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Results tagged with elliptic-curves
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user 26522
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.
6
votes
Order of torsion group
Mazur's theorem is not needed here, and besides, it applies only to elliptic curves themselves defined over $\mathbb{Q}$ rather than the said compositum $K^{(2)}$ of all quadratic extensions of $\math …
- 13.8k
4
votes
1
answer
360
views
The elliptic Lehmer problem for several independent algebraic points
The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then …
- 13.8k
4
votes
Accepted
Orders of reductions of rational points on elliptic curves
For example, is it known that there is an infinite sequence of rational primes $p_i$ and primes $P_i$ of (the ring of integers of) $K$ such that $p_i$ divides the order of the reduction of $x$ modu …
- 13.8k
3
votes
0
answers
202
views
Lang's height conjecture over $\mathbb{F}_q(T)$?
Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or perh …
- 13.8k
7
votes
Integral points on a particular family of curves
Erdos and Selfridge have proved that the product of two or more consecutive non-zero integers is never a power (Illinois J. Math, vol. 19, no. 2, 1975). This implies in particular that for $n > 1$ a p …
- 13.8k
9
votes
What is the chromatic number of the "conic hypergraph" on a non-singular plane cubic?
You mean the six points to be distinct, of course (or not all six points to be the same point).
Fixing the analytic identification $(\wp(z),\wp'(z))$ with $T = \mathbb{C}/\mathbb{\Lambda}$, the Abel- …
- 13.8k
7
votes
CM $j$-invariants in $p$-adic fields
All accumulation points of $J_p$ in $\mathbb{C}_p$ are roots of degree two monic equations over $\mathbb{Z}_p$, and their approximants are necessarily supersingular at $p$. Moreover, there exist accum …
- 13.8k
10
votes
Points of elliptic curves over cyclotomic extensions
Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer.
Amoroso and Dvornicich discovered (A lower bound on the height in abelian ext …
- 13.8k