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

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Meromorphic function written as Weierstrass Elliptic Function [closed]

Let $\Lambda$ be a lattice in the complex plane. And Weierstrass Elliptic Function $$\wp(z)=\frac{1}{z^2}+\sum_{\omega \in \Lambda - \{0\}}\frac{1}{(z-\omega)^2}-\frac{1}{\omega ^2}$$ How can I ...
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26 views

advice for curve fitting

I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
4
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1answer
53 views

Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
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Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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1answer
23 views

Elliptic Curves and Mod P

I'm trying to figure out why the number of points (Np) equals any Prime (P) when: P (is congruent) 2 (mod 3) To the Elliptic Curve y^2=x^3+17 Does anyone know why this is?
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Why are supersingular elliptic curves useful for cryptography?

I don't know very much about cryptography and would like to learn more. I know the basics of RSA alogrithm and how elliptic curves over finite fields can be used to do something similar. But I would ...
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2answers
131 views

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

This question concerns the equation $$\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$$ and the possibility of rational number solutions with $abc \ne 0$. In comments arising from: Using ...
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0answers
43 views

Surjectivity of good reduction maps of elliptic curves

For simplicity, Let $E/\mathbb{Q}$ be an elliptic curve with good reduction, call it $E'$, at $p$. We know that the reduction map $E(\mathbb{Q}_p)\to E'(\mathbb F_p)$ is surjective, but ...
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33 views

Elliptic curve of points order 4

So how do you find the points on the eliptic curve $y^2=x^3+ax$ of order 4, where $4\mid a$ but $4^n$ does not divide $a$ for $n>1$. We proved that for $(x,y)=2(u,v)$, we must have ...
2
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1answer
96 views

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational ...
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The Arithmetic of Elliptic Curves, Exercise 1.3

Let $V\subset \mathbb{A}^n$ be a variety given by a single equation. Prove that a point $P\in V$ is nonsingular if and only if $$\text{dim}_{\bar{K}}M_P/M_P^2=\text{dim}V.$$ For a general variety ...
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44 views

elliptic curves and order of elements

The problem is this: Show that any elliptic curve over $\mathbb Z_{83}$ has an element of order > 30. I don't quite know which way to go on this one. We could use Hasse's thm. to show that the ...
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1answer
88 views

Adding points on an elliptic curve

I'm trying to work out a problem from a previous exam in Cryptography regarding elliptic curves. I can add points on an EC using the formulas given, but the suggested solution to this exam problem I ...
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1answer
75 views

The translation map between elliptic curves is a rational map

I want to see a reference or a prove that the following map is a rational map: Let E be an elliptic curve,$P\in E$ and $T_p$ defined as $T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$. It is important ...
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47 views

Finding number of points on elliptic curve

I'm working on a previous exam problem, and my solution does not match with the given one, and I don't know why. I have the elliptic curve $$E: Y^{2} = X^{3} + X + 46$$ over $\mathbb{F_{101}}$. We're ...
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1answer
112 views

Infinitely rational points in $y^2 = x^3 - 4$?

If the $x$-coordinate of a rational point $P$ of $y^2 = x^3 - 4$ is given by $m/n$, the $x$-coordinate of $2P$ is given by$${{(m^3 + 32n^3)m}\over{4(m^3 - 4n^3)n}}.$$Using this fact, how do I show ...
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1answer
40 views

Why is the Frob in elliptic curve not called an automorphism

Please apologize, if that's a stupid question. Why is the Frobenius Endomorphism of an elliptic curve over a finite field not regarded as an automorphism? Since it is an Isogeny, it is surjective ...
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1answer
95 views

Formal expansion of differential form on elliptic curves

First of all everything i'm asking about comes from the beginning of Katz and Mazur's book : Arithmetic moduli of elliptic curves (which you can find here). I'm considering an elliptic curve $f : E ...
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95 views

Fermat's Last Theorem

Is there any relationship between BSD conjecture and Fermat's Last Theorem? What is the importance of the analytic degree of the L-function of an elliptic curve stated in the BSD conjecture in ...
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1answer
73 views

Are elliptic curves algebraic varieties?

I got a short question. Are elliptic cubes also algebraic varietes? Say we have $E:y^2=x^3+5x=:f(x)$ Then we can $f(x)=x(x^2+5)$ So it can't be an algebraic variety.. I feel like I am totally ...
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1answer
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How can we continuously deform a height 1 formal group law into a height 2 formal group law?

A Quick Review: The complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ may be rewritten using the exponential, $\text{exp(}{2 \pi i \tau}) =: q$ as $\mathbb{C}^\times/q^\mathbb{Z}$ . ...
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1answer
53 views

Rational maps between elliptic curves

I dont understand the definition of rational maps. Here is the definition: Let $E_1$ and $E_2$ be elliptic curves over a field $K$. (projectively written). A rational map $\Phi:E_1\rightarrow E_2$ ...
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1answer
33 views

Example of Constant Morphism

I am reading Basic Theory of Elliptic Curves, there I came about a statement saying : A Morphism of curves is either Surjective or Constant. While studying Isogenes I came across examples of ...
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1answer
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Isogenous elliptic curve.

I'm studying elliptic curves and I have a question Take two $k$-isogenous elliptic curves defined over a number field $k$ and fix a place $v$ of good reduction. Are the reduced curves ...
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1answer
28 views

Absolute value in Hasse's theorem

The Hasse's theorem says that for an elliptic curve $E$ defined on $\mathbb{F}_p$ where $p$ is a prime number, we have: $|n-(p+1)| < 2\sqrt{p}$ with $n$ the order of $E$. I am wondering why the ...
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2answers
78 views

What will be a good source for learning elliptic curves and what viewpoints can I adopt?

I'm attending a research seminar on elliptic curves in my university where the professor is currently presenting a proof of Mordell's theorem (for elliptic curves over $\mathbb Q)$. The professor says ...
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49 views

Reduction of an elliptic curve defined over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\ell$ be a prime number such that the reduced curve $\tilde E_{\ell}$ is non singular. Assume that $\tilde E_{\ell}$ admits a subspace ...
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44 views

De Rham-Etale comparison isomorphism for elliptic curves

I can't find anywhere a proof of the following comparison isomorphishm: $$H^1_{dR}(E)\otimes \mathbb{C}=H^1_{et}(E)\otimes \mathbb{C}$$ where $E$ is an elliptic curve over $\mathbb{C}$. Any ...
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70 views

Tate curve and action of inertia group

I read the answers to this question Clarifying a comment of Serre. However I miss a passage of the second answer and since I can't comment there I have should post a new question. I don't understand ...
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60 views

Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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element of an $\ell$-adic Galois representation with certain eigenvalues.

Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation $$ \varphi_{\ell}: \mathscr{G} \longrightarrow ...
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1answer
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Showing $P=(o,m) \not\in 2E_m(\mathbb{Q})$ for an elliptic curve $E_m$

I have been having trouble with this question. Let $m\in \mathbb{Z}$ with $m > 0$ and define $E_m : y^2 = x^3 −x+m^2$ Then $E_m$ is an elliptic curve Determine the group sturcture of ...
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30 views

intersection of plane elliptic curve with coordinate hyperplanes

Let $E: y^2z = x^3 - Axz^2 - Bz^3$ be a plane elliptic curve. I want to calculate the intersection of $E$ with the coordinate hyperplanes $H_i = \{x_i = 0\}$, $i=1,2,3$. I write $H_x = \{x=0\}, H_y = ...
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106 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
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2answers
127 views

Rational solutions to $x^4+y^4=cz^2$

Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve? What I really want to know is how to transform this into an elliptic ...
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105 views

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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1answer
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The sum of an isogeny and its dual for the Frobenius homeomorphism

This is from page 150 of Silverman's "The Arithmetic of Elliptic Curves". Any my only questions is: How you can conclude that $[a]=\phi+\hat{\phi}$? I tried to use the formula on page 85 which ...
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1answer
123 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
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75 views

Isogenies between elliptic curves with specified torsion groups

For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert ...
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1answer
23 views

Order's size (in bits) of an elliptic curve

I am trying to prove that, given an Elliptic Curve defined on $\mathbb{F}_p$ with $p$ a prime number, the order $q$ verifies: $|p| \le |q| \leq |p|+1$ where $|x|$ denotes the length in bits of $x$. ...
2
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1answer
46 views

Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
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Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
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1answer
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Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
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The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
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1answer
28 views

Addition of x-coordinate on elliptic curve given by Möbius Transformation

Consider the elliptic curve $y^2=(x-\alpha)(x^2+ax+b)=x^3+(a-\alpha)x^2+(b-a\alpha)x-\alpha b$ over the field $K$ with $\text{char}\ K\not= 2$. The questions I am doing asks for a formula for the ...
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2answers
104 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
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0answers
47 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is. One of the ...
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2answers
37 views

Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or ...
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Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?