Chen, Shih-Yu; Cheng, Yao

On Deligne's Conjecture for Certain Automorphic \(L\)-Functions for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\) and \(\mathrm{GL}(4)\)

Doc. Math. 24, 2241-2297 (2019)
DOI: 10.25537/dm.2019v24.2241-2297
Communicated by Don Blasius


We prove Deligne's conjecture for certain automorphic \(L\)-functions for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\) and \(\mathrm{GL}(4)\). The proof is based on rationality results for central critical values of triple product \(L\)-functions, which follow from establishing explicit Ichino's formulae for trilinear period integrals for Hilbert cusp forms on totally real étale cubic algebras over \(\mathbb{Q}\).

Mathematics Subject Classification

11F41, 11F67


Deligne's conjecture, automorphic \(L\)-function, Ichino's formula


  • [AC89]. J. Arthur and L. Clozel. Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. Number 120 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989. DOI 10.1515/9781400882403; zbl 0682.10022; MR1007299.
  • [BFH05]. B. Brubaker, S. Friedberg, and J. Hoffstein. Cubic twists of GL(2) automorphic \(L\)-functions. Inventiones mathematicae, 160:31-58, 2005. DOI 10.1007/s00222-004-0398-8; zbl 1145.11041; MR2129707.
  • [BJ79]. A. Borel and H. Jacquet. Automorphic forms and automorphic representations. In Automorphic forms, representations, and \(L\)-functions, volume 33 of Proceedings of Symposia in Pure Mathematics, pages 189-202, 1979. Part 1. zbl 0414.22020; MR0546598.
  • [Bla87]. D. Blasius. Appendix to Orloff, Critical values of certain tensor product \(L\)-functions. Inventiones mathematicae, 90:181-188, 1987. DOI 10.1007/BF01389037; zbl 0625.10022; MR1554038.
  • [BSP96]. S. Bocherer and R. Schulze-Pillot. On the central critical value of the triple product \(L\)-function. In Number theory, volume 235 of London Mathematical Society Lecture Note Series, pages 1-46. Cambridge Univ. Press, Cambridge, 1996. zbl 0924.11034; MR1628792.
  • [Bum98]. D. Bump. Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge, first edition, 1998. zbl 0868.11022; MR1431508.
  • [Cas73]. W. Casselman. On some results of Atkin and Lehner. Mathematische Annalen, 201:301-314, 1973. DOI 10.1007/BF01428197; zbl 0239.10015; MR0337789.
  • [CCI19]. S. Y. Chen, Y. Cheng, and I. Ishikawa. Gamma factors for Asai representations of GL(2). J. Number Theory, 2019. DOI 10.1016/j.jnt.2019.08.008; arxiv 1810.12561.
  • [Che19]. S.-Y. Chen. Pullback formulas for nearly holomorphic Saito-Kurokawa lifts. Manuscripta Math., 2019. DOI 10.1007/s00229-019-01111-2; arxiv 1806.09467.
  • [FH95]. S. Friedberg and J. Hoffstein. Nonvanishing theorems for automorphic \(L\)-functions on GL(2). Annals of Mathematics, 142(2):385-423, 1995. DOI 10.2307/2118638; zbl 0847.11026; MR1343325.
  • [Fla79]. D. Flath. Decomposition of representations into tensor products. In Automorphic forms, representations, and \(L\)-functions, volume 33, Proc. Symp. Pure Math., Part 1, pages 179-183, 1979. zbl 0414.22019; MR0546596.
  • [FM14]. M. Furusawa and K. Morimoto. On special values of certain \(L\)-functions. American Journal of Mathematics, 136(5):1385-1407, 2014. DOI 10.1353/ajm.2014.0032; zbl 1317.11050; MR3263901.
  • [FM16]. M. Furusawa and K. Morimoto. On special values for certain \(L\)-functions. II. American Journal of Mathematics, 138(4):1117-1166, 2016. DOI 10.1353/ajm.2016.0037; zbl 1404.11064; MR3538153.
  • [Gar87]. P. Garrett. Decomposition of Eisenstein series: Rankin triple products. Annals of Mathematics, 125(2):209-235, 1987. DOI 10.2307/1971310; zbl 0625.10020; MR0881269.
  • [GH93]. P. Garrett and M. Harris. Special values of triple product \(L\)-functions. American Journal of Mathematics, 115(1):161-240, 1993. DOI 10.2307/2374726; zbl 0776.11027; MR1209238.
  • [GJ72]. R. Godement and H. Jacquet. Zeta functions of simple algebras, volume 260 of Lecture Notes in Mathematics. Springer, 1972. DOI 10.1007/BFb0070263; zbl 0244.12011; MR0342495.
  • [GJ78]. S. Gelbart and H. Jacquet. A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. Ecole Norm. Sup., 11(4):471-542, 1978. DOI 10.24033/asens.1355; zbl 0406.10022; MR0533066.
  • [GK92]. B. Gross and S. Kudla. Heights and the central critical values of triple product \(L\)-functions. Compositio Mathematica, 81(2):143-209, 1992. zbl 0807.11027; MR1145805.
  • [Har90]. M. Harris. Automorphic forms of \(\overline{\partial} \)-cohomology type as coherent cohomology classes. Journal of Differential Geometry, 32:1-63, 1990. DOI 10.4310/jdg/1214445036; zbl 0711.14012; MR1064864.
  • [Har93]. M. Harris. \(L\)-functions of \(2 \times 2\) unitary groups and factorization of periods of Hilbert modular forms. Journal of American Mathematical Society, 6(3):637-719, 1993. DOI 10.2307/2152780; zbl 0779.11023; MR1186960.
  • [Har94]. M. Harris. Period invariants of Hilbert modular forms. II. Compositio Mathematica, 94:201-226, 1994. zbl 0871.11037; MR1302316.
  • [Hen01]. G. Henniart. Sur la conjecture de Langlands locale pour \(GL_n\). Journal de Théorie des Nombres de Bordeaux, 13(1):167-187, 2001. DOI 10.5802/jtnb.313; zbl 1048.11093; MR1838079.
  • [Hid93]. H. Hida. Elementary theory of \(L\)-functions and Eisenstein series, volume 26 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1993. zbl 0942.11024; MR1216135.
  • [HK91]. M. Harris and S. Kudla. The central critical value of a triple product \(L\)-function. Annals of Mathematics, 133(3):605-672, 1991. DOI 10.2307/2944321; zbl 0731.11031; MR1109355.
  • [HK04]. M. Harris and S. Kudla. On a conjecture of Jacquet, pages 355-371. Johns Hopkins Univ. Press, Baltimore, 2004. Reviewer: Henry H. Kim. zbl 1080.11039; MR2058614; arxiv math/0111238.
  • [Ich05]. A. Ichino. Pullbacks of Saito-Kurokawa lifts. Inventiones mathematicae, 162:551-647, 2005. DOI 10.1007/s00222-005-0454-z; zbl 1188.11020; MR2198222.
  • [Ich08]. A. Ichino. Trilinear forms and the central values of triple product \(L\)-functions. Duke Mathematical Journal, 145(2):281-307, 2008. DOI 10.1215/00127094-2008-052; zbl 1222.11065; MR2449948.
  • [II10]. A. Ichino and T. Ikeda. On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geometric and Functional Analysis, 19:1378-1425, 2010. DOI 10.1007/s00039-009-0040-4; zbl 1216.11057; MR2585578.
  • [Ike89]. T. Ikeda. On the functional equations of the triple \(L\)-functions. Kyoto Journal of Mathematics, 29:175-219, 1989. DOI 10.1215/kjm/1250520261; zbl 0701.11017; MR1015866.
  • [Ike98]. T. Ikeda. On the gamma factor of the triple \(L\)-function. II. Journal fur die reine und angewandte Mathematik, 499:199-223, 1998. DOI 10.1515/crll.1998.058; zbl 0923.11078; MR1631136.
  • [IP18]. A. Ichino and K. Prasanna. Periods of quaternionic Shimura varieties. I. Mem. Amer. Math. Soc., 2018. to appear. arxiv 1610.00166.
  • [Jan16]. F. Januszewski. On period relations for automorphic L-functions. II. 2016. arxiv 1604.04253.
  • [Jan18]. F. Januszewski. On period relations for automorphic L-functions. I. Trans. Am. Math. Soc., 371(9):6547-6580, 2019. DOI 10.1090/tran/7527; zbl 07050823; MR3937337; arxiv 1504.06973.
  • [JL70]. H. Jacquet and R. Langlands. Automorphic Forms on GL(2), volume 114 of Lecture Notes in Mathematics. Springer-Verlag, Berlin and New York, 1970. DOI 10.1007/BFb0058988; zbl 0236.12010; MR0401654.
  • [JST19]. D. Jiang, B. Sun, and F. Tian. Period relations for standard \(L\)-functions of symplectic type. 2019. arxiv 1909.03476.
  • [KS02]. H. H. Kim and F. Shahidi. Functorial products for \(GL_2 \times GL_3\) and the symmetric cube for \(GL_2\). Annals of Mathematics, 155:837-893, 2002. DOI 10.2307/3062134; zbl 1040.11036; MR1923967.
  • [Nel11]. P. Nelson. Equidistribution of cusp forms in the level aspect. Duke Mathematical Journal, 160(3):467-501, 2011. DOI 10.1215/00127094-144287; zbl 1273.11069; MR2852367; arxiv 1011.1292.
  • [Orl87]. T. Orloff. Special values and mixed weight triple products. Inventiones mathematicae, 90:169-180, 1987. With appendix by Don Blasius. DOI 10.1007/BF01389036; zbl 0625.10021; MR0906584.
  • [Pra90]. D. Prasad. Trilinear forms for representations of GL(2) and local \(\epsilon \)-factors. Compositio Mathematica, 75(1):1-46, 1990. zbl 0731.22013; MR1059954.
  • [Pra92]. D. Prasad. Invariant linear forms for representations of GL(2) over a local field. American Journal of Mathematics, 114:1317-1363, 1992. DOI 10.2307/2374764; zbl 0780.22004; MR1198305.
  • [PSP08]. D. Prasad and R. Schulze-Pillot. Generalised form of a conjecture of Jacquet and a local consequence. Journal fur die reine und angewandte Mathematik, (616):219-236, 2008. DOI 10.1515/CRELLE.2008.023; zbl 1221.11129; MR2369492; arxiv math/0606515.
  • [PSR87]. I. Piatetski-Shapiro and S. Rallis. Rankin triple \(L\)-functions. Compositio Mathematica, 64(1):31-115, 1987. zbl 0637.10023; MR0911357.
  • [Rag09]. A. Raghuram. On the special values of certain Rankin-Selberg \(L\)-functions and applications to odd symmetric power \(L\)-functions of modular forms. International Mathematics Research Notices, 2010(2):334-372, 2009. DOI 10.1093/imrn/rnp127; zbl 1221.11128; MR2581038; arxiv 0811.4435.
  • [Sat87]. T. Satoh. Some remarks on triple \(L\)-functions. Mathematische Annalen, 276:687-698, 1987. DOI 10.1007/BF01456996; zbl 0607.10020; MR0879546.
  • [Shi76]. G. Shimura. The special values of the zeta functions associated with cusp forms. Communications on Pure and Applied Mathematics, pages 783-804, 1976. DOI 10.1002/cpa.3160290618; zbl 0348.10015; MR0434962.
  • [Shi77]. G. Shimura. On the periods of modular forms. Mathematische Annalen, 229:211-221, 1977. DOI 10.1007/BF01391466; zbl 0363.10019; MR0463119.
  • [Shi78]. G. Shimura. The special values of the zeta functions associated with Hilbert modular forms. Duke Mathematical Journal, 45(3):637-679, 1978. DOI 10.1215/S0012-7094-78-04529-5; zbl 0394.10015; MR0507462.
  • [Shi81]. G. Shimura. On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra. Annals of Mathematics, 114(3):569-607, 1981. DOI 10.2307/1971302; zbl 0486.10021; MR0634429.
  • [Stu80]. J. Sturm. Special values of zeta functions, and Eisenstein series of half integral weight. American Journal of Mathematics, 102(2):219-240, 1980. DOI 10.2307/2374237; zbl 0433.10015; MR0564472.
  • [Stu89]. J. Sturm. Evaluation of the symmetric square at the near center point. American Journal of Mathematics, 111(4):585-598, 1989. DOI 10.2307/2374814; zbl 0705.11027; MR1011550.
  • [Wal85]. J.-L. Waldspurger. Sur les valeurs de certaines fonctions \(L\) automorphes en leur centre de symetrie. Compositio Mathematica, 54(2):173-242, 1985. zbl 0567.10021; MR0783511.
  • [Xue19]. H. Xue. Central values of degree six \(L\)-functions. J. Number Theory, 203:350-359, 2019. DOI 10.1016/j.jnt.2019.03.007; zbl 07081800; MR3991408.


Chen, Shih-Yu
Institute of Mathematics, Academia Sinica, No. 1, Sec. 4, Roosevelt R., Taipei 10617, Taiwan, ROC
Cheng, Yao
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge R., Singapore 119076