Trilinear forms and the central values of triple product \(L\)-functions. (English) Zbl 1222.11065

The author and Ikeda have a beautiful conjecture explicating the Gross-Prasad conjecture. In this paper the author proves a low rank case of the Ichino-Ikeda conjecture. The main result is an explicit formula relating the global trilinear form on a quaternion algebra and the Petersson inner product. As a corollary, this implies the relation between the global trilinear form and the central value of the triple product \(L-\)function, which is the content of Jacquet’s conjecture, originally established by Harris-Kudla.
The idea of the proof is to apply the theta correspondence and Siegel-Weil formula, and express both the trilinear form and the Petersson product using the Whittaker functions of automorphic forms of \(\text{GL}_2\). The trilinear form part was done in Harris-Kudla’s work, while the Petersson product part was done in Waldspurger’s work on toric periods. The extra work needed is to establish a local identity on Whittaker functions, where the main difficulty lies in convergence issues.


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
Full Text: DOI


[1] S. Böcherer and R. Schulze-Pillot, “On the central critical value of the triple product \(L\)-function” in Number Theory (Paris, 1993–1994.) , London Math. Soc. Lecture Note Ser. 235 , Cambridge Univ. Press, Cambridge, 1996, 1–46. · Zbl 0924.11034
[2] P. B. Garrett, Decomposition of Eisenstein series: Rankin triple products , Ann. of Math. (2) 125 (1987), 209–235. JSTOR: · Zbl 0625.10020
[3] B. H. Gross and S. S. Kudla, Heights and the central critical values of triple product \(L\)-functions , Compositio Math. 81 (1992), 143–209. · Zbl 0807.11027
[4] B. H. Gross and D. Prasad, On the decomposition of a representation of \(\mathrm{SO}_n\) when restricted to \(\mathrm{SO}_{n-1}\) , Canad. J. Math. 44 (1992), 974–1002. · Zbl 0787.22018
[5] M. Harris and S. S. Kudla, The central critical value of a triple product \(L\)-function , Ann. of Math. (2) 133 (1991), 605–672. JSTOR: · Zbl 0731.11031
[6] -, “On a conjecture of Jacquet” in Contributions to Automorphic Forms, Geometry, and Number Theory (Baltimore, 2002) , Johns Hopkins Univ. Press, Baltimore, 2004, 355–371. · Zbl 1080.11039
[7] K. Hiraga and H. Saito, On \(L\)-packets for inner forms of \(\mathrm{SL}_n\) , preprint, 2007.
[8] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture , preprint, 2007. · Zbl 1216.11057
[9] T. Ikeda, On the location of poles of the triple \(L\)-functions , Compositio Math. 83 (1992), 187–237. · Zbl 0773.11035
[10] H. H. Kim and F. Shahidi, Functorial products for \(\mathrm{GL}_2 \times \mathrm{GL}_3\) and the symmetric cube for \(\mathrm{GL}_2\) , Ann. of Math. (2) 155 (2002), 837–893. JSTOR: · Zbl 1040.11036
[11] S. S. Kudla and S. Rallis, A regularized Siegel-Weil formula: The first term identity , Ann. of Math. (2) 140 (1994), 1–80. · Zbl 0818.11024
[12] H. Y. Loke, Trilinear forms of \(\mathfrak{gl}_2\) , Pacific J. Math. 197 (2001), 119–144. · Zbl 1049.22007
[13] I. Piatetski-Shapiro and S. Rallis, Rankin triple \(L\) functions , Compositio Math. 64 (1987), 31–115. · Zbl 0637.10023
[14] D. Prasad, Trilinear forms for representations of \(\mathrm{GL}(2)\) and local \(\epsilon\)-factors , Compositio Math. 75 (1990), 1–46. · Zbl 0731.22013
[15] -, Invariant forms for representations of \(\mathrm{GL}_2\) over a local field , Amer. J. Math. 114 (1992), 1317–1363. JSTOR: · Zbl 0780.22004
[16] D. Prasad and R. Schulze-Pillot, Generalised form of a conjecture of Jacquet and a local consequence , J. Reine Angew. Math. 616 (2008), 219–236. · Zbl 1221.11129
[17] H. Shimizu, Theta series and automorphic forms on \(\mathrm{GL}_2\) , J. Math. Soc. Japan 24 (1972), 638–683. · Zbl 0241.10016
[18] J.-L. Waldspurger, Sur les valeurs de certaines fonctions \(L\) automorphes en leur centre de symétrie , Compositio Math. 54 (1985), 173–242. · Zbl 0567.10021
[19] T. C. Watson, Rankin triple products and quantum chaos , to appear in Ann. of Math. (2), Ph.D. dissertation, Princeton University, Princeton, 2002.
[20] A. Weil, Sur certains groupes d’opérateurs unitaires , Acta Math. 111 (1964), 143–211. · Zbl 0203.03305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.