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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

##### MSC:
 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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