All Questions
Tagged with galois-representations elliptic-curves
73 questions
4
votes
0
answers
127
views
mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
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 $...
8
votes
1
answer
363
views
Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
- 83
5
votes
1
answer
253
views
Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$
Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$.
Here is my ...
- 622
2
votes
0
answers
60
views
Local property of residual representations attached to elliptic curves over rational numbers
I found the following claim - without reference - in the (famous) book ''Modular Forms and Fermat’s Last Theorem'':
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. Let $\Delta_E$ be the ...
- 1,428
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
7
votes
0
answers
377
views
Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
- 20.4k
2
votes
0
answers
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
1
vote
0
answers
107
views
Ramification of mod $\ell$ representation of elliptic curves [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers.
Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
- 622
0
votes
0
answers
89
views
Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...
- 637
2
votes
1
answer
401
views
Irreducibility of Tate module (as a Galois representation) of elliptic curves with good reduction
This question is following the previous question.
Definitions:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote ...
- 775
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
2
votes
1
answer
410
views
Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
- 2,907
2
votes
0
answers
240
views
Inertia group representation from $p^{n}$-torsion of ordinary elliptic curve
Let $K$ be a complete local field. Suppose that $K$ is an unramified extension of $\mathbb{Q}_{p}$ and let $E$ be an elliptic curve over $K$ with good ordinary reduction. Let $G_{K}=\text{Gal}(\...
- 305
2
votes
1
answer
368
views
Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan
I am trying to understand the following. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication given by the ring of integers $\mathcal{O}_K$. We are given a fixed rational prime $p$ ...
- 637
4
votes
0
answers
214
views
Effect of the surjectivity of Galois representation
Let $K$ be any arbitrary number field and $E$ be any elliptic curve over it. For any integer $m,$ consider the well-known Galois representation
$$\rho:\text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{...
- 521
2
votes
1
answer
159
views
Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
- 309
2
votes
0
answers
230
views
Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
In Siksek's notes The modular approach to Diophantine equations he uses the following result:
Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for ...
- 131
3
votes
0
answers
106
views
A uniform version of Bashmakov's theorem for elliptic curves
Let $E/\mathbb Q$ be an elliptic curve. Serre's open image theorem is the statement that the image of the Galois group $G_{\mathbb Q}$ into $GL_2(\mathbb Z/n\mathbb Z)$ by it's action on the torsion ...
- 7,746
0
votes
0
answers
75
views
An invariant subspace under $G_Q$ action , and BSD-rank
Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
- 547
6
votes
0
answers
230
views
Modularity switching for primes $p>7$
In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,...
- 311
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
0
votes
0
answers
214
views
Elliptic curves with the same Galois representation
Fix a prime $p$. If two elliptic curves over $\mathbb{Q}$ have the same p-adic Galois representation, then what relatinships do we know between them? Any references are welcome.
- 806
4
votes
1
answer
388
views
Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?
I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{...
user130124
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
16
votes
5
answers
689
views
Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?
I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.
Let $n>1$ be an integer. If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) ...
- 1,499
2
votes
0
answers
222
views
Exact ramification information of mod $p$ Galois representation
Let $E$ be an elliptic curve over a number field $F$. As usual, let
$\bar{\rho}:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(E[p])$ be the mod $p$ Galois representation associated to $E$. It is known ...
- 1,428
3
votes
0
answers
301
views
Is $E[p]$ always irreducible for an elliptic curve $E$ with supersingular reduction at an odd prime $p$ $?$
Let $E$ be an elliptic curve defined over a number field $F$ with supersingular reduction at an odd prime $p$. Let $E[p]$ denotes the set of $p$-torsion points of $E$ over an algebraic closure of $F$. ...
- 303
4
votes
1
answer
823
views
Frobenius at ramified primes
Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...
- 1,422
8
votes
0
answers
329
views
Elliptic curves and the $\ell$-adic image of the decomposition group
Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the ...
- 10.5k
11
votes
1
answer
820
views
Galois representations for the curve $y^{2} = x^{3} - x$
Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T_{2}(...
- 1,298
6
votes
1
answer
587
views
realizing uniform boundedness of Galois representations associated to elliptic curves
This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an ...
- 1,298
4
votes
0
answers
339
views
Minimal discriminant of an elliptic curve in terms of its Galois representation
From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...
- 871
7
votes
1
answer
646
views
Serre's surjective theorem importance
I'm studying Serre's paper in wich he shows the following theorem:
Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar K/K)\...
- 205
13
votes
2
answers
781
views
Elliptic curves and supercuspidal representations of conductor $p^2$
Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
- 20.8k
4
votes
2
answers
682
views
n torsion groups of quadratic twists of elliptic curves
If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
15
votes
1
answer
1k
views
What are the strongest conjectured uniform versions of Serre's Open Image Theorem?
This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
- 1,499
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
7
votes
0
answers
267
views
Invariant obstructions to gluing Galois representations on elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...
- 148k
5
votes
0
answers
721
views
The Modularity Theorem and Serre's/Faltings's Isogeny Theorem
Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. ...
- 59
3
votes
2
answers
752
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
Q:...
- 101
5
votes
1
answer
755
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) = {\...
- 101
3
votes
1
answer
437
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 ...
- 193
7
votes
1
answer
914
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 ...
- 945
7
votes
1
answer
440
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 $GL_2(\hat{\mathbb{...
- 71
4
votes
1
answer
491
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 $\overline{\mathbb{Q}}$ ...
- 1,499
13
votes
2
answers
1k
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 $\...
- 26k
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
6
votes
1
answer
355
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
...
- 7,062