Abstract
We study the algebraicity of the central critical values of twisted triple product L-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result with Cheng on Deligne’s conjecture for certain automorphic L-functions for \({\text {GL}}_3 \times {\text {GL}}_2\).
Résumé
Nous étudions l’algébraicité des valeurs critiques centrales des fonctions L à triple produit torsadé associées aux formes de cuspides de Hilbert motiviques sur une algèbre cubique étale totalement réelle dans le cas totalement déséquilibré. L’algébraicité est exprimée en termes de période cohomologique construite via la théorie de la cohomologie cohérente sur les variétés quaternioniques de Shimura développée par Harris. Comme application, nous généralisons notre résultat précédent avec Cheng sur la conjecture de Deligne pour certaines fonctions L automorphes pour \({\text {GL}}_3 \times {\text {GL}}_2\).
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References
D. Blasius, M. Harris, and D. Ramakrishnan. Coherent cohomology, limits of discrete series, and Galois conjugation. Duke Math. J., 73(3):647–685, 1994.
D. Blasius. Appendix to Orloff, Critical values of certain tensor product \(L\)-functions. Invent. Math., 90:181–188, 1987.
A. Borel. Automorphic \(L\)-functions. In Automorphic forms, representations, and L-functions, volume 33, Part 2, pages 27–61. Proceedings of Symposia in Pure Mathematics, 1979.
A. Borel and N. Wallach. Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, volume 67 of Mathematical Surveys and Monographs. Amer. Math. Soc., 2000.
S.-Y. Chen and Y. Cheng. On Deligne’s conjecture for certain automorphic L-functions for \({{\rm {GL}}}(3)\times {{\rm {GL}}}(2)\) and \({{\rm {GL}}}(4)\). Doc. Math., 24:2241–2297, 2019.
S.-Y. Chen. Gamma factors for the Asai cube representation. Math. Z., 297:747–773, 2021.
S.-Y. Chen. Pullback formulas for nearly holomorphic Saito-Kurokawa lifts. Manuscripta Math., 161:501–561, 2020.
L. Clozel. Motifs et Formes Automorphes: Applications du Principe de Fonctorialité. In Automorphic Forms, Shimura Varieties, and L-functions, Vol. I, Perspectives in Mathematics, pages 77–159, 1990.
P. Deligne. Valeurs de fonctions \(L\) et périodes d’intégrales. In Automorphic Forms, Representations and L-Functions, volume 33, pages 313–346. Proceedings of Symposia in Pure Mathematics, 1979. Part 2.
S. Friedberg and J. Hoffstein. Nonvanishing Theorems for Automorphic \(L\)-Functions on \({\rm GL}(2)\). Ann. of Math., 142(2):385–423, 1995.
P. Garrett and M. Harris. Special values of triple product \(L\)-functions. Amer. J. Math., 1993.
S. Gelbart and H. Jacquet. A relation between automorphic representations of \({\rm GL}(2)\) and \({\rm GL}(3)\). Ann. Sci. Éc. Norm. Supér., 11:471–542, 1978.
M. Harris. Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Invent. Math., 82:151–189, 1985.
M. Harris. Arithmetic vector bundles and automorphic forms on Shimura varieties, II. Compos. Math., 60(3):323–378, 1986.
G. Harder. Eisenstein cohomology of arithmetic groups The case \({\rm GL}_2\). Invent. Math., 89:37–118, 1987.
M. Harris. Period invariants of Hilbert modular forms, I: Trilinear differential operators and L-functions. In Cohomology of Arithmetic Groups and Automorphic Forms, volume 1447 of Lecture Notes in Mathematics, pages 155–202. Springer-Verlag, 1989.
M. Harris. Automorphic forms and the cohomology of vector bundles on shimura varieties. In Automorphic Forms, Shimura Varieties, and L-functions, Vol. II, Perspectives in Mathematics, pages 41–91, 1990.
M. Harris. Automorphic forms of \({\overline{\partial }}\)-cohomology type as coherent cohomology classes. J. Differential Geom., 32:1–63, 1990.
M. Harris. Periods invariants of Hilbert modular forms, II. Compos. Math., 94:201–226, 1994.
G. Henniart. Une preuve simple des conjectures de Langlands pour \({\rm GL}(n)\) sur un corps \(p\)-adique. Invent. Math., 139:439–455, 1999.
G. Henniart. Sur la conjecture de Langlands locale pour \({\rm GL}_n\). J. Théor. Nombres Bordeaux, 13(1):167–187, 2001.
M. Harris and S. S. Kudla. The central critical value of a triple product \(L\)-function. Ann. of Math., 133:605–672, 1991.
M. Harris and R. Taylor. On the geometry and cohomology of some simple Shimura varieties. . Princeton University Press, 2001.
A. Ichino. Pullbacks of Saito-Kurokawa lifts. Invent. Math., 162:551–647, 2005.
A. Ichino. Trilinear forms and the central values of triple product L-functions. Duke Math. J., 145(2):281–307, 2008.
A. Ichino and T. Ikeda. On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal., 19:1378–1425, 2010.
T. Ikeda. On the functional equations of the triple \(L\)-functions. Kyoto J. Math., 29:175–219, 1989.
A. Ichino and K. Prasanna. Period of quaternionic Simura varieties. I. Mem. Amer. Math. Soc., 2018. to appear.
F. Januszewski. On Period Relations for Automorphic L-functions I. Trans. Amer. Math. Soc., 371:6547–6580, 2019.
H. Jacquet and R. Langlands. Automorphic Forms on\({\rm GL}(2)\), volume 114 of Lecture Notes in Mathematics. Springer-Verlag, Berlin and New York, 1970.
S. S. Kudla and S. Rallis. Ramified degenerate principal series representations for \({\rm Sp}(n)\). Israel J. Math., 78:209–256, 1992.
H. H. Kim and F. Shahidi. Functorial products for \({{\rm {GL}}}_2 \times {{\rm {GL}}}_3\) and the symmetric cube for \({{\rm {GL}}}_2\). Ann. of Math., 155(2):837–893, 2002.
H. Y. Loke. Trilinear forms of \({\mathfrak{gl}}_2\). Pacific J. Math., 197(1):119–144, 2001.
S. T. Lee and C. B. Zhu. Degenerate principal series and local theta correspondence II. Israel J. Math., 100:29–59, 1997.
J. S. Milne. The action of an automorphism of \({\mathbb{C}}\) on a Shimura variety and its special points. In: Arithmetic and Geometry, Volume I, volume 35 of Progress in Mathematics, pages 239–265. Birkhauser, 1983.
A. Pal and C. de Vera-Piquero. Pullbacks of Saito-Kurokawa lifts and a central value formula for degree \(6\)\(L\)-series. Doc. Math., 24:1935–2036, 2019.
D. Prasad. Trilinear forms for representations of \({\rm GL}(2)\) and local \(\epsilon \)-factors. Compos. Math., 75(1):1–46, 1990.
D. Prasad. Invariant linear forms for representations of \({{{\rm GL}}}(2)\) over a local field. Amer. J. Math., 114:1317–1363, 1992.
I.I. Piatetski-Shapiro and S. Rallis. Rankin triple \(L\) functions. Compos. Math., 64:31–115, 1987.
A. Raghuram. Critical values for Rankin-Selberg \(L\)-functions for \({\rm GL}_n \times {\rm GL}_{n-1}\) and the symmetric cube \(L\)-functions for \({\rm GL}_2\). Forum Math., 28:457–489, 2016.
J. Repka. Tensor products of unitary representations of \({\rm SL}_2({\mathbb{R}})\). Bull. Amer. Math. Soc., 82(6):930–932, 1976.
A. Raghuram and N. Tanabe. Note on the arithmetic of Hilbert modular forms. J. Ramanujan Math. Soc., 26(3):261–319, 2011.
G. Shimura. On the periods of modular forms. Math. Ann., 229:211–221, 1977.
G. Shimura. The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J., 45(3):637–679, 1978.
G. Shimura. On certain zeta functions attached to two Hilbert modular forms: II. The case of automorphic forms on a quaternion algebra. Ann. of Math., 114:569–607, 1981.
G. Shimura. On the critical values of certain Dirichlet series and the periods of automorphic forms. Invent. Math., 94:245–305, 1988.
C.L. Siegel. Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Göttingen, pages 87–102, 1969.
K. Takase. On the trace formula of the Hecke operators and the special values of the second \(L\)-functions attached to the Hilbert modular forms. Manuscripta Math., 55:137–170, 1986.
J. Tate. Number theoretic background. In Automorphic forms, representations, and L-functions, volume 33, pages 3–26. Proceedings of Symposia in Pure Mathematics, 1979. Part 2.
J.-L. Waldspurger. Quelques propriétés arithmétiques de certaines formes automorphes sur \({{\rm {GL}}}(2)\). Compos. Math., 54:121–171, 1985.
H. Xue. Central values of degree six \(L\)-functions. J. Number Theory, 203:350–359, 2019.
H. Yoshida. On the zeta functions of Shimura varieties and periods of Hilbert modular forms. Duke Math. J., 75(1):121–191, 1994.
Acknowledgements
This work was initiated during the author’s visit to Kyoto University supported by the Graduate Students Study Abroad Program sponsored by Ministry of Science and Technology. The author would like to thank Atsushi Ichino for suggesting the problem and his advice and encouragement. Finally, the author is grateful to the referees for the suggestions and comments on the improvement of the manuscript.
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Chen, SY. Algebraicity of the central critical values of twisted triple product L-functions. Ann. Math. Québec 47, 403–442 (2023). https://doi.org/10.1007/s40316-021-00169-3
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DOI: https://doi.org/10.1007/s40316-021-00169-3