Automorphic period and the central value of Rankin-Selberg \(L\)-function. (English) Zbl 1294.11069
The present paper studies the relations between period integrals related to automorphic forms and certain \(L\)-values. Subject to some local conditions, it establishes a refinement of the global Gan-Gross-Prasad conjecture for unitary groups.
In the 1980’s, J. L. Waldspurger [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)] proved his celebrated formula that relates certain toric periods to the central values of \(L\)-functions on \(\mathrm{GL}_2\). Later, B. H. Gross and D. Prasad [Can. J. Math. 44, No. 5, 974–1002 (1992; Zbl 0787.22018)] generalized Waldspurger’s formula to higher rank orthogonal groups, and conjectured that periods on \(\mathrm{SO}_n\times \mathrm{SO}_{n-1}\) restricted to the diagonal \(\mathrm{SO}_{n-1}\) are nonzero exactly when certain automorphic \(L\)-functions at the central critical point are non-vanishing. W. T. Gan et al. in [Astérisque 346, 1–109 (2012; Zbl 1280.22019)] further generalized this conjecture to classical groups, including unitary groups and symplectic groups. In a recent paper [“Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups”, Ann. Math. (2) 180, No. 3, 971–1049 (2014)], the author confirmed the Gan-Gross-Prasad conjecture for unitary groups under some local restrictions.
The refinements for the Gross-Prasad conjecture and for the Gan-Gross-Prasad conjecture seek to establish explicit identities between (the absolute value square of) the period integrals and the critical values of the corresponding \(L\)-functions. The study in this direction was pioneered by A. Ichino and T. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010; Zbl 1216.11057)] for orthogonal groups. As the sequel to his above-mentioned work [loc. cit.] and subject to some local conditions, the author uses the Jacquet-Rallis relative trace formula approach to further establish the refinement of the global Gan-Gross-Prasad conjecture for unitary groups, as formulated by R. N. Harris [“A refined Gross-Prasad Conjecture for unitary groups”, PhD Thesis, San Diego: University of California, San Diego (2011), http://arxiv.org/abs/1201.0518]. As an application, the author proves the positivity for certain Rankin-Selberg central \(L\)-values.
In the 1980’s, J. L. Waldspurger [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)] proved his celebrated formula that relates certain toric periods to the central values of \(L\)-functions on \(\mathrm{GL}_2\). Later, B. H. Gross and D. Prasad [Can. J. Math. 44, No. 5, 974–1002 (1992; Zbl 0787.22018)] generalized Waldspurger’s formula to higher rank orthogonal groups, and conjectured that periods on \(\mathrm{SO}_n\times \mathrm{SO}_{n-1}\) restricted to the diagonal \(\mathrm{SO}_{n-1}\) are nonzero exactly when certain automorphic \(L\)-functions at the central critical point are non-vanishing. W. T. Gan et al. in [Astérisque 346, 1–109 (2012; Zbl 1280.22019)] further generalized this conjecture to classical groups, including unitary groups and symplectic groups. In a recent paper [“Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups”, Ann. Math. (2) 180, No. 3, 971–1049 (2014)], the author confirmed the Gan-Gross-Prasad conjecture for unitary groups under some local restrictions.
The refinements for the Gross-Prasad conjecture and for the Gan-Gross-Prasad conjecture seek to establish explicit identities between (the absolute value square of) the period integrals and the critical values of the corresponding \(L\)-functions. The study in this direction was pioneered by A. Ichino and T. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010; Zbl 1216.11057)] for orthogonal groups. As the sequel to his above-mentioned work [loc. cit.] and subject to some local conditions, the author uses the Jacquet-Rallis relative trace formula approach to further establish the refinement of the global Gan-Gross-Prasad conjecture for unitary groups, as formulated by R. N. Harris [“A refined Gross-Prasad Conjecture for unitary groups”, PhD Thesis, San Diego: University of California, San Diego (2011), http://arxiv.org/abs/1201.0518]. As an application, the author proves the positivity for certain Rankin-Selberg central \(L\)-values.
Reviewer: Qiao Zhang (Fort Worth)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Keywords:
automorphic period; Rankin-Selberg L-function; global Gan-Gross-Prasad conjecture; Ichino-Ikeda conjecture; Jacquet-Rallis relative trace formula; spherical character; local character expansion; regular unipotent orbital integralReferences:
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