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**The central critical value of a triple product \(L\)-function.**
*(English)*
Zbl 0731.11031

This paper establishes two important theorems concerning the triple product \(L\)-function associated to three holomorphic elliptic modular forms. The first gives a criterion for the nonvanishing of this \(L\)-function at the center of the critical strip in terms of the nonvanishing of certain integrals; this verifies (for holomorphic automorphic forms over \(\mathbb Q)\) a conjecture of Jacquet, motivated by Jacquet’s vision of the relative trace formula. The second states that the central value can be expressed, up to a product of nonzero local factors from the bad primes, as the product of a period factor and a square in the extension of \(\mathbb Q\) generated by the Hecke eigenvalues of the forms. This is analogous to results of J.-L. Waldspurger [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)]. On the road to proving the second theorem, the authors also strengthen earlier results of G. Shimura [Ann. Math. (2) 111, 313–375 (1980; Zbl 0438.12003)] concerning canonical models: they show that the (inverse of the) Shimizu correspondence taking automorphic forms on \(\mathrm{GL}(2)\) to automorphic forms on the group of orthogonal similitudes of an indefinite quaternion algebra [H. Shimizu, J. Math. Soc. Japan 24, 638–683 (1972; Zbl 0241.10016)] actually preserves arithmeticity over \(\mathbb Q\).

The ingredients used in the proof of the first main theorem include the integral representation of Garrett for the triple product \(L\)-function [P. Garrett, Ann. Math. (2) 125, 209–235 (1987; Zbl 0625.10020)] as generalized by I. I. Piatetski-Shapiro and S. Rallis, [Compos. Math. 64, 31–115 (1987; Zbl 0637.10023)], the extension by Kudla and Rallis of the Weil-Siegel formula beyond the range of absolute convergence [S. S. Kudla and S. Rallis, J. Reine Angew. Math. 387, 1–68 (1988; Zbl 0644.10021), 391, 65–84 (1988; Zbl 0644.10022)], S. Kudla’s method of seesaw dual reductive pairs [Automorphic Forms of Several Variables, Taniguchi Symp., Katata 1983, Prog. Math. 46, 244–268 (1984; Zbl 0549.10017)], and the results of D. Prasad concerning the existence of a nontrivial invariant trilinear form on \(\mathrm{GL}(2)\times\mathrm{GL}(2)\times \mathrm{GL}(2)\) [Compos. Math. 75, 1–46 (1990; Zbl 0731.22013)]. The key observation is that a special value of the symplectic Eisenstein series which enters into Garrett’s integral may be expressed in terms of the theta functions attached to the norm forms of quaternion algebras over \(\mathbb Q\).

To prove the second theorem, the authors use these results and the methods of Shimura and Waldspurger. Additional ingredients include Harris’ refinement of Shimura’s definition of arithmetic automorphic forms on the Shimura curves attached to indefinite quaternion algebras [M. Harris, Compos. Math. 60, 323–378 (1986; Zbl 0612.14019), Invent. Math. 82, 151–189 (1985; Zbl 0598.14019)], and work of C. Goldstein and N. Schappacher [C. R. Acad. Sci., Paris, Sér. I 296, 615–618 (1983; Zbl 0553.12003)] and D. Blasius [Ann. Math. (2) 124, 23–63 (1986; Zbl 0608.10029)] on the special values of \(L\)-functions attached to algebraic Hecke characters.

The ingredients used in the proof of the first main theorem include the integral representation of Garrett for the triple product \(L\)-function [P. Garrett, Ann. Math. (2) 125, 209–235 (1987; Zbl 0625.10020)] as generalized by I. I. Piatetski-Shapiro and S. Rallis, [Compos. Math. 64, 31–115 (1987; Zbl 0637.10023)], the extension by Kudla and Rallis of the Weil-Siegel formula beyond the range of absolute convergence [S. S. Kudla and S. Rallis, J. Reine Angew. Math. 387, 1–68 (1988; Zbl 0644.10021), 391, 65–84 (1988; Zbl 0644.10022)], S. Kudla’s method of seesaw dual reductive pairs [Automorphic Forms of Several Variables, Taniguchi Symp., Katata 1983, Prog. Math. 46, 244–268 (1984; Zbl 0549.10017)], and the results of D. Prasad concerning the existence of a nontrivial invariant trilinear form on \(\mathrm{GL}(2)\times\mathrm{GL}(2)\times \mathrm{GL}(2)\) [Compos. Math. 75, 1–46 (1990; Zbl 0731.22013)]. The key observation is that a special value of the symplectic Eisenstein series which enters into Garrett’s integral may be expressed in terms of the theta functions attached to the norm forms of quaternion algebras over \(\mathbb Q\).

To prove the second theorem, the authors use these results and the methods of Shimura and Waldspurger. Additional ingredients include Harris’ refinement of Shimura’s definition of arithmetic automorphic forms on the Shimura curves attached to indefinite quaternion algebras [M. Harris, Compos. Math. 60, 323–378 (1986; Zbl 0612.14019), Invent. Math. 82, 151–189 (1985; Zbl 0598.14019)], and work of C. Goldstein and N. Schappacher [C. R. Acad. Sci., Paris, Sér. I 296, 615–618 (1983; Zbl 0553.12003)] and D. Blasius [Ann. Math. (2) 124, 23–63 (1986; Zbl 0608.10029)] on the special values of \(L\)-functions attached to algebraic Hecke characters.

Reviewer: Solomon Friedberg (Santa Cruz)

### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F27 | Theta series; Weil representation; theta correspondences |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11F11 | Holomorphic modular forms of integral weight |