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

Ichino, Atsushi

Duke Math. J. 145, No. 2, 281-307 (2008).

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.

Reviewer: Zhengyu Mao (Newark)

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

Keywords:

triple product L-function; trilinear form

References:

[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