Subconvexity bounds for triple \(L\)-functions and representation theory. (English) Zbl 1225.11068

Bernstein, Joseph; Reznikov, Andre

Ann. Math. (2) 172, No. 3, 1679-1718 (2010).

The subconvexity problem is in the core of modern analytic number theory [H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of \(L\)-functions. GAFA 2000. Visions in mathematics – Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 705–741 (2000; Zbl 0996.11036)]. In the paper under review the authors study the subconvexity problems for triple \(L\)–functions for \(\text{PGL}_2(\mathbb R)\) using their new method of estimating the trilinear period of automorphic representations of \(\text{PGL}_2(\mathbb R)\). Their method is based on the study of analytic structure of the corresponding unique trilinear functional on unitary representations of \(\text{PGL}_2(\mathbb R)\). Their work is based on earlier works of A. Ichino [“Trilinear forms and the central values of triple product \(L\)-functions.” Duke Math. J. 145, No. 2, 281–307 (2008; Zbl 1222.11065)], M. Harris and S. S. Kudla, [“The central critical value of a triple product \(L\)-function,” Ann. Math. (2) 133, No. 3, 605–672 (1991; Zbl 0731.11031)], and Watson [“Rankin triple products and quantum chaos,” Ann. Math. (2) (to appear)].

Reviewer: Goran Muic (Zagreb)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

subconvexity bounds; triple \(L\)–functions; representation theory

Citations:

Zbl 0996.11036; Zbl 0731.11031; Zbl 1222.11065

References:

[1]	V. I. Arnol\('\)d, S. M. Gusevin-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Boston: Birkhauser, 1985, vol. 82. · Zbl 0554.58001
[2]	J. Bernstein and A. Reznikov, ”Analytic continuation of representations and estimates of automorphic forms,” Ann. of Math., vol. 150, iss. 1, pp. 329-352, 1999. · Zbl 0934.11023 · doi:10.2307/121105
[3]	J. Bernstein and A. Reznikov, ”Estimates of automorphic functions,” Mosc. Math. J., vol. 4, iss. 1, pp. 19-37, 310, 2004. · Zbl 1081.11037
[4]	W. Casselman, ”Canonical extensions of Harish-Chandra modules to representations of \(G\),” Canad. J. Math., vol. 41, iss. 3, pp. 385-438, 1989. · Zbl 0702.22016 · doi:10.4153/CJM-1989-019-5
[5]	I. M. Gel\('\)fand and G. E. Shilov, Generalized Functions. Vol. I: Properties and Operations, New York: Academic Press, 1964. · Zbl 0115.33101
[6]	I. M. Gel\('\)fand, M. I. Graev, and Y. N. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, New York: Academic Press, 1966. · Zbl 0144.17202
[7]	I. M. Gel\('\)fand, M. Graev, and I. Piatetski-Shapiro, Generalized Functions. Vol. 6: Representation Theorem and Automorphic Forms, Phila., PA: W. B. Saunders, 1969. · Zbl 0177.18003
[8]	A. Good, ”Cusp forms and eigenfunctions of the Laplacian,” Math. Ann., vol. 255, iss. 4, pp. 523-548, 1981. · Zbl 0439.30031 · doi:10.1007/BF01451932
[9]	B. H. Gross, ”Some applications of Gel\('\)fand pairs to number theory,” Bull. Amer. Math. Soc., vol. 24, iss. 2, pp. 277-301, 1991. · Zbl 0733.11018 · doi:10.1090/S0273-0979-1991-16017-9
[10]	A. Ichino, ”Trilinear forms and the central values of triple product \(L\)-functions,” Duke Math. J., vol. 145, iss. 2, pp. 281-307, 2008. · Zbl 1222.11065 · doi:10.1215/00127094-2008-052
[11]	H. Iwaniec, ”Small eigenvalues of Laplacian for \(\Gamma_0(N)\),” Acta Arith., vol. 56, iss. 1, pp. 65-82, 1990. · Zbl 0702.11034
[12]	H. Iwaniec and P. Sarnak, ”Perspectives on the analytic theory of \(L\)-functions,” Geom. Funct. Anal., iss. Special Volume, Part II, pp. 705-741, 2000. · Zbl 0996.11036
[13]	L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, New York: Springer-Verlag, 2003. · Zbl 1028.35001
[14]	M. Harris and S. S. Kudla, ”The central critical value of a triple product \(L\)-function,” Ann. of Math., vol. 133, iss. 3, pp. 605-672, 1991. · Zbl 0731.11031 · doi:10.2307/2944321
[15]	B. Krötz and R. J. Stanton, ”Holomorphic extensions of representations. I. Automorphic functions,” Ann. of Math., vol. 159, iss. 2, pp. 641-724, 2004. · Zbl 1053.22009 · doi:10.4007/annals.2004.159.641
[16]	H. Y. Loke, ”Trilinear forms of \(\mathfrak{gl}_2\),” Pacific J. Math., vol. 197, iss. 1, pp. 119-144, 2001. · Zbl 1049.22007 · doi:10.2140/pjm.2001.197.119
[17]	H. Maass, ”Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen,” Math. Ann., vol. 121, pp. 141-183, 1949. · Zbl 0033.11702 · doi:10.1007/BF01329622
[18]	W. Magnus, Formulas and Theorems for the Special Functions, New York: Springer-Verlag, 1966. · Zbl 0143.08502
[19]	B. Malgrange, Ideals of Differentiable Functions, Bombay: Tata Institute of Fundamental Research, 1967, vol. 3. · Zbl 0177.17902
[20]	P. Michel, ”Familles de fonctions \(L\) de formes automorphes et applications,” J. Théor. Nombres Bordeaux, vol. 15, iss. 1, pp. 275-307, 2003. · Zbl 1056.11027 · doi:10.5802/jtnb.403
[21]	P. Michel and A. Venkatesh, ”Equidistribution, \(L\)-functions and ergodic theory: on some problems of Yu. Linnik,” in International Congress of Mathematicians, Eur. Math. Soc., Zürich, 2006, vol. II, pp. 421-457. · Zbl 1157.11019
[22]	V. F. Molvcanov, ”Tensor products of unitary representations of the three-dimensional Lorentz group,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 43, iss. 4, pp. 860-891, 967, 1979.
[23]	A. I. Oksak, ”Trilinear Lorentz invariant forms,” Comm. Math. Phys., vol. 29, pp. 189-217, 1973.
[24]	D. Prasad, ”Trilinear forms for representations of \({ GL}(2)\) and local \(\epsilon\)-factors,” Compositio Math., vol. 75, iss. 1, pp. 1-46, 1990. · Zbl 0731.22013
[25]	I. I. Piatetski-Shapiro, ”Euler subgroups,” in Lie Groups and their Representations, New York: Halsted, 1975, pp. 597-620. · Zbl 0329.20028
[26]	R. Rankin, ”Contributions to the theory of Ramanujan’s function \(\tau(n)\),” Proc. Cambridge Philos. Soc., vol. 35, pp. 357-372, 1939. · Zbl 0021.39202
[27]	A. Reznikov, ”Non-vanishing of periods of automorphic functions,” Forum Math., vol. 13, iss. 4, pp. 485-493, 2001. · Zbl 0970.11019 · doi:10.1515/form.2001.018
[28]	A. Reznikov, ”Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms,” J. Amer. Math. Soc., vol. 21, iss. 2, pp. 439-477, 2008. · Zbl 1188.11024 · doi:10.1090/S0894-0347-07-00581-4
[29]	P. Sarnak, ”Integrals of products of eigenfunctions,” Internat. Math. Res. Notices, iss. 6, p. 251, 1994. · Zbl 0833.11020 · doi:10.1155/S1073792894000280
[30]	A. Selberg, On the Estimation of Fourier Coefficients, New York: Springer-Verlag, 1989.
[31]	C. D. Sogge, ”Concerning the \(L^p\) norm of spectral clusters for second-order elliptic operators on compact manifolds,” J. Funct. Anal., vol. 77, iss. 1, pp. 123-138, 1988. · Zbl 0641.46011 · doi:10.1016/0022-1236(88)90081-X
[32]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43. · Zbl 0821.42001
[33]	A. Venkatesh, ”Sparse equidistribution problems, period bounds, and subconvexity,” Ann. of Math., vol. 172, pp. 989-1094, 2010. · Zbl 1214.11051 · doi:10.4007/annals.2010.172.989
[34]	N. R. Wallach, Real Reductive Groups. I, Boston, MA: Academic Press, 1988, vol. 132. · Zbl 0666.22002
[35]	T. C. Watson, Rankin triple products and quantum chaos.