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Results tagged with elliptic-curves
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user 5498
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.
6
votes
Accepted
Image of isogeny of elliptic curves determines kernel?
It is possible. Suppose $E$ has complex multiplication defined over a number field $F$ by the ring of integers $\mathcal{O}_K$ in a quadratic imaginary field $K$ (and we take $K$ that only has $\pm 1$ …
- 5,009
6
votes
Root number of a quadratic twist of an elliptic curve
I will prove a stronger
Claim: Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$, let $N_A$ be its conductor, let $D$ be a squarefree integer, let $\psi_D$ be the Dirichlet character ass …
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1
vote
Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C i...
It is a general fact that if $K$ is a number field, $K'/K$ is a quadratic extension, $E \rightarrow \mathrm{Spec}\, K$ is an elliptic curve, and $E' \rightarrow \mathrm{Spec}\, K'$ is its quadratic tw …
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8
votes
2
answers
1k
views
Distance functions on elliptic curves over number fields
My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a numbe …
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15
votes
Accepted
Do there exist elliptic curves over schemes which have all primes as residue characteristics?
Yes, there can. Choose any elliptic curve $E$ over $\mathbb{Q}$ with potential good reduction (for instance, a curve with potential CM) and pass to a number field $K$ over which the reduction is every …
- 5,009
2
votes
Accepted
Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve
This is explained in the proof of Theorem 3 of section 15 of Mumford's "Abelian varieties" (pages 138-140 in the new edition) in the case when the base is an algebraically closed field of characterist …
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13
votes
Accepted
$\lambda$-invariant is constant for isogenous elliptic curves
Given the brevity of the question, I am not sure about the precise setup, so let me assume that you are looking at some $\mathbf{Z}_p$-extension and that you assume (or know in the situation at hand) …
- 5,009
4
votes
Accepted
Rational points and torsion points of CM elliptic curve
See Cor. 5.18 of Rubin "Elliptic curves with complex multiplication" in LNM 1716 for a proof (that uses the main theorem of complex multiplication).
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10
votes
Elliptic curve and Galois representation
None of these conditions implies that $E$ has good reduction at $p$. Consider, for instance, the elliptic curve $E = X_0(11)$, for which $E[5] \cong \mathbb{Z}/5\mathbb{Z} \oplus \mu_5$. Then for $l = …
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6
votes
Elliptic curve E and Galois representation
Both questions have incorrect expectations. This has already been noted for the second question by S. Carnahan in the comments. For the first question, take any elliptic curve for which the semisimpli …
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5
votes
Explicit calculation of Weil Deligne representations
Yes, it is possible. For all this explained clearly and in detail, see David Rohrlich's paper "Elliptic curves and the Weil-Deligne group" along with the accompanying "Student's supplement to "Ellipti …
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