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**On the critical values of Hecke L-series.**
*(English)*
Zbl 0608.10029

The author proves P. Deligne’s rationality conjecture \([= 2.8\) of Proc. Symp. Pure Math. 33, No.2, 313-346 (1979; Zbl 0449.10022)] for all critical values of all algebraic Hecke characters of CM-fields. Thus he establishes the fact that - up to an algebraic number whose Galois behaviour is that of the Hecke character \(\chi\) in question - the values L(\(\chi\),k), for \(k\in {\mathbb{Z}}\) such that the \(\Gamma\)-factors on either side of the functional equation have no pole at k, are (either zero or) certain periods derived from abelian varieties with complex multiplication.

The idea of proof is, in the first place, a refinement of the method of Eisenstein-Damerell-Shimura: write the L-value as a linear combination of (values of in general non-holomorphic derivatives of) Eisenstein- Kronecker-Lerch-Hecke-Kloosterman series which are Hilbert modular forms relative to the totally real subfield of the CM-field for which \(\chi\) is an algebraic Hecke character [see R. M. Damerell, Acta Arith. 17, 287-301 (1970; Zbl 0209.246); ibid. 19, 311-317 (1971; Zbl 0229.12015); A. Weil, Elliptic functions according to Eisenstein and Kronecker (1976; Zbl 0318.33004); G. Shimura, Ann. Math., II. Ser. 91, 144- 222 (1970; Zbl 0237.14009)].

However, in order to refine Shimura’s result (which is up to an unspecified factor in \({\bar {\mathbb{Q}}})\) to the precise rationality statement of Deligne’s conjecture, the author had to overcome the notorious difficulty that Deligne’s construction of periods looks prima facie quite incompatible with the period that comes naturally with Shimura’s proof [see P. Deligne, loc. cit., 8.14-8.21; G. Harder and the reviewer, Lect. Notes Math. 1111, 17-49 (1985; Zbl 0561.10012)].

The author does this by brilliantly combining both Shimura’s reciprocity laws for special values of Hilbert modular forms and the recent generalization of the Shimura-Taniyama reciprocity law for CM-abelian varieties due to Tate and Deligne [see S. Lang, Complex multiplication (1983; Zbl 0536.14029), chap. 7] with his own remodelling of Deligne’s period definition inside the category of motives (for absolute Hodge cycles constructed from CM-abelian varieties).

This seems to be the first publication where an existing and manageable theory of motives [specifically: the one derived from P. Deligne’s theorem on absolute Hodge cycles on abelian varieties; see Lect. Notes Math. 900 (1982), chap. I, p. 9-100 (Zbl 0537.14006)and chap. IV, p. 261- 279 (Zbl 0499.16001)] is essentially used to prove a theorem about values of L-functions.

G. Harder has announced (in Harder-Schappacher, loc. cit.), but not yet published in detail, a method which would allow to carry over the author’s result to Hecke characters of arbitrary number fields.

The idea of proof is, in the first place, a refinement of the method of Eisenstein-Damerell-Shimura: write the L-value as a linear combination of (values of in general non-holomorphic derivatives of) Eisenstein- Kronecker-Lerch-Hecke-Kloosterman series which are Hilbert modular forms relative to the totally real subfield of the CM-field for which \(\chi\) is an algebraic Hecke character [see R. M. Damerell, Acta Arith. 17, 287-301 (1970; Zbl 0209.246); ibid. 19, 311-317 (1971; Zbl 0229.12015); A. Weil, Elliptic functions according to Eisenstein and Kronecker (1976; Zbl 0318.33004); G. Shimura, Ann. Math., II. Ser. 91, 144- 222 (1970; Zbl 0237.14009)].

However, in order to refine Shimura’s result (which is up to an unspecified factor in \({\bar {\mathbb{Q}}})\) to the precise rationality statement of Deligne’s conjecture, the author had to overcome the notorious difficulty that Deligne’s construction of periods looks prima facie quite incompatible with the period that comes naturally with Shimura’s proof [see P. Deligne, loc. cit., 8.14-8.21; G. Harder and the reviewer, Lect. Notes Math. 1111, 17-49 (1985; Zbl 0561.10012)].

The author does this by brilliantly combining both Shimura’s reciprocity laws for special values of Hilbert modular forms and the recent generalization of the Shimura-Taniyama reciprocity law for CM-abelian varieties due to Tate and Deligne [see S. Lang, Complex multiplication (1983; Zbl 0536.14029), chap. 7] with his own remodelling of Deligne’s period definition inside the category of motives (for absolute Hodge cycles constructed from CM-abelian varieties).

This seems to be the first publication where an existing and manageable theory of motives [specifically: the one derived from P. Deligne’s theorem on absolute Hodge cycles on abelian varieties; see Lect. Notes Math. 900 (1982), chap. I, p. 9-100 (Zbl 0537.14006)and chap. IV, p. 261- 279 (Zbl 0499.16001)] is essentially used to prove a theorem about values of L-functions.

G. Harder has announced (in Harder-Schappacher, loc. cit.), but not yet published in detail, a method which would allow to carry over the author’s result to Hecke characters of arbitrary number fields.

Reviewer: N.Schappacher

### MSC:

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

11G15 | Complex multiplication and moduli of abelian varieties |

14A20 | Generalizations (algebraic spaces, stacks) |

14K22 | Complex multiplication and abelian varieties |

11R42 | Zeta functions and \(L\)-functions of number fields |

11S40 | Zeta functions and \(L\)-functions |

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