Weak abelian direct summands and irreducibility of Galois representations
Let $$\rho _\ell $$be a semisimple $$\ell $$-adic representation of a number field K that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $$\rho _\ell $$and completely characterize them, for example, if the algebraic monodromy of $$\rho _\ell $$is co...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
19 August 2025
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| In: |
Mathematische Annalen
Year: 2025, Volume: 393, Pages: 543-569 |
| ISSN: | 1432-1807 |
| DOI: | 10.1007/s00208-025-03252-0 |
| Online Access: | Verlag, kostenfrei, Volltext: https://doi.org/10.1007/s00208-025-03252-0
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| Author Notes: | Gebhard Böckle, Chun-Yin Hui |
MARC
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| 520 | |a Let $$\rho _\ell $$be a semisimple $$\ell $$-adic representation of a number field K that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $$\rho _\ell $$and completely characterize them, for example, if the algebraic monodromy of $$\rho _\ell $$is connected. If $$\rho _\ell $$is in addition E-rational for some number field E, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when K is totally real and $$\rho _\ell $$is the three-dimensional $$\ell $$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $$\pi $$of $$\textrm{GL}_3(\mathbb {A}_K)$$together with an isomorphism $$\mathbb {C}\simeq {\overline{\mathbb {Q}}}_\ell $$, we prove that $$\rho _\ell $$is irreducible. We deduce in this case also some $$\ell $$-adic Hodge theoretic properties of $$\rho _\ell $$if $$\ell $$belongs to a Dirichlet density one set of primes. | ||
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