Local constancy in families of non-abelian Galois representations

2000 ◽  
Vol 233 (2) ◽  
pp. 347-363 ◽  
Author(s):  
Mark Kisin
Keyword(s):  
Galois Representations ◽  
Local Constancy

scholarly journals Explicit Calculations of Automorphic Forms for Definite Unitary Groups

2008 ◽  
Vol 11 ◽  
pp. 326-342 ◽  
Author(s):  
David Loeffler
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Level Structure ◽  
Unitary Groups ◽  
3 Dimensional ◽  
Local Constancy

I give an algorithm for computing the full space of automor-phic forms for definite unitary groups over ℚ, and apply this to calculate the automorphic forms of level G(hat{Z}) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic form of weight (3, 3) corresponding to a family of 3-dimensional irreducible ℓ-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.

scholarly journals Independence of algebraic monodromy groups in compatible systems

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Federico Amadio Guidi
Keyword(s):  
Galois Representations ◽  
Global Function ◽  
Irreducible Decomposition ◽  
Global Function Fields ◽  
Normal Varieties ◽  
Independence Result ◽  
Monodromy Groups ◽  
Independence Results ◽  
Lisse Sheaves ◽  
General Method

AbstractIn this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

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