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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.
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Torsion points with exactly one singular prime (on elliptic curves)
The status of your "implicit assumption" isn't quite clear -- do you want all primes where $E$ has bad reduction to be in $S_P$, or only a subset of them?
If you take E to be Cremona's 27a1, $y^2 + y …
- 37k
4
votes
Accepted
Can we find a set of elliptic curves over rationals associated with $f$?.
This has already had some votes to close, but I'll see if I can answer it anyway...
The answer is "no". There are lots of motivic L-functions that are not elliptic curve L-functions, just because the …
- 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
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
8
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0
answers
1k
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Torsion points of CM elliptic curves
Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.
Th …
- 37k
9
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2
answers
715
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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
24
votes
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Supersingular elliptic curves over $\mathbb{Q}$
In fact, this cannot happen: an elliptic curve over $\mathbb{Q}_p$ is supersingular if and only if its associated mod $p$ Galois representation is irreducible, but if it is irreducible as a representa …
- 37k
17
votes
Accepted
Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?
Serre has shown that there exists a complementary subspace invariant under the Lie algebra $\mathfrak{g}$ if and only if E has complex multiplication. Otherwise the image of Galois is open in the Bore …
- 37k
8
votes
The existence of an elliptic curve with a specific Galois representation induced by a character
In this context, if $\rho$ is a mod $\ell$ representation of $Gal(\overline{F} / F)$, and $A$ is an elliptic curve over an extension $F' / F$, then the statement "$A[\ell] \cong \rho$" needs a little …
- 37k
16
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Growth of Coefficients of cusp forms
It's worth distinguishing between the prime coefficients $a_p$, and the coefficients $a_n$ for general $n$. Let's look at $a_p$ first.
Firstly: for elliptic curves, it is fairly easy and elementary t …
- 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
6
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Accepted
What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}...
Theorem 14.4 of Kato's paper in Asterisque 295 (2004), on page 236, says:
Let $A$ be an abelian variety over $\mathbf{Q}$ such that there is a surjective homomorphism $J_1(N) \to A$ for some integ …
- 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
18
votes
Accepted
Extensions of the modularity theorem
Yes, this is a very active area -- one of the major themes of current research in number theory.
Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than …
- 37k
2
votes
Accepted
How do you calculate the Euler factors of the imprimitive symmetric square at primes with ba...
The factor $D_r$ is easy to compute (much easier than $\mathcal{D}_r$). Basically, you just need to find the eigenvalues $\lambda_i$ of Frobenius on $H^1_\ell(E)^{I_r}$ (i.e. the reciprocal roots of t …
- 37k