All Questions
Tagged with galois-representations elliptic-curves
73 questions
13
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0
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Why doesn't functoriality immediately imply the modularity theorem?
Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
- 7,062
6
votes
3
answers
486
views
Example of a diophantine application of an open image theorem
I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model ...
- 1,905
3
votes
3
answers
557
views
Another question related to the isogeny theorem for elliptic curves
I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer:
"Note also that the situation is ...
- 1,905
9
votes
5
answers
2k
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The significance of modularity for all Galois representations
On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...
- 7,062
31
votes
1
answer
5k
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Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
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11
votes
2
answers
1k
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Serre's Open Image Theorem Without Shafarevich's Theorem
In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ associated to an elliptic ...
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9
votes
0
answers
596
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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
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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
2
votes
1
answer
771
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Serre's open image theorem for products of elliptic curves over function fields via specialization
In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6
′′, p. 325):
Let $K$ be a number field and let
$K^...
- 1,905
6
votes
2
answers
589
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
2
votes
1
answer
555
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How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?
The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square.
Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...
- 821
8
votes
0
answers
832
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Semistable Elliptic Curves and irreducible Galois representations
I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that
$$p>c(E,K)\...
- 313
13
votes
2
answers
1k
views
Are Kato's zeta elements integral?
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms"...
- 37k
18
votes
1
answer
5k
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About isogeny theorem for elliptic curves
$K$ a number field, $G_K$ its Galois group, $E_1, E_2$ two elliptic curves defined over $K$. The isogeny theorem says that if for some prime number $\ell$, The Tate modules (tensored with $\mathbb{Q}$)...
- 1,503
15
votes
2
answers
2k
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Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction.
The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a 2-...
- 1,500
9
votes
1
answer
1k
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Images of action of Galois on the Tate module of Elliptic Curve,
Let E be an elliptic curve over the rationals, and let $TE = \lim_\leftarrow E[n]$ be the Tate module of the elliptic curve. The action of the Galois group of $\bf Q$ gives rise to a representation $\...
- 818
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 ...
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12
votes
3
answers
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What is the etymology for the term conductor?
This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...
- 3,268
7
votes
1
answer
220
views
Cyclic extensions coming from E[p] \equiv F[p],
Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of elliptic curves F/L, ...
- 818
14
votes
2
answers
2k
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Galois theory and rational points on elliptic curves
I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses ...
12
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3
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1k
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Motivation for uniform surjectivity of mod l representations associated to elliptic curves
Background
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $G_{\mathbb{Q}}$ be the absolute Galois group $Aut(\overline{\mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup $E[...
- 10.5k
5
votes
1
answer
836
views
An inverse problem: Number fields attached to elliptic curves over Q
If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/...
- 7,062
11
votes
3
answers
1k
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Does Ribet's level lowering theorem hold for prime powers?
I often use the following theorem (that one can state more generally) in my research.
Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or ...
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