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Results tagged with elliptic-curves
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user 5513
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.
3
votes
Example of a diophantine application of an open image theorem
Well, this isn't explicitly diophantine, but here goes:
If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation
$\rho_f:G_{\mathbb{Q}} …
- 4,790
3
votes
The significance of modularity for all Galois representations
Proving modularity of finite image Galois representations seems to be the most feasible way of proving the Artin conjecture. In fact, this was one of Langlands' original motivations.
- 4,790
7
votes
Isogenous elliptic curve with integral j-invariant
This will never be the case.
By the criterion of Neron-Ogg-Shafarevich, an elliptic curve has good reduction if and only if its $\ell$-adic Tate module is unramified for $\ell\neq p$, where $p$ is t …
- 4,790