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# 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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### Monogenic cubic rings and elliptic curves

By an elliptic curve over $\mathbb{Q}$, we mean a genus 1 curve with a $\mathbb{Q}$-point. By a monogenic cubic order, we mean a unital cubic ring $R$ isomorphic to $\mathbb{Z}^3$ as a $\mathbb{Z}$-...
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### Nice forms for the Weierstrass equation

The motivation for this question comes from the study of elliptic curves but the question itself is about choosing convenient algebraic transformations. Fix a base ring (commutative, with a ...
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### Height of point elliptic curve finite field

Is there any way to know the height of point from elliptic curve over finite field? Is there any ways to lift the point first to rational or number field and find the height from there? Thanks before
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### Texts on moduli of elliptic curves

I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves. I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...
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Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$E\!:y_1^2 = ... 1answer 96 views ### Generators of a graded algebra defining bundle over elliptic curve I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425): We consider an elliptic curve X and a line bundle (=invertible sheaf) L on X. Then,... 1answer 118 views ### Formal group and formal completion of an elliptic curve Let A be a ring, S the spectrum of A, f : E \to S an elliptic curve. Then assuming f_*\Omega_{E/A} is free over S, \hat{E} (the formal completion along the 0-section.)  \cong \... 3answers 358 views ### Is there a way to find any non-trivial \mathbb{F}_p(t)-point on the given elliptic curve? Consider a finite field \mathbb{F}_p (where p \equiv 1 \ (\mathrm{mod} \ 3), p \equiv 3 \ (\mathrm{mod} \ 4)) and the elliptic curve$$ E\!:y^2 = x^3 + (t^6 + 1)^2  over the univariate ...
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### Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties?

I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I ...

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