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

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Elliptic curve question

Let $P$ be a point on an elliptic curve over $\mathbb{R}$. Give a geometric condition that is equivalent to P being a point of order (a) $2$ , (b) $3 $ , (c) $ 4$ . Could someone explain this to ...
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2answers
627 views

Prove that the equation $y^2=x^3-73$ has no integer solutions

Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
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2answers
206 views

Exposition on Modular Curves

I was recently reading this paper by Weston, whereby he talks about the modular curves $X_0(11)$ and $X_1(11)$. I was wondering if anyone can recommend a more general exposition of modular curves ...
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2answers
119 views

Bound for number of points on surface over $\mathbb{F}_p$

I know of the bound for the number of points on an elliptic curve over a finite field: $$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$ where this includes the point at infinity. I have been told that ...
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35 views

Elliptic curve linear recurrence proof

Let $E/\mathbb{F}_q$ be an elliptic curve with $q=p^m$ for some prime $p$. Let $a_n=q^n+1-\#E(F_{q^n})$ and by convention we let $a_0=2$. Prove that $a_{n+2}=a_1a_{n+1}-qa_n$ for all $n>0$. ...
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80 views

Third point of intersection is also a point of inflection?

Let $C \subset \mathbb{P}_2$ be a nonsingular cubic. If $L$ is a line through two distinct points of inflection on $C$, how do I show that the third point of intersection is also a point of ...
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2answers
180 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
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70 views

The definition of an elliptic curve?

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other ...
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100 views

Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
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Silverman Adv. Topics example

I would like to refer you to Silverman's Advanced Topics in the Arithmetic of Elliptic Curves example 10.6: Let $D$ be a nonzero integer, $E:y^2=x^3+D$ with complex multiplication by $\mathcal{O}_K$ ...
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64 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
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1answer
203 views

Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
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277 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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175 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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655 views

Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
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178 views

Cohomology group and elliptic curve

Let $E$ be an elliptic curve with a 3-torsion point $P$ and $G = \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $X = \{O, P, -P\}$ where $O$ is the point at infinity and $X$ is a ...
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854 views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
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1answer
204 views

division polynomials of elliptic curve as function on $\mathbb{C}$

I have a question about exercise 6.15 of Silverman's book AEC. Suppose that $E$ is a nonsingular elliptic curve over $\mathbb{C}$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Then we ...
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94 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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2answers
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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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67 views

Help with proving that the torsion subgroup of $y^2=x^3+x$ is $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$

Let $E: y^2= x^3 + x$ be an elliptic curve over $\mathbb{Q}$. I'm trying to prove that $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$. In order to do that, I've already shown that ...
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114 views

In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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96 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
4
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1answer
157 views

First problem in Silverman's Arithmetic of Elliptic Curves

I started working through Silverman's Arithmetic of Elliptic Curves. For some reason it looks like the first problem in the first chapter is the hardest problem in the whole chapter or I'm completely ...
4
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1answer
130 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
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153 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
4
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1answer
102 views

Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
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197 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
4
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1answer
182 views

Computing cohomology groups of elliptic curves

I'm skimming through Silverman's text to recall some theory of elliptic curves that I've learned in undergrad. In practice however, I'm having trouble actually computing the cohomology groups. For ...
4
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1answer
192 views

Discriminant of isogenous elliptic curves

Let $E$ be an elliptic curve with a rational $p$-torsion point $P$. Then $E$ is isogenous to the elliptic curve $E' := E/\langle P \rangle$ via the mod $P$ map. I know that the conductor of $E$ and ...
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1answer
264 views

Global minimal model of elliptic curve over $\mathbb{Q}$

I am basically trying to solve the cannonball problem using elliptic curves. In other words I have to show that the only integer points on the "elliptic curve" $6y^2 = 2x^3 + 3x^2 + x$ are $(0,0), ...
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154 views

How to show that the image of a certain projective embedding is an algebraic curve?

I found the following claim in a paper by Griffiths and Harris : Start with a complex torus $\mathbb{C}/\Lambda$. The vector space of meromorphic functions having period lattice $\Lambda$ and a pole ...
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1answer
129 views

Hyperelliptic curve order

How to compute order of a hyperelliptic curve ($y^2=f(x)$, $deg(f)=2 \cdot g+1$, $g=4$), over $F_p$ for small $p$ ($p$ prime)? Are there any efficient algorithms to do so? Is it possible with ...
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119 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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73 views

Does there have to be a point on elliptic curve over $\mathbb{C}(t)$

Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation. $$ y^2 = x^3 + A x + B$$ Where $A, B \in \mathbb{C} (t)$. Question: ...
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1answer
132 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
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1answer
77 views

Formal Group for the Elliptic Curve $Y^2=X^3+AX$

I'm trying to solve the following problem without resorting to a direct calculation: Let $E : Y^2 = X^3 + AX$, where $A \in \mathbb{Z}$ and $A \ne 0$. Let $F(X, Y )$ be the formal group associated to ...
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1answer
500 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
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66 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
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140 views

The group structure of elliptic curve over $\mathbb F_p$

I want to find the group of the elliptic curve $y^2=x^3-x$ over $\mathbb F_p$ for all primes $p \le 29$. But I know only 1 fact about the structure of this group: $E(\mathbb F_p)=\mathbb Z/m \mathbb Z ...
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Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
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Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...
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1answer
80 views

Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC

I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves. 2.30(b)[(c) in errata]: Suppose $\mathfrak{P}$ remains inert in $L'$, say ...
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1answer
84 views

How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
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352 views

Proving that the differential on an elliptic curve $E$ given by $\omega=\frac{dx}{y}$ is translation invariant

I'm taking a course on elliptic curves and I'm stuck on a line in a proof. We're assuming we're in an algebraically closed field $K$ and char($K)\not=2$. We have our elliptic curve ...
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1answer
91 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
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1answer
398 views

The Process of Choosing Projective Axes to Put an Elliptic Curve into Weierstrass Normal Form

I'm reading the book "Rational Points on Elliptic Curves" and on page 23 the author takes an arbitrary (non-singular) elliptic curve in the projective plane and finds a rational point $O$, referring ...
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138 views

A question on level structures on elliptic curves

I have a question on $\mathbb{H}/\Gamma(N)$, which parametrizes level $N$ structures on elliptic curves. Let $Y(N)$ be the set of isomorphism classes of such objects, then, according to Fact 2 on page ...
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239 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
4
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1answer
287 views

Extend an holomorphic function defined on a torus

Suppose we have an holomorphic function $$ f : \frac{\mathbb{C}}{\Lambda} \mapsto \frac{\mathbb{C}}{\Lambda} $$ where $\Lambda$ is a lattice. Is it always possible to find another function $\psi : ...