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Results tagged with elliptic-curves
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user 2481
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
Finiteness and bounds for elliptic curves realizing a given galois representation
The set $\mathcal{L}_{\rho}$ is either empty, or a singleton finite. This follows from Faltings' isogeny theorem, which states that for any two elliptic curves (or, more generally, abelian varieties) …
- 37k
6
votes
Accepted
Evidence for the equivariant BSD conjecture with higher multiplicity
You might want to study the work of Darmon--Lauder--Rotger, notably this paper: https://web.mat.upc.edu/victor.rotger/docs/DLR1.pdf
They study cases of the equivariant BSD conjecture where $\rho$ is a …
- 37k
10
votes
Accepted
Definition of modular curve associated to $\Gamma(N)$
This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).
Your $S(N)$ is naturally a scheme over $\mathbb …
- 37k
2
votes
Accepted
Why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$ ? ($\psi$ is Hecke character of ellipt...
This is essentially the same as your other recent CM-theory question, in a mild disguise; for both questions the point is that $\psi(I)$ is a generator of $I$. This follows easily from the fact that $ …
- 37k
9
votes
Accepted
Non-modular elliptic curves
It is a widely believed conjecture that all elliptic curves, over any number field $K$, are modular (in the sense that there exists an automorphic representation [*] $\pi$ of $\operatorname{GL}_2 / K$ …
- 37k
11
votes
Accepted
Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathc...
Here is an example.
Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C_2 \times C_2$ and its Hilbert class field is $H_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $ …
- 37k
4
votes
Accepted
Rationalizing and minimizing elliptic curve coefficients
If I understand correctly: you are starting with $a = 10656\sqrt{-23} +13600$, and you want to choose $f$ in the same number field such that $af^2$ is "as nice as possible", ideally lying in $\mathbf{ …
- 37k
5
votes
Frobenius actions on de Rham cohomology of ordinary elliptic curves
It's important to be clear that this map on $H^1_{\mathrm{dR}}$ overlies a highly non-trivial map on the base-ring $R$. You can imagine a case where $R$ is something like $\mathbf{Z}_p\langle X \rangl …
- 37k
7
votes
Accepted
Global section of vertical differential 1 forms on universal elliptic curve
What you're looking for is a section of the sheaf $\omega = \pi_* \Omega^1_{E/B}$, where $\pi: E \to B$ is the structure map, which is a line bundle on $B$. This line bundle $\omega$ has a canonical …
- 37k
23
votes
Accepted
Quaternionic and octonionic analogues of the Basel problem
This isn't really a full answer, but it's too long for a comment, and perhaps it's informative all the same.
Your sum $S_k[\mathcal{O}]$ can be written as the value at $s = k$ of the sum
$$\sum_{0 \ne …
- 37k
9
votes
2
answers
715
views
Q-curves and twisting
An elliptic curve $E$ over $\overline{\mathbb{Q}}$ is called a $\mathbb{Q}$-curve if it is isogenous (over $\overline{\mathbb{Q}}$) to all its Galois conjugates -- see Are Q-curves now known to be mod …
- 37k
4
votes
Accepted
Modular symbols associated to Rankin Selberg convolutions and the symmetric square
I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of th …
- 37k
5
votes
Accepted
Vanishing of L-function of elliptic curve over $\mathbb{Q}$
Just to pick up on something mentioned in @MyNinthAccount's answer:
If you just want to determine whether or not $L(E, 1) = 0$, then there is another approach, which doesn't involve computing $L(E, …
- 37k
6
votes
Accepted
Existence of newforms which are non-ordinary at a given prime
Given p and k, it's clear we can find a CM-type newform of weight k and some level which is supersingular at p (just choose an imaginary quadratic field in which p is inert, and some sufficiently larg …
- 37k
14
votes
Accepted
BSD conjecture for rank 1 elliptic curves
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other techni …
- 37k