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Results tagged with elliptic-curves
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user 5015
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.
8
votes
Accepted
Cohomology of elliptic curves
More generally let k be a local field of residual characteristic $\ell$ and $E/k$ an elliptic curve with good reduction. Then for any prime $p\neq\ell$, there is a perfect duality between $E(k)/p^n$ a …
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3
votes
Accepted
Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \nmid \infty} H^1\left(K_{q},E_{p...
The answer is "no" in general.
By the definition of the Selmer group, you can replace the target of the map by the product of $E(K_q)/p^n E(K_q)$. Now $E(K)/p^n E(K)$ is a subgroup of the Selmer gro …
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12
votes
Accepted
Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\m...
Let me summarise the comments above that give a full answer (correct me if I am wrong).
The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.
If $r=0$, the fact …
2
votes
For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p...
$\newcommand{\FF}{\mathbb{F}}\DeclareMathOperator{\SL}{SL}$
OK, let me try.
Write $M=E[p]$. If the order of $G$ is coprime to $p$, then $H^1(G,M)=0$. Assume that $p$ divides the order of $G$. Now by …
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0
votes
Accepted
How to conclude good reduction from $\mathcal{P} \nmid m$?
(Not sure this question is suitable, but I reply anyway).
Here is a simple counter-example. Take a cm curve like 27a1 over $\mathbb{Q}$ and adjoin the $2$-torsion points. This will be a sextic extens …
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5
votes
Accepted
What is the exact meaning of the real period in the $p$-adic formulation of BSD?
Mazur-Tate-Teitelbaum have written their p-adic BSD paper for a modular form of even weight $k\geq 2$ so unless we are dealing with a elliptic curve, we might want to avoid choosing a period. They vie …
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6
votes
Accepted
Determining $\mu$-invariant of elliptic curves over $\mathbb{Q}$
At what prime ? Well, let $p$ be an odd prime at which $E$ has good ordinary reduction. There are two ways:
Use sage or magma to compute the analytic $p$-adic L-function via modular symbols. In sage …
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6
votes
Accepted
Isogeny classes and reduction types of elliptic curves at primes of bad reduction
Let $E$ be an elliptic curve over a $p$-adic field $k$. Let $\varphi: E \to E'$ be an isogeny defined over $k$. Write minimal Weierstrass equations with integer coefficients for both curves. Write $\ …
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6
votes
How to get explicit unramified covers of an elliptic curve?
Let me expand Felipe's answer a bit.
Vélu's formulae given in the very readable short paper [1] are very easy to use. For instance if you are given a $n$-torsion point one can immediately write down …
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3
votes
Ramification index and additive reduction of elliptic curves
This minimal ramification index is the order of the Serre-Tate group $\Phi$, defined in their article "Good reduction of abelian varieties". It is shown in the proof of theorem 2 there that $\Phi$ is …
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13
votes
Torsion subgroups in families of twists of elliptic curves
Let me expand my comment above. While we believe that we expect this very very frequently, it is not always the case.
As I commented, we have
$$ E_d(\mathbb{Q})[n]\oplus E(\mathbb{Q})[n] = E(\mathbb …
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3
votes
Accepted
Wiman's method for bounding the rank of an elliptic curve
I have not read all the details of the article, but most of what I see is just descent by the isogeny $[2]$. The map is the Kummer map $$E(\mathbb{Q})\to \mathbb{Q}^{\times}/\square \times \mathbb{Q}^ …
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3
votes
Anomalous elliptic curves over finite rings
The condition $\#E(\mathbb{Z}/n\mathbb{Z}) = n$ is not enough, I believe. You would need that $E(\mathbb{Z}/p\mathbb{Z})$ is cyclic of order $p$ for all prime divisors $p$ of $n$.
By the Chinese Rem …
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8
votes
Example of elliptic curve with CM (complex multiplication) by \sqrt{-7}
I am sure others (magma, etc) can do the same. With sage the following lines will produce the rational functions $\bigl(f(x,y), g(x,y)\bigr)$ representing the multiplication by $(1\pm\sqrt{-7})/2$ on …
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5
votes
Accepted
How is the period of an elliptic curve defined exactly?
The comments above give already the answer, but for the sake of completeness let us be a bit more precise.
Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element i …
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