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.

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.

Reviewer: W.W.J.Hulsbergen (Breda)

##### 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. |