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**“1-motifs” et formes automorphes (Théorie arithmétique des domaines de Siegel).**
*(French)*
Zbl 0565.14001

Journées automorphes, Dijon 1981, Publ. Math. Univ. Paris VII, No. 15, 43-106 (1983).

[For the entire collection see Zbl 0516.00011.]

The author’s main objective is to apply absolute Hodge cycles in Deligne’s well-known Hodge theory in algebraic varieties and 1-motifs, to the theory of moduli spaces and the arithmetic of automorphic forms on Shimura varieties. While the former imparts a purely algebraic sense to Hodge cycles on an abelian variety in characteristic 0 (and \(\ell\)-adic realisations) and leads, moreover, to an interpretation of canonical models of Shimura varieties (for reductive groups) studied by Kuga, as moduli spaces for polarised abelian varieties (with level structure and absolute Hodge cycles), the latter is useful in studying the ’formal completion’ (along a boundary component in the Baily-Borel compactification) of a Shimura variety. Defining Hodge cycles and absolute Hodge cycles for 1-motifs M (over \({\mathbb{C}})\), the author shows that they coincide and further constructs moduli spaces for polarised 1- motifs (with absolute Hodge cycles) which seem to generalize Shimura varieties. (A 1-motif M over a field k just consists of an abelian variety A over k, a k-split torus T, a free abelian group X of finite rank, a k-algebraic group G that is an extension of A by T and a morphism (”of periods”) \(u: X\to G(k).)\) After dealing with degeneracy of families of 1-motifs, boundary components of Shimura varieties and a comparison with the ’classical’ situation, the author obtains, by geometric methods, the arithmetic structure of local rings on the Baily-Borel compactification, Fourier-Jacobi expansions of automorphic forms relative to boundary components and results on the field of definition for the Mumford compactifications of Shimura varieties.

The author’s main objective is to apply absolute Hodge cycles in Deligne’s well-known Hodge theory in algebraic varieties and 1-motifs, to the theory of moduli spaces and the arithmetic of automorphic forms on Shimura varieties. While the former imparts a purely algebraic sense to Hodge cycles on an abelian variety in characteristic 0 (and \(\ell\)-adic realisations) and leads, moreover, to an interpretation of canonical models of Shimura varieties (for reductive groups) studied by Kuga, as moduli spaces for polarised abelian varieties (with level structure and absolute Hodge cycles), the latter is useful in studying the ’formal completion’ (along a boundary component in the Baily-Borel compactification) of a Shimura variety. Defining Hodge cycles and absolute Hodge cycles for 1-motifs M (over \({\mathbb{C}})\), the author shows that they coincide and further constructs moduli spaces for polarised 1- motifs (with absolute Hodge cycles) which seem to generalize Shimura varieties. (A 1-motif M over a field k just consists of an abelian variety A over k, a k-split torus T, a free abelian group X of finite rank, a k-algebraic group G that is an extension of A by T and a morphism (”of periods”) \(u: X\to G(k).)\) After dealing with degeneracy of families of 1-motifs, boundary components of Shimura varieties and a comparison with the ’classical’ situation, the author obtains, by geometric methods, the arithmetic structure of local rings on the Baily-Borel compactification, Fourier-Jacobi expansions of automorphic forms relative to boundary components and results on the field of definition for the Mumford compactifications of Shimura varieties.

Reviewer: S.Raghavan

### MSC:

14A20 | Generalizations (algebraic spaces, stacks) |

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

32N10 | Automorphic forms in several complex variables |

14K15 | Arithmetic ground fields for abelian varieties |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |