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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.
7
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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
votes
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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2
votes
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Super-singular reduction at a given prime
This follows from Deuring's work on endomorphism rings (M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272). I couldn't …
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1
vote
A question on degeneration of elliptic curves with actions.
Concerning your general question, any finite order group of translations can be degenerated into an action on some Néron $n$-gon. An $n$-gon is made by taking $n$ copies of $\mathbb{P}^1$, parametriz …
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1
vote
Express Weierstrass' g_2 and g_3 in terms of theta-functions of the periods
The formulas at the Wikipedia article on the J-invariant seem to be correct. At least, the formula $g_2 = \frac{2\pi^4}{3}(\theta_{00}^8 + \theta_{01}^8 + \theta_{10}^8)$ yields $g_2 = 60G_4 = \frac{ …
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2
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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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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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4
votes
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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6
votes
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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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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Choosing tau for elliptic curves over the rational numbers with prescribed ramification data
I am having difficulty making sense of question 1. If you already know $f$, then you have $E$, which gives you an $SL_2(\mathbb{Z})$-orbit of values of $\tau$. This seems to be the best you can do.
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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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11
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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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answers
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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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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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