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Results tagged with elliptic-curves
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user 1384
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.
22
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1
answer
749
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Low-level proof of identity related to Weierstrass P-function
A theorem which can be extracted from Theorem V.1.1 of Silverman's "advanced topics in the theory of elliptic curves" is the following. Here $\mathbb{Q}(u)$ denotes rational functions in a variable $u …
- 41.4k
2
votes
Accepted
Isogeny from kernel in higher dimensional abelian varieties
This question seems a bit confused.
If $D$ is an arbitrary point in the Jacobian then one cannot construct an isogeny with kernel the subgroup generated by $D$ -- as this subgroup is typically infini …
- 41.4k
9
votes
What is the smallest positive integer for which the congruent number problem is unsolved?
Kazuo Matsuno writes (personal communication):
"I verified (10 years ago) by using mwrank and magma that E_N:y^2=x^3-N^2x
has a non-torsion point if N<=10^6 is congruent to 1,2,3 modulo 8 and
the ana …
- 41.4k
17
votes
2
answers
2k
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What is the smallest positive integer for which the congruent number problem is unsolved?
The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History …
- 41.4k
18
votes
Why does the definition of modularity demand weight 2?
There has been a lot written already about this question. but here is a simple answer. The Hodge--Tate weights of the Tate module of an elliptic curve are 0 and 1. The Hodge--Tate weights of the Galoi …
- 41.4k
27
votes
2
answers
2k
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How to explicitly compute lifting of points from an elliptic curve to a modular curve?
Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ther …
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22
votes
2
answers
2k
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unboundedness of number of integral points on elliptic curves?
If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and th …
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2
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Is it possible for the repeated doubling of a non torsion point of an elliptic curve stays b...
EDIT: this answer is wrong. I misread the question as looking at the group generated by P, not the points obtained by repeated doubling. I would be OK if the subset of S^1 generated by taking a non-to …
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15
votes
1
answer
843
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components of E[p], E universal in char p.
I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his construct …
- 41.4k
10
votes
How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstras...
http://en.wikipedia.org/wiki/Division_polynomials
That's not the best wikipedia page. "The division polynomials form an elliptic divisibility sequence." is mentioned well before the far more importan …
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9
votes
Accepted
How to find all integer points on an elliptic curve?
Finding all the integral points on an elliptic curve is a non-trivial computational problem. You say you are a "non-professional" so here is a non-professional answer: get hold of some mathematical so …
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3
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Does Ribet's level lowering theorem hold for prime powers?
If you put yourself in a position where an R=T theorem holds at level N/p (e.g.E[ell] irreducible, big image, ell>2), then you'll get a map from a Hecke algebra at level N/p to Z/ell^nZ. But in genera …
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11
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The class number formula, the BSD conjecture, and the Kronecker limit formula
Not a great answer, but some comments that hopefully push in the right direction.
For a number field $K$, there is naturally a finite-dimensional complex vector space associated to it, namely the spa …
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19
votes
1
answer
1k
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Are Q-curves now known to be modular?
I really should know the answer to this, but I don't, so I'll ask here.
A Q-curve is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is modular if it's isoge …
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