2
votes
2answers
342 views

Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$? Next ...
6
votes
1answer
341 views

Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by $\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...
3
votes
1answer
150 views

Galois representation attached to $3$-torsion points of an elliptic curve

Let $ E $ - Elliptic curve defined over $ {\mathbb{Q}} $. $G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$. $ E[3] $ - $3$-torsion points ...
7
votes
1answer
354 views

Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult ...
5
votes
0answers
144 views

When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$. Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...
3
votes
1answer
236 views

Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely: Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...
9
votes
2answers
457 views

Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations ...
8
votes
2answers
541 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, ...
6
votes
1answer
242 views

The existence of an elliptic curve with a specific Galois representation induced by a character

In Kevin Buzzard's survey article on potential modularity Buzzard writes: Let us say that we have an elliptic curve $E$ over a totally real field $F$, and we want to prove that $E$ is ...
10
votes
0answers
1k views

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. ...
6
votes
3answers
322 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 ...
3
votes
3answers
430 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 ...
9
votes
5answers
1k views

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 ...
18
votes
1answer
2k views

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 ...
10
votes
1answer
718 views

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 ...
7
votes
0answers
296 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 ...
3
votes
1answer
506 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
votes
1answer
495 views

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 ...
4
votes
2answers
476 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 ...
2
votes
1answer
277 views

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 ...
7
votes
0answers
559 views

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 ...
10
votes
2answers
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 ...
10
votes
1answer
3k views

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 ...
8
votes
2answers
861 views

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 ...
8
votes
1answer
622 views

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 ...
7
votes
1answer
371 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 ...
9
votes
3answers
1k views

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 ...
6
votes
1answer
211 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, ...
10
votes
2answers
1k views

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 ...
10
votes
3answers
833 views

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 ...
3
votes
1answer
627 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 ...
8
votes
3answers
782 views

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 ...