Irreducibility of automorphic Galois representations of low dimensions

Xia, Yuhou

doi:10.1007/s00208-018-1786-5

Cited by 7 publications

(18 citation statements)

References 13 publications

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“…Proposition 1.5 ([Xia19]). Fixing Π and letting p vary, the set of rational primes p such that V P (Π) is irreducible for all P | p has density 1.…”

Section: Setupmentioning

confidence: 93%

Euler Systems and Selmer Bounds for GU(2,1)

Manji¹

2022

Preprint

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We investigate properties of the Euler system associated to certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field E, constructed by Loeffler-Skinner-Zerbes. By adapting Mazur and Rubin's Euler system machinery we prove one divisibility of the "rank 1" Iwasawa main conjecture under some mild hypotheses. When p is split in E we also prove a "rank 0" statement of the main conjecture, bounding a particular Selmer group in terms of a p-adic distribution conjecturally interpolating complex L-values. We then prove descended versions of these results, at integral level, where we bound certain Bloch-Kato Selmer groups. We will also discuss the case where p is inert, which is a work in progress.

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“…Proposition 1.5 ([Xia19]). Fixing Π and letting p vary, the set of rational primes p such that V P (Π) is irreducible for all P | p has density 1.…”

Section: Setupmentioning

confidence: 93%

Euler Systems and Selmer Bounds for GU(2,1)

Manji¹

2022

Preprint

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“…If we fix and let p vary, then [ 30 , Theorem 2] shows that there is a density 1 set of rational primes p such that is irreducible for all (and hence unique up to isomorphism).…”

Section: Mapping To Galois Cohomologymentioning

confidence: 99%

An Euler system for GU(2, 1)

2021

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We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) .

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“…Two impetuses of the problem are the Mumford‐Tate conjecture that concerns the

\lambda

‐independence of

\mathbf {G}_\lambda ^\circ

for motivic compatible systems and the irreducibility conjecture that predicts the absolute irreducibility of

\rho _\lambda

for compatible systems attached to algebraic cuspidal automorphic forms of

\mathrm{GL}_n(\mathbb {A}_K)

. There have been many studies on the irreducibility and big images of compatible system since 1970s, assuming that

\lbrace \rho _\lambda \rbrace _\lambda

is motivic, see [41, 44, 47] for abelian varieties, [13, 18] for

n=3

, and [14, 15] for

n=4

, automorphic, see [17, 42, 43, 53] for (Hilbert) modular forms, [2, 13] for

n=3

, [11, 12, 16, 40, 57] for

n=4

, [4, 25, 58] for

n\leqslant 6

, and [1] in general, and more recently, a weakly compatible system (Definition 2.1), see [1, 35, 38, 39], and [18]. …”

Section: Introductionmentioning

confidence: 99%

Monodromy of four‐dimensional irreducible compatible systems of Q$\mathbb {Q}$

Hui

2023

Bulletin of London Math Soc

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Let 𝐹 be a totally real field and 𝑛 ⩽ 4 a natural number. We study the monodromy groups of any 𝑛-dimensional strictly compatible system {𝜌 𝜆 } 𝜆 of 𝜆-adic representations of 𝐹 with distinct Hodge-Tate numbers such that 𝜌 𝜆 0 is irreducible for some 𝜆 0 . When 𝐹 = ℚ, 𝑛 = 4, and 𝜌 𝜆 0 is fully symplectic, the following assertions are obtained.(i) The representation 𝜌 𝜆 is fully symplectic for almost all 𝜆. (ii) If in addition the similitude character 𝜇 𝜆 0 of 𝜌 𝜆 0 is odd, then the system {𝜌 𝜆 } 𝜆 is potentially automorphic and the residual image ρ𝜆 (Gal ℚ ) has a subgroup conjugate to Sp 4 (𝔽 𝓁 ) for almost all 𝜆.M S C 2 0 2 0 11F80, 11F70, 11F22, 20G05 INTRODUCTION Main resultsLet 𝐾, 𝐸 be number fields, and {𝜌 𝜆 } 𝜆 an 𝐸-rational compatible system of semisimple 𝑛dimensional 𝜆-adic representations of 𝐾 in the sense of Serre (Definition 2.2). Denote by Γ 𝜆 the monodromy (i.e., image) of 𝜌 𝜆 and by 𝐆 𝜆 the algebraic monodromy group of 𝜌 𝜆 † . A fundamental † The Zariski closure of the monodromy Γ 𝜆 in GL 𝑛,𝐸 𝜆 .In honor of Professor Michael Larsen's 60th birthday.

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Irreducibility of automorphic Galois representations of low dimensions

Cited by 7 publications

References 13 publications

Euler Systems and Selmer Bounds for GU(2,1)

Euler Systems and Selmer Bounds for GU(2,1)

An Euler system for GU(2, 1)

Monodromy of four‐dimensional irreducible compatible systems of Q$\mathbb {Q}$

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