# Questions tagged [elliptic-curves]

An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.

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### Factoring integers of the form $n=p q^2$ using elliptic curves

We got argument and strong experimental support that integers of the form $n=p q^2$ can be factored using elliptic curves easier than general integers Q1 Is this known? Added This is known since at ...
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### Torsion points on $E/\mathbb{Q}$ with large coordinates

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points. What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
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### Common prime of the finite number of order of imaginary quadratic field

This is from Silverman's 'the arithmetic of elliptic curves', exercise 5.5. Let $K$ be an imaginary quadratic field, and let $R_1...R_n$ be orders in $K$. I would like to prove that there are more ...
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### Trace map on rational points of elliptic curves

Let $L/K$ be finite Galois ext. of number fields and $E/K$ an elliptic curve. Define trace $$Tr : E(L) \rightarrow E(K), \;\; P \mapsto \sum_{\sigma \in G_{L/K}} P^{\sigma}$$ When is this map ...
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### An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
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### Finite quotients of the absolute Galois group of $\mathbb{Q}$ via torsion of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime. Then there is an action of the absolute Galois group of $\mathbb{Q}$ on $E[p]$ that factors through a finite quotient. Does any finite ...
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### Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...