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

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arxiv 2205.15382 v4 pith:5IBKUMHN submitted 2022-05-30 math.NT

Shih-Yu Chen This is my paper · ORCID
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keywords algebraicityfunctionscriticalconjecturevaluesapplicationsassociateddeligne
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abstract

In this paper, we prove Deligne's conjecture on the algebraicity of the critical values of symmetric power $L$-functions associated with modular forms of weight at least 5. We also establish new cases of Blasius' conjecture on the algebraicity of the critical values of tensor product $L$-functions associated with modular forms. Additionally, we prove an algebraicity result for the critical values of Rankin--Selberg $L$-functions for $\GL_n \times \GL_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for ${\rm SO}(V) \times \GL_2$. These results are applications of our main theorem on the algebraicity of cross ratios of Rankin--Selberg $L$-functions at critical points.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$

    math.NT 2025-06 conditional novelty 8.0 of 10

    Period relations for critical values of GL(n)xGL(n) Rankin-Selberg L-functions are proved over number fields containing a CM field.

  2. Betti-Whittaker periods of the contragredient representations for $\textrm{GL}(n)$

    math.RT 2026-06 unverdicted novelty 6.0 of 10

    Defines Betti-Whittaker periods for a broad class of cohomological automorphic representations of GL(n) and establishes a relation to their contragredients, extending Chen's result on cuspidal cases.

  3. Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields

    math.NT 2025-09 conditional novelty 6.0 of 10

    Under regularity assumptions and an unproved archimedean rationality conjecture, the authors factor automorphic periods on unitary groups and identify the factors with motivic periods, giving a conditional proof of De...

  4. On rationality of certain Eisenstein cohomology

    math.NT 2025-06 conditional novelty 6.0 of 10

    For degenerate principal series of GL_n satisfying a balanced condition, the Eisenstein map on the bottom-degree cohomology is Aut(C)-equivariant, yielding a rational structure on this Eisenstein cohomology.

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