All Questions
11 questions
2
votes
1
answer
147
views
Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
1
vote
0
answers
76
views
Global minimal discriminants of elliptic curves and Galois representations
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if ...
- 1,428
4
votes
1
answer
412
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
user141691
8
votes
2
answers
403
views
Mazur's Question on Mod $N$ Galois representations
In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\...
- 901
3
votes
1
answer
223
views
Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms
I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
- 2,227
11
votes
1
answer
762
views
Eichler-Shimura congruence
I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T_p$ mod $p$ seem to be:
A) ...
- 113
10
votes
2
answers
961
views
What is a(n algebro-geometric) family of modular forms?
We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
- 15.4k
9
votes
0
answers
596
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 ...
- 1,694
5
votes
2
answers
944
views
Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$
Characteristic polynomials of Hecke operators $T_\ell$, with $\ell$ prime, acting on cusp forms $S_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).
This can be ...
- 2,406
6
votes
2
answers
590
views
Intersection of field extensions of torsion points of non-isogenous elliptic curves
Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\...
- 1,905
8
votes
1
answer
470
views
Is there an R=T type result for modular forms with additive reduction?
Let E be an elliptic curve over the rationals with conductor $Mp^2$ with p>5 and M and p coprime, and let $\rho$ be the Galois representation attached to the p-torsion points of E. Is there a way to ...
- 818