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Results tagged with galois-representations
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user 1384
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.
49
votes
Accepted
Are there Maass forms where the expected Galois representation is $\ell$-adic?
Here's some piece of the bigger picture. Maass forms and holomorphic modular forms are both automorphic representations for $GL(2)$ over the rationals. An automorphic representation is a typically hug …
- 41.4k
5
votes
Quotients of Tate modules
Although this question isn't really well-defined (you'd surely need to be more precise about the word "canonical" in the comment under the question) let me make two comments which hopefully put this t …
- 41.4k
8
votes
Number of modular lifts with prescribed parameters
I can give you a "formula" in the sense that I can give you an algorithm to compute the number in any given case. If $\ell\not=p$ is prime then an old result of Carayol and Livn\'e says that the condu …
- 41.4k
5
votes
Image of complex conjugation by modular representations in characteristic 2
Joel -- it's difficult to work out what you're asking. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if y …
- 41.4k
17
votes
Accepted
Why is there a weight 2 modular form congruent to any modular form
By "level $\ell$" I assume you mean "level $\Gamma_1(\ell)$".
Here's a proof. By the Eichler-Shimura theorem, the system of eigenvalues associated to the modular form shows up in $H^1(SL(2,\mathbf{Z} …
- 41.4k
2
votes
Companion forms
It might all depend on precisely what you mean by Serre's conjecture. Various versions are in print. Serre's original conjecture stayed away from $k=1$ and K-W resolved this version of the conjecture …
- 41.4k
4
votes
Semisimple Weil-Deligne representations
Is this really a sensible question to ask, I wonder?
Here's a guess as to what the answer might look like. The Weil-Deligne group comes in three pieces. First there's inertia. Then there's a copy of …
- 41.4k
86
votes
8
answers
13k
views
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$...
Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader kno …
- 41.4k
15
votes
Accepted
Galois representations attached to newforms
The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forg …
- 41.4k
3
votes
Does Ribet's level lowering theorem hold for prime powers?
If you put yourself in a position where an R=T theorem holds at level N/p (e.g.E[ell] irreducible, big image, ell>2), then you'll get a map from a Hecke algebra at level N/p to Z/ell^nZ. But in genera …
- 41.4k
10
votes
Accepted
Level raising by prime powers
Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial.
Let me also assume ell isn't p.
If the form g is new at p, and has level Gamma0(p^3) at p, then …
- 41.4k