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Results tagged with elliptic-curves
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user 3384
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
Quadratic twist of an elliptic curve given by non-Weierstrass model
Classical invariant theory gives you the answer given by Rene. I add this one only to show how very straightforward it is. We assume the characteristic of the field is not 2 or 3. In this case every e …
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7
votes
Accepted
Two questions on isomorphic elliptic curves
Question 1: Putting both curves in say, Legendre Normal Form (or else appealing the lefschetz principle) shows that if the two curves are isomorphic over $\mathbf{C}$ then they are isomorphic over $\o …
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5
votes
Accepted
Uniform bounds for the order of a rational torsion point on CM elliptic curves
Hey, I probably should have answered this one some time ago. It was proved in 1989 by J.L. Parish that the order of an $H$-rational torsion point is 1,2,3,4 or 6, and this also can be deduced from wor …
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3
votes
elliptic curve with j-invariant T
I exceedingly concur with Emerton on the nontriviality of this problem. The ideas for its solution take up the bulk of chapter 7 in Diamond and Shurman's book on modular forms.
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11
votes
Atkin-Lehner involution and class number
Given that I don't know exactly which relation you're talking about, I'll give you something old and something new:
A priori, asking for a formula for the number of fixed points of Atkin-Lehner is as …
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4
votes
Isogeny classes and elliptic curves over finite fields
If in (2) you are asking whether the conductor entirely determines the number of points on all reductions, the answer is most assuredly not. If that were the case then there would only be one cusp for …
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8
votes
Accepted
Supersingular Elliptic Curves with rational isogeny?
You can't prove it because it is untrue.
Let $E$ be an elliptic curve with CM by $\mathbf{Z}[\sqrt{-p}]$ defined over a number field $K$ which
Contains $\mathbf{Q}(\sqrt{-p})$ so that the action of …
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12
votes
Accepted
Average rank of elliptic curves over $\mathbb{Q}$
Color me surprised if people no longer believe that the distribution of elliptic curves is half rank zero, half rank one, and a density zero subset of higher rank curves. To my knowledge this is still …
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6
votes
Accepted
elliptic curves with and without CM
1)If an elliptic curve has integral $j$-invariant it absolutely DOES NOT NEED to have CM. The class of curves with integral $j$-invariant (let's call that the class of IM Elliptic curves for Integral …
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3
votes
Curves of higher genus
It should be noted that Murabayashi determined that there should be finitely many (whose moduli lie in the rational numbers) for $g=2$ over the complex numbers and (mostly) explicitly determined them. …
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16
votes
Accepted
Isogeny classes of elliptic curves
We say that an elliptic curve $E$ over a number field $K$ is an elliptic $\mathbf{Q}$-curve if it is is isogenous to its Galois conjugates $E^\sigma$. These were first studied by Benedict Gross, but w …
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2
votes
Imaginary quadratic field contained in Hecke orbit field?
"No" for both questions about CM elliptic curves and "I'm not even sure I know what the question would be" about general Shimura varieties.
Basic idea: If $j$ is the $j$-invariant of a CM elliptic cu …
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10
votes
Accepted
The significance of modularity for all Galois representations
Your question reminds me of a current strain of research whose starting point is Serre's conjecture, now the Khare-Wintenberger Theorem:
any continuous odd irreducible two-dimensional Galois repre …
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13
votes
Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?
Thinking about $X_0(N)$ as a bunch of enhanced elliptic curves is a red herring as far as a description of the modular parametrization goes. For instance, you can obtain a similar "modular parametriza …
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8
votes
Accepted
Intersection of Hilbert class fields of imaginary quadratic fields
The generalization of the phenomena you see is genus theory. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j_d)$ then $H$ contains the Genus field $G$.
If $d = \prod_{i=1}^n p_i$ is squarefree (and odd f …
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