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**Eisenstein series on Shimura varieties.**
*(English)*
Zbl 0589.10030

In a previous paper [Invent. Math. 63, 305–310 (1981; Zbl 0452.10031)] the author proved rationality properties of the Fourier coefficients of holomorphic Eisenstein series attached to cusp forms on boundary components of Siegel’s upper half plane of degree \(n\). The proof – which did not use any explicit knowledge of the Fourier coefficients – was based on the fact that these Eisenstein series (at least in the range of absolute convergence) are completely characterized by their Hecke eigenvalues.

The purpose of the paper under review is to generalize that important result. The author starts from a bounded symmetric domain \(X\) and an arithmetic subgroup \(\Gamma\) of the group of holomorphic automorphisms of \(X\); to a cusp form on a boundary component of \(X\) one can associate a holomorphic Eisenstein series \(E(f)\) on \(X\). Suppose now that on \(X\) one has a theory of automorphic forms rational over \(K\) \((K\) a number field). Then the author proves that – under some additional conditions – the following principle holds: ”If \(f\) is rational over \(K\) and \(E(f)\) is in the range of absolute convergence then \(E(f)\) is rational over \(K\)”.

The main idea of proof is the same as in the previous paper. However two new difficulties arise in this general context: The notion of rationality of automorphic forms on \(X\) has to be treated carefully. Here the author uses G. Shimura’s theory of canonical models [Ann. Math. (2) 91, 144–222 (1970; Zbl 0237.14009); and ibid. 92, 528–549 (1970; Zbl 0237.14010)] as interpreted by P. Deligne [Sémin. Bourbaki 1970/71, Exp. No. 389, Lect. Notes Math. 244, 123–165 (1971; Zbl 0225.14007)]. The second new point is that instead of elementary estimates of Hecke eigenvalues (which were used in the previous paper) the author uses I. G. Macdonald’s work [”Spherical functions on a group of \(p\)-adic type”, Ramanujan Inst. 2, 79 p. (1971; Zbl 0302.43018)].

The purpose of the paper under review is to generalize that important result. The author starts from a bounded symmetric domain \(X\) and an arithmetic subgroup \(\Gamma\) of the group of holomorphic automorphisms of \(X\); to a cusp form on a boundary component of \(X\) one can associate a holomorphic Eisenstein series \(E(f)\) on \(X\). Suppose now that on \(X\) one has a theory of automorphic forms rational over \(K\) \((K\) a number field). Then the author proves that – under some additional conditions – the following principle holds: ”If \(f\) is rational over \(K\) and \(E(f)\) is in the range of absolute convergence then \(E(f)\) is rational over \(K\)”.

The main idea of proof is the same as in the previous paper. However two new difficulties arise in this general context: The notion of rationality of automorphic forms on \(X\) has to be treated carefully. Here the author uses G. Shimura’s theory of canonical models [Ann. Math. (2) 91, 144–222 (1970; Zbl 0237.14009); and ibid. 92, 528–549 (1970; Zbl 0237.14010)] as interpreted by P. Deligne [Sémin. Bourbaki 1970/71, Exp. No. 389, Lect. Notes Math. 244, 123–165 (1971; Zbl 0225.14007)]. The second new point is that instead of elementary estimates of Hecke eigenvalues (which were used in the previous paper) the author uses I. G. Macdonald’s work [”Spherical functions on a group of \(p\)-adic type”, Ramanujan Inst. 2, 79 p. (1971; Zbl 0302.43018)].

Reviewer: Siegfried Böcherer

### MSC:

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

14G35 | Modular and Shimura varieties |

14K15 | Arithmetic ground fields for abelian varieties |