On Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms

Chen, Shih‐Yu

doi:10.48550/arxiv.2101.07507

Cited by 4 publications

(6 citation statements)

References 71 publications

(35 reference statements)

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“…More precisely, under the assumptions that κ ě 6, ω " 1, and AutpCq replaced by AutpC{Eq for some bi-quadratic extension E{Q, Morimoto proves that (1.1) holds up to square. In [Che21d], we refine the result of Grobner and Lin, and prove the conjecture for n " 4 assuming κ ě 3. Actually, we have analogous conjecture for symmetric power L-functions associated to normalized Hilbert cusp newforms over a totally real number field F. The conjecture holds for n " 1, 2, 3, 4, 6 by authors mentioned above under similar assumptions.…”

Section: Introductionsupporting

confidence: 65%

“…Note that the results are special cases of Theorem 4.5. On the other hand, when we take Σ be the Kim-Ramakrishnan-Shahidi lift of Π , the algebraicity of these critical values can be expressed in terms of powers of }f Π } using the results [GH93], [GL16], [Mor18], [Che21c], [Che21d], [Har21], [JST21]. We then obtain the period relation (1.3).…”

Section: χ " | | 3wmentioning

confidence: 99%

“…As for the left-half critical points, it follows from the global functional equation for GL 7 pA F q applied to the regular C-algebraic irreducible cuspidal automorphic representation Sym 6 pΠ q. We refer to [Che21d,Proposition A.4] for the precise statement.…”

Section: 21mentioning

confidence: 99%

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On Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms

Chen

2021

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In this paper, we prove Deligne's conjecture for symmetric sixth L-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic C-algebraic cuspidal automorphic representations of GSp 4 over totally real number fields whose archimedean components are (limits of) discrete series representations. We show that the algebraicity of critical L-values for GSp 4 ˆGL 2 can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of GL 2 , we establish period relations between the automorphic periods and powers of Petersson norm of Hilbert modular forms. The conjecture for symmetric sixth L-functions then follows from these period relations and our previous work on the algebraicity of critical values for the adjoint L-functions for GSp 4 .

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Section: Introductionsupporting

confidence: 65%

Section: χ " | | 3wmentioning

confidence: 99%

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On Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms

Chen

2021

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“…For r " 2, we prove the conjecture assuming κ ě 6 in [Che22b]. For symmetric even power L-functions, we have the results [Stu80] and [Stu89] of Sturm for Sym 2 , and [Mor21], [Che21b], [Che21c] for Sym 4 and Sym 6 due to Morimoto and the author. For higher r, our result is as follows:…”

Section: Note Thatmentioning

confidence: 55%

Algebraicity of ratios of special values of Rankin-Selberg $L$-functions and applications to Deligne's conjecture

Chen¹

2022

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In this paper, we prove new cases of Blasius' and Deligne's conjectures on the algebraicity of critical values of tensor product L-functions and symmetric odd power L-functions associated to modular forms. We also prove an algebraicity result on critical values of Rankin-Selberg L-functions for GLn ˆGL 2 in the unbalanced case, which extends the previous results of Furusawa and Morimoto for SOpV q ˆGL 2 . These results are applications of our main result on the algebraicity of ratios of special values of Rankin-Selberg L-functions. Let Σ , Σ 1 , Π , Π 1 be algebraic automorphic representations of general linear groups over Q such that Σ8 " Σ 1 8 and Π8 " Π 1 8 . Based on conjectures of Clozel and Deligne, and Yoshida's computation of motivic periods, we expect the ratio Lps, Σ ˆΠ q ¨Lps, Σ 1 ˆΠ 1 q Lps, Σ ˆΠ 1 q ¨Lps, Σ 1 ˆΠ q to be algebraic and Galois-equivariant at critical points. We show that this assertion holds under certain parity and regularity assumptions on the archimedean components. Our second main result is to prove an automorphic analogue of Blasius' conjecture on the behavior of critical values of motivic L-functions upon twisting by Artin motives. We consider Rankin-Selberg L-functions twisted by finite order Hecke characters.

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“…‚ n " 4, 6: Morimoto [Mor21], C.- [Che21b], [Che21c]. In this paper, we consider the case when n " 5.…”

Section: Introductionmentioning

confidence: 99%

On Deligne's conjecture for symmetric fifth $L$-functions of modular forms

Chen¹

2021

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On Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms

Cited by 4 publications

References 71 publications

On Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms

On Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms

Algebraicity of ratios of special values of Rankin-Selberg $L$-functions and applications to Deligne's conjecture

On Deligne's conjecture for symmetric fifth $L$-functions of modular forms

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