# 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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### Pro-representability and the obstruction to deformations of “stable curve of genus one + a section”

I have 2 questions about the theorem III.1.2 of Deligne-Rapoport's "Les shemas de modules de courbes elliptiques". 1. Let $k$ be a field, $\Lambda$ a complete noetherian local ring with the ...
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### Supersingular elliptic curves and their automorphisms

If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology). Do all ...
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### reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$

In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...
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To avoid X-Y problem I am going to write my problem down in detail, so plz bear with me. The elliptic curve over $Q$ given by a Weierstrass equation is - $E := y^2 +a_1 xy +a_3 y = x^3 + a_2 x^2+... 1answer 216 views ### Rank 3 Lagrangian vector bundles on an elliptic curve Let$k$be an algebraically closed field of characteristic zero (feel free to assume$k= \mathbb{C}$) and$E$an elliptic curve over$k$with identity$P \in E(k)$. I am interested in certain ... 2answers 660 views ### How can I see the relation between shtukas and the Langlands conjecture? The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about. Drinfeld ... 1answer 192 views ### The$S$-unit equation for functions on curves Let$X$be a smooth projective connected curve over a number field$k$, and let$S \neq \emptyset$be a finite set of closed points of$X$. The curve$Y = X \setminus S$is affine, and we denote by$R$... 1answer 109 views ### Clarification: Using Hensel's Lemma to determine$K_v$-rational points on a curve From Silverman's AEC page 332: I need to understand why the determination of the following local kernel $$ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big)$$ is straightforward. The ... 0answers 116 views ### The Picard scheme of an ordinary singular curve Let$k$be an algebraically closed field,$C$a proper reduced connected scheme over$k$of dimension 1, whose singularity is at worse ordinary,$\pi : \tilde{C} \to C$the normalization of$C$and$...
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For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square. I start with $y^2=(5cx+100)(5cx-64c+36)$ and ...
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### Indecomposable vector bundles on elliptic curves

Let $X$ be a smooth complex projective curve of genus 3, $E$ an elliptic curve, and $f: X \to E$ a finite map of degree 2. Let $L$ be a line bundle on $X$, and $R^0f_*(L)$ its direct image. Question: ...