = 1$?

(Note: This has been updated to be similar with this MO post.) There are exactly 22 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{11}$. Noam Elkies showed that $(1)$ as, ...
9
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174 views

Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
8
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0answers
162 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
8
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109 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
7
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78 views

Motivation for Weil pairing

The Weil pairing $$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$ for an elliptic curve is defined as follows. Let $\phi:E\to E'$ be an isogeny of degree $n$ and $\hat\phi:E'\to E$ be the dual ...
7
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77 views

Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
7
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152 views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic ...
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119 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
6
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110 views

Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be ...
5
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130 views

Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a ...
5
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128 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
5
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111 views

Why is this a characterization of isogenies of elliptic curves? (From Silverman)

In the proof of Theorem III.6.2 (c) in Silverman's The Arithmetic Of Elliptic Curves it says: Let $x_1, y_1 \in K(E_1)$ and $x_2, y_2 \in K(E_2)$ be Weierstrass coordinates. We start by looking at ...
4
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90 views

A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$

Update (Dec. 22): I have already solved this question with Magma. Recently, I read a paper [1] and saw the following equation: $$u^3−2u^2−2v^3−20v^2+16v=0.$$ The author then got a Weierstrass ...
4
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35 views

Distribution of points in ellliptic curves over finite fileds

Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in ...
4
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50 views

There are no elliptic curves over $\mathbb{F}_8$ satisfying either $\#E(\mathbb{F}_8)=7$ or $\#E(\mathbb{F}_8)=11$

This is taken from The Arithmetic of Elliptic Curves by Silverman on page 154, Q5.10(f). One way of directly solving this problem is to work out on sage all 8^5 possibilities of elliptic curves and ...
4
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125 views

Complex multiplication - Ray class fields

I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin. According to ...
4
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105 views

Complex Multiplication/ Elliptic Curves Question

I'm working on the following problem from Silverman's advanced topic in the arithmetic of elliptic curves: Let $E$ be an elliptic curve defined over a number field $L$ with complex multiplication by ...
4
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97 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
3
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59 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
3
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28 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
3
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61 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
3
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50 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
3
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0answers
29 views

Ample non-flex on elliptic curve

Say $E$ is a plane cubic, and $p$ is a point on $E$. Riemann-Roch tells us that $\mathcal O_E(3p)$ is very ample. If $p$ is a flex, it's easy to write down the three sections giving an embedding of ...
3
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49 views

Klein's invariant and negative discriminant

Let $J = J(\tau)$ be Klein's invariant and let $0 < k < 1$ be the elliptic modulus. It is known that $$J = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2},$$ where ...
3
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0answers
57 views

Elliptic curves as $\mathbb{C}^*/\mathbb{Z}$

I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where ...
3
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0answers
77 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
3
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70 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
3
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31 views

does every elliptic curve E/S have infinitely many sections after passing to an etale extension of S?

Let E/S be an elliptic curve, where S is any scheme. Must there exist a scheme $S'$, etale and surjective over $S$, such that the pullback $E' := E\times_S S'$ has infinitely (or even > 1) many ...
3
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0answers
49 views

Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
3
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0answers
47 views

Automorphisms of elliptic curves with a point of order N

Probably a stupid question, but I'm trying to figure out the following: Suppose we have an element of $X_1(N)$, i.e. an elliptic curve $E$ with a point $p$ of order $N$. If we have another such curve ...
3
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87 views

Frobenius Endomorphism

I had a lecture last week which dealt with the Frobenius Endomorphism on elliptic curves. The lecturer showed an example at the end of the lecture, when almost out of time and I don't quite understand ...
3
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0answers
61 views

Weil operator of elliptic curve

Let $V$ be a $1$-dimensional $\mathbb{C}$-vector space and $\Lambda \subset V$ be an elliptic curve (=lattice). Let $C : V \rightarrow V$ be the multiplication by $i$. Consider the two following ...
3
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199 views

Every smooth cubic curve has a flex point

I want to show that every smooth irreducible plane cubic $C$ has a flex point, i.e. a point $P$ with $i_P(C, T_C(P)) = 3)$ (where $T_C(P)$ is the tangent to $C$ at $P$). I know how to do this in ...
3
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0answers
105 views

Good source of problems for Knapp's Elliptic Curves?

I'm studying elliptic curves (and eventually modular forms) out of Knapp's book because of the softer algebraic geometry prereqs. It's incredibly accessible but the problem is that I don't know I can ...
3
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0answers
102 views

Ribet's proof of open image for elliptic curves

In http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183555477, Ribet gives a proof of Serre's open image theorem for elliptic curves using ...
3
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0answers
53 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
3
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0answers
107 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
3
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0answers
131 views

when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?

If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
2
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0answers
65 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
2
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0answers
95 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
2
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0answers
60 views

Parametrization of this elliptic curve

What's the simplest way to parametrize the curve given by the equation $$y^2 = (x^2-a^2)^2 - b^2,$$ namely simple functions (polynomials?) $x(z)$, $y(z)$, that would satisfy the above relation. This ...
2
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0answers
51 views

Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an ...
2
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0answers
104 views

Using an elliptic curve to create pseudo random number

I recently started learning about encryption. I read about how elliptic curves can be used to create pseudo random numbers (and how the nsa might have abused this fact to create a backdoor in ...
2
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0answers
35 views

Inflection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is ...
2
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55 views

Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in ...
2
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0answers
110 views

Representing Points of Jacobian in Magma

Although I understand the Mumford representation of points on the Jacobian (of a genus 2 hyperelliptic curve), I don't understand how Magma represents such points. I would guess the confusion arises ...
2
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0answers
49 views

Supersingular elliptic curves- Invariant differential exact proof question

I'm writing a minor thesis about different criteria of supersingularity and I wanted to show the following from Husemöller's Elliptic Curves [Prop. 13.3.8]: An elliptic curve $E$ in characteristic ...
2
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0answers
44 views

Question about paper on Selmer groups

Let $\textrm{Sel}_{n}E$ denote the $n$-Selmer group and $\textrm{Sel}_{p^{\infty}}E = \varinjlim_{n}\textrm{Sel}_{p^{n}}E$. Proposition 5.10 of this paper http://arxiv.org/abs/1304.3971 states that ...
2
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0answers
35 views

number of solutions eqauation on supersingular elliptic curve

To Frobenius endomorphism on supersingular elliptic curve I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point $A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular. Is it ...