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Results tagged with elliptic-curves
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user 70751
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.
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Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of charac …
- 1,805
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0
answers
248
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Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable imag …
- 37k
5
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0
answers
323
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Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $K/ …
CommunityBot
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0
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Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$...
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ con …
- 4,745
-3
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1
answer
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Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$?
- 156k
6
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1
answer
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Relationship between Tate-Shafarevich group and the BSD conjecture
The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken …
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0
answers
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Existence of infinite or bad places that ramify in $K(p^{-1}E(K))/K$ where $p$ is a prime of...
Let $E$ be an elliptic curve, $K$ a number field so that $\operatorname{rank}_{K}(E)\geq 1,$ $p$ a prime of $\mathbb{Q}$ at which $E$ reduces well.
We know (see for instance Silverman, The Arithmetic …
- 1,805
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Is $(pE(L))^{\operatorname{Gal}(L/K)}/pE(K)=0$ for almost all $p\geq 5$ if $rank(E)\geq 1$, ...
Let $K$ be a number field (possibly of infinite degree over $\mathbb{Q}$) and $E$ an elliptic curve without complex multiplication.
Let $L:= K(E_{5^{\infty}7^{\infty}11^{\infty}...})$ be the field ob …
- 1,805
3
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1
answer
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Selmer and free rank of Elliptic Curves
If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer conj …
- 17.6k
2
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0
answers
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Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension
If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\operato …
- 1,805
3
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1
answer
318
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Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\s...
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-dime …
- 1,805
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Accepted
Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\s...
Pasten has answered the question: Murty (MS1106677, Corollary to Theorem 2) has shown that the quadratic twist of $E$ by a prime $q$ has rank zero for infinitely many primes $q$, if GRH holds.
- 1,805
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2
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Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5}...
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is infini …
CommunityBot
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1
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Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \nmid \infty} H^1\left(K_{q},E_{p...
If $K$ is a number field, $E$ an elliptic curve and $p$ a prime, does the Selmer group
$$H^{1}_{\operatorname{Sel}}\left(K,E_{p^{n}}\right)$$
always inject into
$$\prod_{q \text{ a nonarchimedean prim …
- 8,945
0
votes
0
answers
387
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Conditions for splitting of short exact sequence?
Assume $K$ is a number field and $E$ is an elliptic curve defined over $K$.
Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarro …
- 8,529