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Results tagged with elliptic-curves
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user 4180
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.
4
votes
Structure of $E(Q_p)$ for elliptic curves with anomalous reduction modulo $p$
Hi, this is only a partial answer to (3), that was too long to be a comment.
In a recent post of mine, Felipe Voloch pointed out a very useful tameness criterion proved by Gross ("A tameness criterio …
- 1,694
3
votes
Tate module of CM elliptic curves
You can find the answer to your question (and learn a whole lot more about complex multiplication) in another book by Joe Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves". See Chapter …
- 1,694
12
votes
Why does the definition of modularity demand weight 2?
The reason why the weight needs to be $k=2$ becomes clear when you consider this (equivalent) version of modularity: $E/\mathbb{Q}$ is modular if there there exists a normalized newform $f$ for $\Gamm …
- 1,694
8
votes
understanding the main theorem of complex multiplication (of elliptic curves)
I would suggest the following reference: Karl Rubin's "Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer". In particular, in Section 5 he states the main theo …
- 1,694
6
votes
Class Field Theory for Imaginary Quadratic Fields
Hello,
In general $K(j,E[c])$ will not be abelian over $K$ (the reason being that $K(j,h(E[c]))$ is the ray class field of $K$ of conductor $c$, therefore maximal for abelian extensions of conductor …
- 1,694
5
votes
0
answers
697
views
Formal groups in the supersingular reduction case
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us …
- 1,694
5
votes
1
answer
371
views
Inertia subgroup in the ordinary reduction case when $p=2$
Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let $\rho:\operatorname{Gal}(\overline{K}/K …
- 1,694
8
votes
Modular curves of genus zero and normal forms for elliptic curves
Hello,
I believe the following results that appear in papers of Rubin and Silverberg can be very useful here. Let $N=3,4,$ or $5$ and let $Y_N$ be the (non-compact) modular curve over $\mathbb{Q}$ wh …
- 1,694
12
votes
1
answer
1k
views
Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper ``Propriétés galoisiennes des points d'o …
- 1,694
2
votes
Proofs of Mordell-Weil theorem
Already mentioned: Silverman and Tate's "Rational Points on Elliptic Curves" (undergraduate level) and Silverman's "The Arithmetic of Elliptic Curves" (graduate level).
Another text at the undergradu …
- 1,694
4
votes
Points of elliptic curves over cyclotomic extensions
As for the torsion subgroup, Michael Chou has classified the possible subgroups that may occur as $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$ for an elliptic curve $E/\mathbb{Q}$. You can find a preprin …
- 1,694
9
votes
0
answers
586
views
Tameness criterion in the reducible case
Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely s …
- 1,694