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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.
86
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answers
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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
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
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
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
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
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
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
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
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
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