×

Heights and the central critical values of triple product \(L\)-functions. (English) Zbl 0807.11027

The \(L\)-function studied here is \(L(f\otimes g\otimes h,s)\), where \(f\), \(g\), \(h\) are three newforms of weight 2 for \(\Gamma_ 0(N)\) and of level \(N\), \(N\) assumed to be square-free. The functional equation is derived, using the integral representation of the \(L\)-function, with explicit local computations. Then the (extended) Siegel-Weil formula is used to study the central critical value. This value is found to be the product of a period and an algebraic number, which is interpreted as a height pairing. A criterion for the vanishing of the critical value follows. Also a conjectural formula for the derivative at the critical point is given.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] A. Beilinson , Higher regulators and values of L-functions , J. Soviet Math. 30 (1985), 2036-2070. · Zbl 0588.14013
[2] B.J. Birch , Hecke actions and classes of ternary quadratic forms , preprint. · Zbl 0748.11023
[3] D. Blasius , Critical values of certain tensor product L-functions , Inv. Math. 90 (1987), 181-188. · Zbl 0625.10022
[4] S. Bloch , Height pairings for algebraic cycles , J. Pure and Applied Algebra 34 (1984), 119-145. · Zbl 0577.14004
[5] W. Casselman , On representations of GL(2) and arithmetic of modular curves, in Modular Forms of One Variable II , Lect. Notes in Math., vol. 349, Springer-Verlag, New York, Heidelberg, Berlin, 1973, 107-141. · Zbl 0302.22020
[6] P. Deligne , Formes modulaires et representations de GL(2) , Lect. Notes in Math. 349, Springer-Verlag, New York, Heidelberg, Berlin, 1973. · Zbl 0271.10032
[7] M. Eichler , Zur Zahlentheorie der Quaternion-Algebren , Crelle J. 195 (1955), 127-151. · Zbl 0068.03303
[8] M. Fichler , Quaternaire quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetatfunction , Arch. Math. 5 (1955), 355-366. · Zbl 0059.03804
[9] P.B. Garrett , Decomposition of Eisenstein series: Rankin triple products , Ann. of Math. 125 (1987), 209-235. · Zbl 0625.10020
[10] P.B. Garrett and M. Harris , Special values of triple product L-functions , preprint (1989). · Zbl 0776.11027
[11] S. Gelbart and H. Jacquet , A relation between automorphic forms on GL(2) and GL(3) , Ann. Sci. E. N. S. 11 (1978), 471-542. · Zbl 0406.10022
[12] R. Godement , Notes on Jacquet-Langlands Theory , Institute for Advanced Study, Princeton, N.J., 1970. · Zbl 0298.12004
[13] B.H. Gross , Heights and special values of L-series , CMS Conf. Proc. 7 (1987), 115-187. · Zbl 0623.10019
[14] B.H. Gross , S. Kudla and D. Zagier , in preparation (1991).
[15] B.H. Gross and C. Schoen , The diagonal on the 3 fold product of a curve , in preparation (1990).
[16] B.H. Gross and D. Zagier , Heegner points and the derivatives of L-series , Inv. Math. 84 (1986). · Zbl 0608.14019
[17] M. Harris and S. Kudla , The central critical value of a triple product L-function , Annals of Math. 133 (1991), 605-672. · Zbl 0731.11031
[18] J. Igusa , Some results on p-adic complex powers , Amer. J. Math. 106 (1984), 1013-1032. · Zbl 0589.14023
[19] T. Ikeda , On the functional equations of the triple L-functions , J. Math. Kyoto Univ. 29-2 (1989), 175-219. · Zbl 0701.11017
[20] T. Ikeda , On the poles of the triple product L-function , preprint (1990).
[21] H. Jacquet and R.P. Langlands , in Automorphic Forms on GL(2) , Lect. Notes in Math. 114, Springer-Verlag, New York, Heidelberg, Berlin, 1970. · Zbl 0236.12010
[22] H. Jacquet , I.I. Piatetski-Shapiro and J. Shalika , On Euler products and the classification of automorphic representations I, II , Amer. J. Math. 103 (1981), 499-558 and 777-815. · Zbl 0491.10020
[23] M. Kashiwara and M. Vergne , On the Segal-Shale-Weil representations and harmonic polynomials , Invent. Math. 44 (1978), 1-47. · Zbl 0375.22009
[24] S.S. Kudla and S. Rallis , On the Weil-Siegel formula , J. Reine Angew. Math. 391 (1988), 65-84. · Zbl 0644.10022
[25] S.S. Kudla and S. Rallis , A regularized Siegel-Weil formula: the first term identity , in preparation (1990). · Zbl 0818.11024
[26] S.S. Kudla and S. Rallis , Poles of Eisenstein series and L-functions, Festschrift in honor of I. Piatetski-Shapiro , Israel Math. Conf. Proc., vol. 3, Weizmann Sci. Press, Israel (1990), 81-110. · Zbl 0712.11029
[27] S.S. Kudla and S. Rallis , Degenerate principal series and invariant distributions , Isr. J. Math. 69 (1990), 25-45. · Zbl 0708.22005
[28] J.-F. Mestre and J. Oesterle , Elliptic Curves of Prime Conductor , preprint.
[29] C. Moeglin and J.-L. Waldspurger , Le spectre résiduel de GL(n) , Ann. Sci. E. N. S. 22 (1989), 605-674. · Zbl 0696.10023
[30] T. Orloff , Special values of mixed weight triple products , Inv. Math. 90 (1987), 169-180. · Zbl 0625.10021
[31] I.I. Piatetski-Shapiro and S. Rallis , Rankin triple L-functions , Compositio Math. 64 (1987), 333-399. · Zbl 0637.10023
[32] D. Prasad , Trilinear forms for GL(2) and local epsilon factors , Compositio Math. 75 (1990), 1-46. · Zbl 0731.22013
[33] S. Rallis , L-Functions and the Oscillator Representation , Lect. Notes in Math. 1245, Springer-Verlag, New York, Heidelberg, Berlin, 1987. · Zbl 0605.10016
[34] S. Rallis , Poles of Standard L functions , Proc. ICM, Kyoto (1990). · Zbl 0744.11028
[35] J.-P. Serre , Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures) , Sem. DPP 19 (1969). · Zbl 0214.48403
[36] F. Shahidi , On certain L-functions , Amer. J. Math. 103 (1981), 297-355. · Zbl 0467.12013
[37] F. Shahidi , On the Ramanunjan conjecture and finiteness of poles for certain L-functions , Ann. Math. 127 (1988), 547-584. · Zbl 0654.10029
[38] F. Shahidi , Third symmetric power L-functions on GL(2) , Compositio Math. 70 (1989), 245-273. · Zbl 0684.10026
[39] G. Shimura , Construction of Class Fields and Zeta Functions of Algebraic Curves , Ann. Math. 85 (1967), 58-159. · Zbl 0204.07201
[40] G. Shimura , Introduction to the Arithmetic Theory of Automorphic Functions , Princeton University Press, 1971. · Zbl 0221.10029
[41] M.F. Vigneras , Arithmétique des Algèbres de Quaternions , Lect. Notes in Math. 800, Springer-Verlag, New York, Heidelberg, Berlin, 1980. · Zbl 0422.12008
[42] A. Weil , Sur certains groupes d’opérateurs unitaires , Acta Math. 111 (1964), 143-211. · Zbl 0203.03305
[43] A. Weil , Basic Number Theory , Springer-Verlag, New York, Heidelberg, Berlin, 1974. · Zbl 0326.12001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.