# Tagged Questions

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

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### Elliptic curve - perfect square discriminant

Given an elliptic curve: $$y^2 = x^3+ax+b$$ where a,b are rational, what can be said about the curve if its elliptic discriminant is a perfect square? $$discriminant= -16(4a^3+27b^2)$$ Any special ...
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### Rational solutions of $y^2 = x^3 - x$

I believe that the only rational solutions of $$y^2 = x^3 - x$$ are the obvious ones $(-1,0)$, $(0,0)$, $(1,0)$, and that this was proved by Fermat using the method of descent. Can anyone outline a ...
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### Finding an Elliptic Curve with 103 points

I am trying to solve the following problem: Find an elliptic curve over F101 with 103 points. I know all of the equations when needing to find alpha, and beta and all that when I am given two points ...
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### A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
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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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### 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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### 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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### 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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### 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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### 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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### The bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
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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 "...