Topics on Modular Galois Representations Modulo Prime Powers

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.

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Galois representations and Hecke operators associated with the mod- $p$ cohomology of $GL(m(p-1),\mathbb{Z})$

Mathematische Zeitschrift ◽

10.1007/pl00004399 ◽

1998 ◽

Vol 227 (4) ◽

pp. 685-703 ◽

Cited By ~ 1

Author(s):

Avner Ash ◽

Rajesh Manjrekar

Keyword(s):

Galois Representations ◽

Hecke Operators ◽

Mod P

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Computing Galois representations of modular abelian surfaces

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000205 ◽

2014 ◽

Vol 17 (A) ◽

pp. 36-48 ◽

Cited By ~ 1

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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