Heights and the central critical values of triple product $L$-functions

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1. 1 A. Beilinson, Higher regulators and values of L-functions, J. Soviet Math.30 (1985), 2036-2070. Zbl0588.14013
2. 2 B.J. Birch, Hecke actions and classes of ternary quadratic forms, preprint. Zbl0748.11023MR1151865
3. 3 D. Blasius, Critical values of certain tensor product L-functions, Inv. Math.90 (1987), 181-188. Zbl0625.10022
4. 4 S. Bloch, Height pairings for algebraic cycles, J. Pure and Applied Algebra34 (1984), 119-145. Zbl0577.14004MR772054
5. 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. Zbl0302.22020MR340257
6. 6 P. Deligne, Formes modulaires et representations de GL(2), Lect. Notes in Math.349, Springer-Verlag, New York, Heidelberg, Berlin, 1973. Zbl0271.10032MR347738
7. 7 M. Eichler, Zur Zahlentheorie der Quaternion-Algebren, Crelle J.195 (1955), 127-151. Zbl0068.03303MR80767
8. 8 M. Fichler, Quaternaire quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetatfunction, Arch. Math.5 (1955), 355-366. Zbl0059.03804MR63406
9. 9 P.B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math.125 (1987), 209-235. Zbl0625.10020MR881269
10. 10 P.B. Garrett and M. Harris, Special values of triple product L-functions, preprint (1989). Zbl0776.11027MR1209238
11. 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. Zbl0406.10022MR533066
12. 12 R. Godement, Notes on Jacquet-Langlands Theory, Institute for Advanced Study, Princeton, N.J., 1970. 
13. 13 B.H. Gross, Heights and special values of L-series, CMS Conf. Proc.7 (1987), 115-187. Zbl0623.10019MR894322
14. 14 B.H. Gross, S. Kudla and D. Zagier, in preparation (1991). 
15. 15 B.H. Gross and C. Schoen, The diagonal on the 3 fold product of a curve, in preparation (1990). 
16. 16 B.H. Gross and D. Zagier, Heegner points and the derivatives of L-series, Inv. Math.84 (1986). Zbl0608.14019MR833192
17. 17 M. Harris and S. Kudla, The central critical value of a triple product L-function, Annals of Math.133 (1991), 605-672. Zbl0731.11031MR1109355
18. 18 J. Igusa, Some results on p-adic complex powers, Amer. J. Math.106 (1984), 1013-1032. Zbl0589.14023MR761577
19. 19 T. Ikeda, On the functional equations of the triple L-functions, J. Math. Kyoto Univ.29-2 (1989), 175-219. Zbl0701.11017MR1015866
20. 20 T. Ikeda, On the poles of the triple product L-function, preprint (1990). 
21. 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. Zbl0236.12010MR401654
22. 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. Zbl0491.10020
23. 23 M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math.44 (1978), 1-47. Zbl0375.22009MR463359
24. 24 S.S. Kudla and S. Rallis, On the Weil-Siegel formula, J. Reine Angew. Math.391 (1988), 65-84. Zbl0644.10022MR961164
25. 25 S.S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity, in preparation (1990). Zbl0818.11024MR1046171
26. 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. Zbl0712.11029MR1159110
27. 27 S.S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Isr. J. Math.69 (1990), 25-45. Zbl0708.22005MR1046171
28. 28 J.-F. Mestre and J. Oesterle, Elliptic Curves of Prime Conductor, preprint. Zbl0693.14004
29. 29 C. Moeglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. E. N. S.22 (1989), 605-674. Zbl0696.10023MR1026752
30. 30 T. Orloff, Special values of mixed weight triple products, Inv. Math.90 (1987), 169-180. Zbl0625.10021MR906584
31. 31 I.I. Piatetski-Shapiro and S. Rallis, Rankin triple L-functions, Compositio Math.64 (1987), 333-399. Zbl0637.10023MR911357
32. 32 D. Prasad, Trilinear forms for GL(2) and local epsilon factors, Compositio Math.75 (1990), 1-46. Zbl0731.22013MR1059954
33. 33 S. Rallis, L-Functions and the Oscillator Representation, Lect. Notes in Math.1245, Springer-Verlag, New York, Heidelberg, Berlin, 1987. Zbl0605.10016MR887329
34. 34 S. Rallis, Poles of Standard L functions, Proc. ICM, Kyoto (1990). Zbl0744.11028MR1159270
35. 35 J.-P. Serre, Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Sem. DPP19 (1969). Zbl0214.48403
36. 36 F. Shahidi, On certain L-functions, Amer. J. Math.103 (1981), 297-355. Zbl0467.12013MR610479
37. 37 F. Shahidi, On the Ramanunjan conjecture and finiteness of poles for certain L-functions, Ann. Math.127 (1988), 547-584. Zbl0654.10029MR942520
38. 38 F. Shahidi, Third symmetric power L-functions on GL(2), Compositio Math.70 (1989), 245-273. Zbl0684.10026MR1002045
39. 39 G. Shimura, Construction of Class Fields and Zeta Functions of Algebraic Curves, Ann. Math.85 (1967), 58-159. Zbl0204.07201MR204426
40. 40 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971. Zbl0221.10029MR314766
41. 41 M.F. Vigneras, Arithmétique des Algèbres de Quaternions, Lect. Notes in Math.800, Springer-Verlag, New York, Heidelberg, Berlin, 1980. Zbl0422.12008MR580949
42. 42 A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math.111 (1964), 143-211. Zbl0203.03305MR165033
43. 43 A. Weil, Basic Number Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1974. Zbl0326.12001MR427267

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