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

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How can I “groupify” an elliptic curve over a non-field?

The book Primes of the form $x^2+ny^2$ by David A. Cox contains the following definitions regarding an elliptic curve (by which he means an equation $y^2=4x^3-g_2x-g_3$ such that $\Delta=g_2^3-27g_3^2$...
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Homogeneous spaces of elliptic curve

Let $d>1$ be a cube free natural number and $a,b,c$ are natural numbers greater than 1 with $abc=d$. How to explain that the curve $D: ax^3+by^3+cz^3=0$ is homogeneous space (over $\mathbb{Q}$) ...
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clarification on the inertia group in the proposition 1.5 chap VIII of Silverman's Arithemic of elliptic curves

Let K be a number field. Let $ Q\in \mathbb {P}^2 (\overline {K})$ and define $K (Q)=$ fixed field of $\{\sigma\in G_{ \overline {K}/K } :Q^\sigma=Q \}$ where $G_{ \overline {K}/K }$ is the ...
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Can a separable isogeny of elliptic curves have an inseparable dual?

Let $\phi: E_1\to E_2$ be an isogeny of elliptic curves over a field $K$ of characteristic $p>0$. Suppose that $\phi$ is separable and let $\hat{\phi}: E_2\to E_1$ denote the dual isogeny. Then $\...
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Intersection of algebraic curves at a point with given multiplicity

I don't know if this question is too basic for MO, so I put it here, but if you think I should migrate the question to MathOverflow please suggest me. Let $C/k$ be a smooth curve over a perfect ...
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Derivative of the Klein j-invariant

To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes). But I ...
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Homomorphism between elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^2+(a^2-4b)x$.

I am reading Rational Points on Elliptic Curves by Silverman and Tate. In Section 3.4, Page 76, the authors defined two elliptic curves elliptic curves $C: y^2=x^3+ax^2+bx$ and $\bar{C}: y^2=x^3-2ax^...
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Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
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$2 \times 2$ matrix representing elliptic curve?

Suppose we have $E/\mathbb{C}$ and we let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda=\mathbb{Z}+\mathbb{Z}\sqrt{5}i$. Suppose also that $\alpha=10+3\sqrt{5}i$. For a basis $\{\...
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Trace and degree of elliptic curve endomorphism?

Let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda = \mathbb{Z} + \mathbb{Z}\sqrt{5}i$. Let $\alpha=10+3\sqrt{5}i$. Show that $\alpha \in$ End$(E)$ and compute the trace and degree of $\...
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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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Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
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The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then $...
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Help needed in determining the singularity

Can someone teach me how to determine the singularity of algebraic curve $y^2 =x^3+x^2$. I'll be really grateful and thanks in advance
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Relation between pullbacks of a degree zero line bundle on an elliptic curve

Let $E$ be an elliptic curve over a field $k$. Let $$\mu:E \times_k E \to E$$ be the addition map on $E$. Furthermore let $p_1,p_2:E \times_k E \to E$ be the two canonical projections and let ${\...
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help to parametrize $y^2 = x^3 -x^2$

I appreciate if someone could help me to parametrize this equation $y^2 = x^3 -x^2$. Thanks in advance. I used maple to find the solution as $(x,y) = ((t^2-1),(t(t^2-1))$
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$p$ adic modular forms and wide open neighbourhood (e.g. Coleman primitive): is it possible to obtain a holomorphic function?

It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, ...
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Show 2-torsion subgroups are equivalent?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero squarefree element in $k$. Let $E_c/k$ be a curve defined by $cy^2=x^3+ax+b$. Show that the 2-...
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Simple automorphism proof with lattice elliptic curves?

For a lattice $\Lambda_1=\mathbb{Z}+\mathbb{Z}i$, find Aut($E_1$) where $E_1(\mathbb{C})=\mathbb{C}/\Lambda_1$. So I know that End($E$) $ =\{\beta \in \mathbb{C} $| $\beta\Lambda \subseteq \Lambda\...
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Elliptic curve n-torsion point?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, we ...
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What is more amazing?

$S$ denotes the set of rational points of any curve in the plane. What is more amazing between a) and b)? a) $S$ is dense in the curve $y^2=x^3-2^4\cdot3^3\cdot7^2$ b) $S=\emptyset$ in the curve $...
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How to show elliptic curve endomorphism is commutative?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-$n$ endomorphism. Show that $[n] \bullet \alpha=\alpha \bullet [n]$, ...
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Show an elliptic curve is a twist of another curve?

Let $E/k$ be an elliptic curve defined by the Weierstrass form $y^2=x^3+ax+b$. Let $c$ be a nonzero square free element in $k$. Let ${E_c}/k$ be a curve defined by $cy^2=x^3+ax+b$. Using a linear ...
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Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
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Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
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Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
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How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over $\mathbb{Q}$...
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Show $(x,y) \rightarrow (x,-y)$ is a group homomorphism?

Show that $(x,y) \rightarrow (x,-y)$ is a group homomorphism from $E$ to itself where $E$ is an elliptic curve in Weierstrass form. So $E$ is of the form $y^2=x^3+ax+b$. Would I just show that any ...
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Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is "...
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Computing the multiplicative inverse for point addition on an elliptic curve

I'm trying to perform point addition on an elliptic for two points taken from an example in the book "Understanding Cryptography by Christof Paar & Jan Pelzl". The points I'm trying to add are: $$...
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Quartic Diophantine equation $ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$

About the quartic Diophantine equation: $$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
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Isogenous elliptic curves over finite fields have the same number of points

I'm stuck in this question, it is the first part of exercise 5.4 from Silverman - The arithmetic of elliptic curves. Let $C,D$ be two isogenous elliptic curves over a finite field $\mathbb{F}_q$. ...
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Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, $...
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What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic $L$-...
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Degree $5$ embeddings of genus 1 curves, plucker embedding of a Grassmannian…

Here is something Ravi Vakil says after 19.9F in his notes. He is talking about embeddings of a genus 1 curve $C$ by complete linear series associated to the divisors $O(nP)$. "The beautiful ...
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Is a projective curve minus finite number of points affine?

The answer to the question in the title is "yes". This is proved for example here, or here (page 5). However, I seem to be able to construct a counterexample. Can you help me find the flaw in my ...
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question about Galois theory and dimension

I try to understand the proof of the lemma 13.7 of the following article: http://www.cs.nyu.edu/courses/spring05/G22.3220-001/ec-intro1.pdf The lemma says that if $r$ is a rational function which ...
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Points of finite order : Size of the group E/E0

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...
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Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
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Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
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Hasse theorem proof

I don't understand the following points 1) why is $E$ isomorphic to the $\ker(\phi - 1)$? 2) Why is $\#\ker(\phi - 1) = \deg(\phi - 1)$? The proof is taken from here page 11 https://www.math....
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ELMO 2012 Shortlist N9

I'll admit that I've made no progress to solve this one. It is way too hard. I guess I must do some stuffs with elliptic curve to solve it but I got nowhere So, here is the problem: Are there ...
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degree of isogeny understanding

In the context below why is the degree of the Frobenius endomorphism p ?
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$p$-depletion of a modular form

Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$. Let $f$ be a modular form with ...
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Mordell-Weil group - 2 descent

On my elliptic curve, I have generator P and 2-torsion point T in general. If i compute points nP and nP+T, and substitute these points in my function,z, I noticed that the non-square part are equal(...
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8 divides $\#C(\mathbb{F}_p)$ for $C$ : $y^2=x(x+1)(x-8)$, $p\geq 5$

I was trying to compute the torsion of $C$ : $y^2=x(x+1)(x-8)$ over $\mathbb{Q}$ by using the fact that the order of the torsion divides the order of $C$ reduced modulo any $p\nmid 2\Delta$. For any ...
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On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
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1answer
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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 ...
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Is $H^{1}(k,E[n])$ a subgroup of $H^{1}(k,E)$?

Let $E/k$ be an elliptic curve. Consider $E[n]$ which is a subgroup of $E$. Is it true that $H^{1}(k,E[n])$ is again a subgroup of $H^{1}(k,E)$ in Galois cohomology? I thought that this was true but I'...