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De Rham cohomology and the Gauss-Manin connection for Drinfeld modules. (English) Zbl 0735.14016
p-adic analysis, Proc. Int. Conf., Trento/Italy 1989, Lect. Notes Math. 1454, 223-255 (1990).
[For the entire collection see Zbl 0707.00010.]
For the function field $$K$$ in one variable over the constant field $$\mathbb{F}_ q$$, $$q=p^ f$$, with fixed place $$\infty$$, $$|.|$$ the associated normalized absolute value, let $$K_{\infty}$$ be the completion of $$K$$ with respect to $$|.|$$ and let $$A$$ be the subring of $$K$$ of functions with poles at most at $$\infty$$. Also let $$C=\overline K$$ denote the completion of $$\overline K_ \infty$$ with respect to the canonical extension of $$|.|$$ to $$K_ \infty$$, i.e. $$C$$ is the smallest field extension of $$K_ \infty$$ that is complete and algebraically closed. For a field $$L$$ of characteristic $$p$$, it is well known that the ring of endomorphisms $$\text{End}_ L(\mathbb{G}_ a)$$ consists of additive polynomials of the form $$\phi(X)=a_ 0X+a_ 1X^ p+a_ 2X^{p^ 2}+\cdots+a_ nX^{p^ n}$$. Such a polynomial is said to have degree $$\deg(\phi)=p^ n$$. A Drinfeld module $$(G,\Phi)$$ of $$\hbox{rank }r$$ over $$L$$ is a ring morphism $$\Phi:A\to\text{End}_ L(G)$$, $$a\mapsto\Phi_ a$$, such that $$G$$ is isomorphic to $$\mathbb{G}_ a$$ and $$\deg(\Phi_ a)=| a|^ r$$ for all non-zero $$a\in A$$. One can also define Drinfeld modules over arbitrary $$A$$-schemes. One defines in a natural way morphisms between Drinfeld modules. Non-vanishing morphisms have finite kernels and are called isogenies. For an ideal $${\mathfrak a}\subset A$$ one defines the scheme of $${\mathfrak a}$$-division points of the Drinfeld module $$\Phi=(\Phi,G)$$ as $$_{\mathfrak a}\Phi=\cap_{{\mathfrak a}\in A}\text{Ker}(\Phi_ a)$$. The theory of Drinfeld modules (over global function fields) has much in common with the one of abelian varieties (over number fields). From a more motivic point of view, one would like to have an interplay of various ‘cohomology theories’ for Drinfeld modules just as for abelian varieties. Substitutes for the Betti and $$\ell$$-adic theories have been known since Drinfeld’s introduction of his modules.
Here a candidate for ‘de Rham cohomology’ $$H^*_{DR}$$ is presented and some of its properties are discussed. Its definition is due to the efforts of several people (e.g. G. Anderson, P. Deligne, J. Yu and the author). For a Drinfeld module $$\Phi=(\Phi,G)$$ of rank $$r$$ over an affine $$A$$-scheme $$S=\text{Spec} (B)$$, $$H^*_{DR}(\Phi)$$ is defined as a $$B$$-module that is the quotient of $$A$$-bimodules $$D(\Phi)$$ (the module of derivations) and the sub-$$A$$-bimodule $$D_{si}(\Phi)$$ (the module of strictly inner derivations). If $$B$$ is an $$A$$-field, then $$H^*_{DR}(\Phi)$$ is an $$r$$-dimensional $$B$$-vector space. More generally, for a Drinfeld module of rank $$r$$ over the $$A$$-scheme $$S$$, $$H^*_{DR}(\Phi)$$ is a locally free sheaf of rank $$r$$ on $$S$$, and its formation commutes with arbitrary base changes. Parallel to the Weierstraß parametrization of an elliptic curve over $$\mathbb{C}$$, one may associate with an $$A$$-lattice in $$C$$ (i.e. a finitely generated discrete $$A$$-submodule $$\Lambda$$ of $$C)$$ a Drinfeld module $$\Phi^ \Lambda$$ and in fact, the association $$\Lambda\mapsto\Phi^ \Lambda$$ gives a bijection between the set of lattices of rank $$r$$ in $$C$$ and the set of structures of Drinfeld module of rank $$r$$ on $$\mathbb{G}_ a\mid C$$. One also has an analytic de Rham functor $$H^{*,an}_{DR}$$ and there is a canonical isomorphism (‘GAGA’) $$H^*_{DR}(\Phi)@>\sim>>H^{*,an}_{DR}(\Phi)$$ whenever $$\Phi$$ is defined over $$C$$. Also, there is a de Rham isomorphism $$DR:H^*_{DR}(\Phi)@>\sim>>H^*_{Betti}(\Phi):=\operatorname{Hom}_ A(\Lambda,C)$$ whenever $$\Phi$$ is defined over $$C$$. The formalism of vanishing cycles for degenerating Drinfeld modules applies. A more geometric description of $$H^*_{DR}(\Phi)$$, $$\Phi=(G,\Phi)$$ defined over the $$A$$-scheme $$S$$ and of rank $$r$$, is obtained via the universal additive extension of $$(G,\Phi)\mid S$$ as the extension by the additive group $$V$$ of the dual sheaf $${\mathcal EXT}((G,\Phi),\mathbb{G}_ a)^ \vee$$, which is locally free of rank $$r-1$$. One has a canonical isomorphism of sheaves $$H^*_{DR}(\Phi)@>\sim>>\text{Lie}(V\oplus G)^ \vee$$. One gets the canonically split short exact sequence $$0\to H^*_ 1(\Phi)\to H^*_{DR}(\Phi)\to H^*_ 2(\Phi)\to0$$, with $$H^*_ 1$$ of rank one and $$H^*_ 2$$ of rank $$r-1$$. To investigate the action of tangent vectors of the base scheme on the sheaf $$H^*_{DR}(\Phi)$$, the Gauss-Manin connection for a Drinfeld module $$\Phi$$ over the scheme $$S$$ can be introduced. This leads to a Kodaira-Spencer map from the tangent bundle to $${\mathcal HOM}(H^*_ 1(\Phi)$$, $$H^*_ 2(\Phi))$$. In case $$\Phi$$ is modular, this map is an isomorphism. The paper closes with a section on Drinfeld modules of rank two, modular schemes and modular forms. A particular role is played by Eisenstein series.

##### MSC:
 14F40 de Rham cohomology and algebraic geometry 14L05 Formal groups, $$p$$-divisible groups 14G20 Local ground fields in algebraic geometry 11G09 Drinfel’d modules; higher-dimensional motives, etc.