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Results tagged with elliptic-curves
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user 2284
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
votes
Accepted
Irreducibility of residual Galois representations attached to an elliptic curve
As Will Sawin points out, the following is mostly an answer to the question "Is it true that the set of primes such that $\rho_p$ is reducible is finite?" which is related but strictly stronger than t …
- 10.9k
3
votes
Accepted
Does $\mu=0$ for an imaginary quadratic field $K$ imply $\mu=0$ for $\mathbf{Q}$?
Under your setting
\begin{equation}
\operatorname{Sel}(E/K_{\infty})\simeq\operatorname{Sel}(E/\mathbb Q_{\infty})\oplus\operatorname{Sel}(E\otimes\chi/\mathbb Q_{\infty})
\end{equation}
where $\chi$ …
- 10.9k
5
votes
Decomposition of Tate-Shafarevich groups in field extensions
First of all, I am not sure I fully agree with the notion that Tamagawa numbers are harmless factors.
What you wish for exists, and here is roughly why. The Birch and Swinnerton-Dyer conjecture is a …
- 10.9k
5
votes
Accepted
Existence of congruences between modular forms / elliptic curves
Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This …
- 10.9k
2
votes
$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
Going in the other direction, the Néron-Ogg-Shafarevich criterion and Weil pairing imply that the Tate module $T_{\ell}E$ is a Galois representation which is ramified at $p$. So if $n$ is large enough …
- 10.9k
12
votes
Accepted
Converse to Modularity I: weight 2 newforms
This should be a comment, but is getting too long.
First, unless you are using the strange convention that Galois representations have by definition complex coefficients, odd, irreducible 2-dimension …
- 10.9k
4
votes
Can you show rank E(Q) = 1 exactly for infinitely many elliptic curves E over Q without usin...
If $K$ is totally real, then the rank part of BSD is known in analytic rank 1 (using CM points for instance) for modular elliptic curves. As potential modularity is also known under the same hypothese …
- 10.9k
7
votes
Is Galois representation induced by semistable elliptic curve semistable?
A Galois representation $\rho_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ can be semistable (technically $B_{st}$-admissible i …
- 10.9k
1
vote
Relation between the Selmer group and the ideal class group
Franz Lemmermeyer's answer points out to a connection. Let me point out in the converse direction that believable heuristics suggest that if $E(\mathbb Q)=0$, then $\operatorname{Sel}_p(E)$ should be …
- 10.9k
4
votes
Accepted
Some questions related to Iwasawa invariants of elliptic curves
1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z_{p}$-extensions. There is only one $\mathbb Z_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more g …
- 10.9k
9
votes
Accepted
Neron models and ramification
This result (and much more) is in Exposé IX Modèles de Néron et monodromie by A.Grothendieck (in SGA7), and more precisely in section 11.1. In particular what you want is exactly Proposition 11.2 ther …
- 10.9k
6
votes
Are Kato's zeta elements integral?
First, let us assume that $p$ is odd for safety.
Then I think your impression is correct: the first statement of 12.5 (4), i.e the integrality of the module generated by the $\mathbf{z}^{(p)}_{\gamma …
- 10.9k
4
votes
Accepted
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at ...
I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$.
In full generality, I believe you are right that none of the properties you state are known for all elliptic …
- 10.9k
5
votes
Accepted
Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its re...
I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous.
First of all, I'm assuming that $E$ has good ordinary reduction (otherwis …
- 10.9k
5
votes
Accepted
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
I don't see any contradiction: the Selmer group also has a contribution of rational points. Indeed, the group of 2-torsion rational points on this elliptic curve is isomorphic to $\mathbb Z/2\mathbb Z …
- 10.9k