# 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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### Reduction types of punctured elliptic curves

Let $E$ be an elliptic curve over a mixed-characteristic local field $K$ with split multiplication reduction. Then the Kodaira symbol of the reduction type of $E$ would be $I_n$, where $n=v_K(j(E))$, ...
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### Ranks of elliptic curves over cubic fields

We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations. D. Jeon,...
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### Explicit equations for the universal vector extension of an elliptic curve

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
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### Torsion of an elliptic curve injects under reduction - question

Let $E/K$ be an elliptic curve over a number field. I am interested in the folowing statement: the map $E(K)[m]\rightarrow \tilde E_v(\tilde k_v)$ is injective for any place of $K$ provided there is ...
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### Finiteness of elliptic curves with trivial conductor over function fields

Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
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### Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$

Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
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### How fast can elliptic curve rank grow in towers of number fields?

Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
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### One unexpected observation related to algebraic curves and their Jacobians

Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
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