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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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      Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields

      Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
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      2answers
      285 views

      Diophantine equations $ax^4+by^2=c$ in rational numbers

      Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or ...
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      244 views

      Max order of an isogeny class of rational elliptic curves is 8?

      I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's Rational Isogenies of Prime Degree. Theorem 5 There is a constant $C$ such that every elliptic ...
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      140 views

      How to find a CM point with the image in the elliptic curve under modular parametrization given

      everyone! Let $E:y^2+y=x^3-61$ be the minimal model of the elliptic curve 243b. How can I find the CM point $\tau$ in $X_0(243)$ such that $\tau$ maps to the point $(3\sqrt[3]{3},4)$ under the modular ...
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      Non-vanishing Taylor coefficients and Poincaré series

      I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had. The table is found in the book "The 1-2-3 of Modular Forms" by Jan ...
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      votes
      1answer
      128 views

      One more generator needed for a Z/6 elliptic curve

      I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in https://arxiv.org/pdf/1503.03667.pdf. Meanwhile, I came across a curve generated by $t=629/...
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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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      43 views

      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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      2answers
      291 views

      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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      42 views

      Is there a non-singular irreducible genus $2$ curve $C/\mathbb{F}_p$ and two $\mathbb{F}_p$-coverings $C \to E$, $C \to E^{(1)}$ of some degree $n$?

      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 = ...
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      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,...
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      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 \...
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      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 ...
      4
      votes
      1answer
      241 views

      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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