# Tagged Questions

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

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### For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
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### Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
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### Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
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### Splitting of a prime and $p$-divisibility on an elliptic curve

Let $K$ be a quadratic imaginary field and let $\lambda$ a prime of norm $l^2$, for a rational prime $l$. We consider $E$ to be an elliptic curve such that $E[p](K)$ is trivial, where $p\neq l$ is a ...
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### Show that division polynomials of elliptic curve $y^2=x^3+x$ are in $\mathbb{Z}[x,y]$.

This is an exercise from Rational Points on Elliptic Curves by Silverman and Tate. Define a sequence of division polynomials $\psi_n\in \mathbb{Z}[x,y]$ for the elliptic curve $y^2=x^3+x$ ...
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### Elliptic curves over $\mathbf{F}_q$ with $q = p^{2m}$

I am reading Washingtons book about elliptic curves and struggling with an exercise there (4.9), which is the following: Let $E$ be an elliptic curve over $\mathbf{F}_q$ with $q = p^{2m}$. Suppose ...
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### Hasse-Weil bound for even degree polynomial

Consider the equation $y^2 = f(x)$, in the finite field $\mathbb{F}_q$($q$ is a prime power) where $f(\cdot)$ is a monic polynoimal of even degree (greater than or equal to $4$) with integer ...
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### Sheeted-covering space degree 2 of Riemann Surfaces

In Milne - Elliptic curves, one finds the following on page 92: Branched-covering maps are not local isomorphisms at the ramified points; so could somebody explain to me what Milne means by 'a ...
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### Solve Elliptic Curve equation

Suppose you have an elliptic curve $E_{p}$: $y^{2} = x^{3} + Ax + B \mod p$ and points $P$ and $Q$ which lie on $E_{p}$. Does there always exist $n$ such that $nP=Q$? If so, how do we solve for it?...
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### Prove that there are only finitely many rational numbers $p/q$ satisfying $|\frac{p}{q}-\sqrt[d]{b}|\leq \frac{C}{q^3}$.

This is a problem from Silverman & Tate's Rational Points on Elliptic Curves. The following is the Diophantine Approximation theorem by Thue which is proved in Chapter 5: Theorem. Let $b$ be a ...
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### Inertia of an elliptic curve with potentially good reduction

Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good ...
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### How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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### Congruent Numbers and Integral Points on Elliptic Curves

As you probably know, congruent numbers $N$ and elliptic curves of the form $$E_N:y^2=x^3-N^2x$$ are intimately connected. While playing around with curves of this form, I found that $E_N$ will have ...