Near holomorphy of some automorphic forms at critical points.

*(English)*Zbl 0708.11028In this long and subtle article some results of G. Shimura [Math. Ann. 278, 1-28 (1987; Zbl 0636.10023)] on near holomorphy of a distinguished automorphic form f on \({\mathcal Z}\times {\mathbb{C}}\) are extended. Here \({\mathcal Z}\) is a (bounded) symmetric domain arising from symmetric bilinear forms S over totally real number fields F. The form f in question always has an integral representation involving a Hilbert modular form of a common subfield E of the F’s, an Eisenstein series and a certain theta series associated to the S’s which allows to get a meromorphic continuation to the whole plane and guarantees real analyticity in \({\mathcal Z}\). The following questions are at the core of investigations:

(1) finiteness and holomorphy of \(w\mapsto f(w,s_ 0).\)

(2) more generally, near holomorphy of \(w\mapsto f(w,s_ 0)\) which amounts to say to be annihilated by a power of a certain non holomorphic differential operator.

(3) nearly holomorphy of the residue in case of a simple pole.

As to the importance of the questions above “an affirmative answer is essential to the study of the arithmetic properties of \(f(w,s_ 0)''\). All these questions are tackled by Shimura (loc.cit.) in the case \(s=0\). His method is extended to nonpositive integers in the right half plane \(s\geq -b/2\) for a certain (half) integer b associated to f. Furthermore under additional assumptions on the common subfield E, the involved Hilbert modular form and the theta series better results are obtained by the author for a product of f by a certain L-series.

(1) finiteness and holomorphy of \(w\mapsto f(w,s_ 0).\)

(2) more generally, near holomorphy of \(w\mapsto f(w,s_ 0)\) which amounts to say to be annihilated by a power of a certain non holomorphic differential operator.

(3) nearly holomorphy of the residue in case of a simple pole.

As to the importance of the questions above “an affirmative answer is essential to the study of the arithmetic properties of \(f(w,s_ 0)''\). All these questions are tackled by Shimura (loc.cit.) in the case \(s=0\). His method is extended to nonpositive integers in the right half plane \(s\geq -b/2\) for a certain (half) integer b associated to f. Furthermore under additional assumptions on the common subfield E, the involved Hilbert modular form and the theta series better results are obtained by the author for a product of f by a certain L-series.

Reviewer: F.W.Knoeller

##### MSC:

11F55 | Other groups and their modular and automorphic forms (several variables) |

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

32D15 | Continuation of analytic objects in several complex variables |

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

##### Keywords:

near holomorphy; integral representation; Hilbert modular form; Eisenstein series; theta series; meromorphic continuation; real analyticity##### References:

[1] | Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015 · doi:10.1002/cpa.3160290618 |

[2] | Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5 |

[3] | Shimura, G.: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups. Ann. Math.111, 313-375 (1980) · Zbl 0438.12003 · doi:10.2307/1971202 |

[4] | Shimura, G.: On certain zeta functions attached to two Hilbert modular forms I, II. Ann. Math.114, 127-164, 569-607 (1981) · Zbl 0468.10016 · doi:10.2307/1971381 |

[5] | Shimura, G.: Algebraic relations between critical values of zeta functions and inner products. Am. J. Math.105, 253-285 (1983) · Zbl 0518.10032 · doi:10.2307/2374388 |

[6] | Shimura, G.: On differential operators attached to certain representations of classical groups. Invent. Math.77, 463-488 (1984) · Zbl 0558.10023 · doi:10.1007/BF01388834 |

[7] | Shimura, G.: On Eisenstein series of half-integral weight. Duke Math. J.52, 281-314 (1985) · Zbl 0577.10025 · doi:10.1215/S0012-7094-85-05216-0 |

[8] | Shimura, G.: On the Eisenstein series of Hilbert modular groups. Rev. Mat. Ib.1 (No. 3) 1-42 (1985) · Zbl 0608.10028 |

[9] | Shimura, G.: On a class of nearly holomorphic automorphic forms. Ann. Math.123, 347-406 (1986) · Zbl 0593.10022 · doi:10.2307/1971276 |

[10] | Shimura, G.: On Hilbert modular forms of half-integral weight. Duke Math. J.55, 765-838 (1987) · Zbl 0636.10024 · doi:10.1215/S0012-7094-87-05538-4 |

[11] | Shimura, G.: Nearly holomorphic functions on hermitian symmetric spaces. Math. Ann.278, 1-28 (1987) · Zbl 0636.10023 · doi:10.1007/BF01458058 |

[12] | Shimura, G.: On the critical values of certain Dirichlet series and periods of automorphic forms. Invent. Math.94, 245-305 (1988) · Zbl 0656.10018 · doi:10.1007/BF01394326 |

[13] | Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton: Princeton University Press 1971 · Zbl 0232.42007 |

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