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Results tagged with galois-representations
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user 2481
The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
4
votes
Finiteness and bounds for elliptic curves realizing a given galois representation
The set $\mathcal{L}_{\rho}$ is either empty, or a singleton finite. This follows from Faltings' isogeny theorem, which states that for any two elliptic curves (or, more generally, abelian varieties) …
- 37k
6
votes
Accepted
Evidence for the equivariant BSD conjecture with higher multiplicity
You might want to study the work of Darmon--Lauder--Rotger, notably this paper: https://web.mat.upc.edu/victor.rotger/docs/DLR1.pdf
They study cases of the equivariant BSD conjecture where $\rho$ is a …
- 37k
10
votes
Accepted
Understanding absolute Galois group from its representations
The slogan "number theorists aim to understand $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$" is one that gets used a lot, but it's perhaps a tiny little bit misleading.
Understanding the s …
- 37k
2
votes
Families of Galois representations over disks
I don't really understand the setup of this question: what is $R\langle x_1/r_1, \dots \rangle$ supposed to mean if $r_i$ is a real number?
That said, if $R = \mathbb{Z}_p$, then $\mathbb{Z}_p[[x_1, \ …
- 37k
5
votes
Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction
Both statements follow readily once you know that the Frobenius map on $D_{\mathrm{cris}}$ satisfies $\varphi^2 - a_p \varphi + p = 0$. This, in turn, can be deduced from the (much stronger) fact that …
- 37k
6
votes
Accepted
Two different local Langlands parameters for quadratic extension
This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves:
Dembélé, Lassina; Loeffler …
- 37k
3
votes
Accepted
$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.
- 37k
2
votes
Accepted
Understand the $p$-adic local Langlands correspondence with examples
Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rh …
- 37k
8
votes
Accepted
Crystalline when restricted to inertial subgroup
This is purely formal. If $V$ is crystalline, then $V \otimes \mathbf{B}_{\mathrm{cris}}$ has a basis as a $\mathbf{B}_{\mathrm{cris}}$-module in which the action of $G_K$ is trivial. Hence a fortiori …
- 37k
3
votes
Accepted
Trianguline representation
No, triangulations are not in general unique.
A simple way of seeing this is to consider the case when $K = \mathbf{Q}_p$, $V$ is 2-dimensional and crystalline with distinct Hodge–Tate weights, say $\ …
- 37k
2
votes
Accepted
On presentations of universal rings of deformations
Doesn't this kind of prove itself? Pick some elements $\alpha_1, \dots, \alpha_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t_i$ to $\alpha_i$ defines a m …
- 37k
5
votes
Galois representations attached to a cusp form for different primes
At the most basic level, $\rho_p$ and $\rho_q$ are "nothing to do with each other". E.g. the kernels of $\rho_p \bmod p$ and of $\rho_q \bmod q$ cut out two finite Galois extensions of $\mathbf{Q}$ wh …
- 6,039
7
votes
Accepted
A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$
You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL_2(\mathbf{Q}_p)$ and $WD(\mathbf{Q}_p …
- 37k
1
vote
Accepted
Restriction of $(\varphi, N)$-modules
Don't confuse $(\phi, N)$-modules (which are finite-dimensional vector spaces over $\mathbf{Q}_p$ with various extra structures) with $(\phi, \Gamma)$-modules (which are modules over a much bigger and …
- 37k
4
votes
Accepted
Eigenvarieties and functoriality
You have asked a lot of questions at once, and it is impossible to give more than a hint at a small subset of these questions.
I think the general theme here is: the existence of eigenvarieties doesn' …
- 37k