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On the de Rham isomorphism for Drinfeld modules. (English) Zbl 0672.14011
Let K be a global function field with a distinguished place “\(\infty ''\), A the subring of functions in K regular away from \(\infty\), and C the completion of the algebraic closure of \(K_{\infty}\), i.e., the substitute for the complex numbers in the analogy of number theory with the arithmetic of K.
For a Drinfeld A-module \(\Phi\) defined over C, one disposes, due to work of P. Deligne and G. Anderson of a de Rham cohomology module \(H^*_{DR}(\Phi)\) that is defined purely algebraically. It is related to the Betti cohomology \(H^*(\Phi)\) by a de Rham morphism DR, some sort of “cycle integration map”. The main results of the paper are:
(a) DR is always bijective;
(b) explicit construction of the inverse of DR;
(c) description of the Hodge decomposition on \(H^*(\Phi)\) induced by DR.
In proving these, for each additive character \(\chi\) of the homology \(H_*(\Phi)\) of \(\Phi\), a series of functions \(F_{h,\chi}\) is introduced and investigated. The crucial point is to verify the functional equations of \(F_{h,\chi}\). All of this should be regarded as the beginning of a theory of additive theta functions over C.
Some concrete arithmetical consequences (e.g. vanishing of certain p- class groups associated with abelian extensions of K) are drawn in the following papers of the author: “Quasi-periodic functions and Drinfeld modular forms”, Compos. Math. 69, 277-23 (1989) and “Some new identities for Bernoulli-Carlitz numbers”, J. Number Theory 33, No.2, 209-219 (1989).
Reviewer: E.-U.Gekele

14F40 de Rham cohomology and algebraic geometry
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14L05 Formal groups, \(p\)-divisible groups
14G25 Global ground fields in algebraic geometry
11F27 Theta series; Weil representation; theta correspondences
11R58 Arithmetic theory of algebraic function fields
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