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

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26
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373 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 ...
10
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246 views

Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
8
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149 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
107 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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137 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 ...
7
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73 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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109 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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109 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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45 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 ...
6
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104 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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126 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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125 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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103 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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117 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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92 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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53 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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43 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 ...
3
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48 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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41 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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84 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
59 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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160 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
84 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
79 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
50 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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103 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
130 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
25 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 ...
2
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46 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
51 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
votes
0answers
44 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
90 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
29 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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51 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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102 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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43 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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34 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 ...
2
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0answers
72 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
2
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0answers
172 views

A strong form of the Nagell-Lutz theorem

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ ...
2
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99 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
2
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76 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
2
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0answers
97 views

Examples of non-minimal Weierstrass equations that make $[E(K):E_0(K)]$ arbitrarily large.

I'm taking a course on elliptic curves, in which the lecturer briefly mentioned that the Tamagawa number $c_K(E)=[E(K):E_0(K)]$ satisfies $c_K(E)=\mbox{ord}(\Delta)$ or $c_K(E) \leq 4$. He said that ...
2
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0answers
48 views

Different elliptic curves over given $\mathbb{F}_q$ can have different orders?

As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders? I assume yes. Let $q:=5$. Wikipedia gives $9$ points for ...
2
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0answers
101 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
2
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0answers
97 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 ...
2
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73 views

Galois action on CM elliptic curves

Let $E$ be an elliptic curve $E$ defined over a number field $K$ such that $E$ has complex multiplication by the maximal order in the ring of integers of an imaginary quadratic field $F$. Let ...
2
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282 views

Elliptic curves, 2-torsion and branch points.

I'm currently reading through Ravi Vakil's notes on Algebraic Geometry. I've been having trouble grasping some things conceptually though and I hope that you can help me. For an elliptic curve (E,p) ...
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0answers
35 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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40 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...