\(t\)-motives.

*(English)*Zbl 0679.14001The paper under review contains answers to the questions arisen as a result of a correspondence of V. G. Drinfel’d and B. Gross. Let \(X\) be a projective smooth curve defined over a finite field of characteristic \(p\), \(\infty\) be the closed point on \(X\), \(A=\Gamma (X\setminus \infty,{\mathcal O}_ X)\). In this situation, the notion of an elliptic \(A\)-module was introduced by Drinfel’d and classification of these modules was given in terms of “lattices” and in terms of “pieces” (“shtuka”) which generates mutual description of “lattices” and “pieces” in terms of each other. In the category of “pieces”, there is the operation of taking a “determinant” which assigns to a “piece” \({\mathcal M}\) of rank \(n\) the “piece” \({\mathcal M}'\) of rank \(1\). The first question can be formulated in the following way. If \({\mathcal L}\) and \({\mathcal L}'\) are the “lattices” corresponding to the “pieces” \({\mathcal M}\) and \({\mathcal M}'\), how can \({\mathcal L}'\) be expressed in terms of \({\mathcal L}\) then? In the paper under review, it is proved that \({\mathcal L}'\simeq Hom_ A(\Omega^{\otimes (n-1)},\bigwedge^ n{\mathcal L})\) where \(\Omega\) is the \(A\)-module of Kähler differentials of the ring \(A\) (the tensor and exterior powers are taken over \(A\)).

The second question is related to the generalization of the notion of an elliptic module to higher dimensions. The Hilbert-Blumenthal-Drinfel’d modules introduced in the paper under review are analogues of the Hilbert-Blumenthal abelian varieties (i.e., abelian varieties endowed with the action of the ring of integers of a totally real field of algebraic numbers of degree equal to the degree of the variety) and possess a series of properties which are analogous to classical properties of these varieties.

The second question is related to the generalization of the notion of an elliptic module to higher dimensions. The Hilbert-Blumenthal-Drinfel’d modules introduced in the paper under review are analogues of the Hilbert-Blumenthal abelian varieties (i.e., abelian varieties endowed with the action of the ring of integers of a totally real field of algebraic numbers of degree equal to the degree of the variety) and possess a series of properties which are analogous to classical properties of these varieties.

##### MSC:

14A20 | Generalizations (algebraic spaces, stacks) |

14L05 | Formal groups, \(p\)-divisible groups |

14G15 | Finite ground fields in algebraic geometry |

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##### References:

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[2] | V. G. Drinfeld, Mar. 1 1981, letter to B. Gross. |

[3] | B. Gross, letter to V. G. Drinfeld , Feb. 5 1981. |

[4] | R. Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie , Invent. Math. 2 (1967), 256-273. · Zbl 0202.20201 · doi:10.1007/BF01425404 · eudml:141857 |

[5] | D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 115-153. · Zbl 0423.14007 |

[6] | U. Stuhler, \(p\)-adic homogeneous spaces and moduli problems , · Zbl 0631.14034 · doi:10.1007/BF01162699 · eudml:173752 |

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