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Results tagged with elliptic-curves
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user 121
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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Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in c...
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves o …
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22
votes
What does "supersingular" mean?
There are many equivalent ways to define supersingularity for an elliptic curve over a characteristic $p$ field. One of them is that the $p$-torsion of the curve is connected, i.e., it is a purely in …
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Elliptic Curves, Lattices, Lie Algebras
No, for any cubic curve in the plane, there is a family of cubic plane curves (3 or 4 dimensional - I forget Edit: 8-dimensional, with a transitive action of PGL3) that are isomorphic as curves. For …
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Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in c...
This will be a summary of the proof referenced by BCnrd in the comments, for the benefit of those that haven't looked through Katz, Mazur, Arithmetic Moduli of Elliptic Curves. A scan is available as …
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Accepted
supersingular elliptic curve in char. 2 or 3
If you have a supersingular curve in general, the $\ell$-adic cohomology is a faithful representation of the endomorphism ring, which is a quaternion order that splits over $\mathbb{Q}_\ell$. In the …
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7
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Ways to characterize supersingular primes?
Supersingular primes are those primes p for which all supersingular elliptic curves over an algebraic closure of Fp have j-invariant in Fp. There is a theorem of Deuring that implies the j-invariant …
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7
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Elliptic curves — general structure of the group
If you don't specify more about the structure of the field $K$, then we can't say much about the structure of the group $E(K)$. There are special cases (described in the Wikipedia article):
If $K$ i …
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7
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Supersingular elliptic curves
This is a theorem of Deuring, 1941. David alluded to this, but Section 5.3 of Silverman's The Arithmetic of Elliptic Curves has a proof that 5 conditions concerning elliptic curves over a characteris …
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6
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Regulators of Number fields and Elliptic Curves
I don't think the two regulators are so different. Both the number field regulator and the elliptic curve regulator measure the volume of a parallelopiped. In both cases, the shape in question is th …
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6
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Accepted
Modular curve parametrizing two cyclic subgroups of an elliptic curve
$Y_0(M,N)$ can be reinterpreted as the moduli space of diagrams $E_1 \to E \leftarrow E_2$ of elliptic curves, where the arrows are cyclic isogenies of degree $M$ and $N$. From this viewpoint, it is …
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4
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Elliptic curve over spectra?
Jacob Lurie has notes on this question: (pdf). The short answer is "an oriented spectral (or derived) elliptic curve." A more primitive answer is given by an elliptic spectrum (due to Hopkins and Mi …
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4
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Mystery of the Monstrous Moonshine
I can give you half of the answer, but the other half is wide open. I will use the characterization of supersingular primes as those primes p for which the normalizer of Gamma0(p) in SL(2,R) acts on …
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2
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Accepted
Where do the product expansions of modular forms come from?
Many "natural" examples of automorphic infinite products (also known as Borcherds products) can be explained using the singular theta lift of Harvey-Moore and Borcherds. These examples have the prope …
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2
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Logarithmic structures on moduli of elliptic curves over Z
If you're working away from the primes dividing the level, your curves have semi-stable reduction, and have canonical log-smooth log structures. For any pair (X,D), where X is smooth and D is a divis …
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2
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Tate uniformization of nonsplit semistable elliptic curves
Most of it makes sense. Elliptic curves with non-split reduction can be analytically uniformized by the norm torus. There is a "nice" picture of this using the Berkovich spectrum for a non-split tor …
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