Arthur’s multiplicity formula for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="bold">GSp</mml:mi> <mml:mn>4</mml:mn> </mml:msub></mml:math> and restriction to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="bold">Sp</mml:mi> <mml:mn>4</mml:mn> </mml:msub></mml:math>

Gee, Toby; Taïbi, Olivier

doi:10.5802/jep.99

Cited by 30 publications

(36 citation statements)

References 70 publications

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“…-The existence of a representation ρ π,p valued in GL 4 (Q p ) and satisfying (2) and (3) is part of [Mok14, Thm. 3.5] (note that the results of [Art04] cited in [Mok14] hold unconditionally by [GT19]). That the representation actually takes values in GSp 4 (Q p ) with the claimed multiplier follows from [BC11, Cor.…”

Section: 6mentioning

confidence: 96%

“…Arthur. -It should be noted that we use Arthur's multiplicity formula for the discrete spectrum of GSp 4 , as announced in [Art04]. A proof of this (relying on Arthur's work for symplectic and orthogonal groups in [Art13]) was given in [GT19], but this proof is only as unconditional as the results of [Art13] and [MW16a,MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.…”

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confidence: 99%

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Abelian surfaces over totally real fields are potentially modular

et al. 2021

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We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ A over ${\mathbf {Q}}$ Q with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

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Section: 6mentioning

confidence: 96%

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confidence: 99%

Abelian surfaces over totally real fields are potentially modular

et al. 2021

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“…This statement can be seen as an integral refinement of the weight-monodromy conjecture because the component group is defined as the cokernel of a certain monodromy operator. [14] prove more general level lowering results for cuspidal automorphic representations of GSp 4 , under the assumption that ρ π, has large image and is 'ordinary', see Theorem 7.5.2 of op. cit.…”

Section: Remark 121mentioning

confidence: 85%

A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds

Hoften

2021

Math. Z.

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We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ GSp 4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $${\text {GSp}}_4$$ GSp 4 .

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“…We recall some facts about Siegel modular forms and their associated Galois representations. By Arthur's classification (see [3] and [18]) cuspidal automorphic representations for GSp 4 (A Q ) fall into different types. Cuspidal automorphic representations whose transfer to GL 4 stays cuspidal are called of "general type" or type (G).…”

Section: Siegel Modular Formsmentioning

confidence: 99%

Irreducibility of limits of Galois representations of Saito–Kurokawa type

Berger

Klosin

2021

Res. number theory

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We prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$ σ 2 of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$ σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$ 1 ⊕ ρ ⊕ χ with $$\rho $$ ρ irreducible, two-dimensional of determinant $$\chi $$ χ , where $$\chi $$ χ is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$ σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$ k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$ ρ ) which we assume are non-zero.

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Arthur’s multiplicity formula for GSp 4 and restriction to Sp 4

Abstract: We prove the classification of discrete automorphic representations of GSp 4 explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp 4 and Sp 4 .

Cited by 30 publications

References 70 publications

Abelian surfaces over totally real fields are potentially modular

Abelian surfaces over totally real fields are potentially modular

A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds

Irreducibility of limits of Galois representations of Saito–Kurokawa type

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