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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.
5
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Questions about elliptic curves with level-$n$ structure
Let $K$ be a finite extension of $\mathbb{Q}_p$ with $p\neq 2$. Suppose that $E/K$ is an elliptic curve with additive reduction and such that $E$ has full $4$-torsion over $K$. By the Kodaira classif …
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7
votes
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About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local ...
The Kodaira symbol characterises the geometry of the special fibre, that is the structure over the algebraically closed field $\bar{\mathbb{F}}_p$. Good reduction means type I${}_0$, multiplicative re …
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4
votes
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Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the con...
The original proof is by Tate in WC groups over $\mathfrak{p}$-adic fields. Nowadays, it is often derived from local Tate duality $H^1\bigl(K_v,E[n]\bigr)\times H^1\bigl(K_v,E[n]\bigr)\to \mathbb{Z}/n …
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2
votes
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Algorithm for computing isogeny class of elliptic curve
Again an early source explaining how this is done in practice is Cremona's book. Specifically section 3.8.
One current implementation for finding the isogeny class of an elliptic curves over a number …
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2
votes
Accepted
Argument for non-existence of elliptic curve over $\mathbb{C}[t, t^{-1}]$
Let me expand that sentence, which may hint at the confusion in this and the subsequent question.
Let $K=\mathbb{C}(t)$ and see the ring $R=\mathbb{C}[t]$ as a Dedekind ring. Let $v$ be a place of $R$ …
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8
votes
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Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$
I fear you wish for too much here.
If $Ш$ is finite, then we can represent each element by a torsor; each torsor has good reduction away from a finite set and the union of all bad places would then be …
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3
votes
Accepted
Difficulties in the proof of finiteness of n-Selmer group using cohomology
(Not sure any of these questions are at the right level for this forum, but here the comments that may help.)
question : Inflation-restriction sequence.
question : The target can be identified with …
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3
votes
Accepted
cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\...
I will assume $p$ is odd to avoid having to think about places at infinity.
Cassels proved more: There is an exact sequence
$$H^1\bigl(F_\Sigma/F,E[p^\infty]\bigr)\longrightarrow \bigoplus_{v\in \Sigm …
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13
votes
Does the number of roots of the modular form associated to an elliptic curve, on the positiv...
Let $Z$ be the number of zeroes of $y\mapsto f(iy)$ for $0<y<\infty$ and let $r$ be the analytic rank of $E$. Then $Z\geqslant r$ and $Z\equiv r \pmod{2}$ as I will explain below. I don't know of an e …
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0
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Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha...
Let $\sigma$ be the non-trivial element of the Galois group of the quadratic extension $L/K$. Let $\phi \colon E \to E_D$ be the isomorphism defined over $L$.
First, if $P \in E(\bar L)$ then $\sigma\ …
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7
votes
Accepted
How fast can elliptic curve rank grow in towers of number fields?
I think it is a good idea to compare the growth of the rank to the degree. I would say that we have excessive growth in an extension $F/K$ if $$\DeclareMathOperator{\rank}{rank}\rank E(F) - \rank E(K) …
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11
votes
Accepted
Cubic twist of elliptic curves and its rank
There is a formula but it involves both cubic twists. Let $E: y^2 = x^3+B$ be an elliptic curve over $\mathbb{Q}$ with $j=0$ as the one in the question. Let $D$ be a cubefree integer. Set $E_1: y^2=x^ …
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4
votes
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Lazard module structure of rings with formal elliptic curve
As far as I understand the question (I know elliptic curves, but I don't know what MU and BP are), the task is to express the coefficients of the Weierstrass equation given the series of multiplicatio …
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5
votes
Is there an elliptic curve over a number field with a point of order 64 and Mordell-Weil ran...
This is not an answer, but an idea how one might get an answer with quite a bit of work. I am not too confident that I have not overlooked something.
Let $p$ be a prime and $k>1$. I am aiming to const …
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11
votes
Accepted
Discrepancy in Magma's calculation and Sage's of elliptic curve?
Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the s …
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