Universal extension crystals of 1-motives and applications
This paper contains the construction of the Marvin map in the context of 1-motives, so providing realizations of the Manin map in the de Rham context, in the context of mixed Hodge structures (MHS) and in the crystalline context. The Manin map is a homomorphism \[ {\mathbf A}(S)\to \text{Ext}^1_S({\mathbf H}^1_{dR}({\mathbf A}/S,\nabla_{{\mathbf A}}),({\mathcal O}_S, d)), \] where \(S\) is a smooth connected curve and \({\mathbf A}\) is an abelian scheme. Here \(\text{Ext}^1_S\) is the group of isomorphism classes of\({\mathcal O}_S\)-modules endowed with an integral connection, extension of \({\mathbf H}^1_{dR}({\mathbf A}/S)\), with its Gauss-Marvin connection \(\nabla_{{\mathbf A}}\), by \({\mathcal O}_S\), with the usual derivation \(d\). A 1-motive \({\mathbf M}\) over the scheme \(S\) is the datum of an \(S\)-group scheme \({\mathbf X}\) locally defined, for the étale topology, by a finite \({\mathbf Z}\)-module, an \(S\)-group scheme \({\mathbf G}\), which is an extension over \(S\) of an abelian scheme \({\mathbf A}\) by a torus \({\mathbf T}\) and a homomorphism of \(S\)-group schemes \(u:{\mathbf X}\to{\mathbf G}\). For a 1-motive \({\mathbf M}\) one can define a Cartier dual \({\mathbf M}^\vee\) and the \(S\)-valued points \({\mathbf M}(S)\) which are the elements \(x\in\Hom({\mathbf Z},{\mathbf M})\) in the bounded derived category of sheaves on the fppf site of \(S\). Then there is an isomorphism of abelian groups \[ {\mathcal M}_{{\mathbf M}}:{\mathbf M}(S)\to \text{Ext}^1_1\text{-mot}({\mathbf M}^\vee,{\mathbf G}_m) \] which is functorial in \({\mathbf M}\) and \(S\). The authors construct a Gauss-Manin connection \(\nabla_{{\mathbf M}}\) on the locally free \({\mathcal O}_S\)-module \({\mathbf T}_{dR}({\mathbf M})\), the de Rham realization of \({\mathbf M}\). This connection is functorial in \({\mathbf M}\) and preserves the weight filtration on \({\mathbf T}_{dR}({\mathbf M})\). Composing \({\mathcal M}_{{\mathbf M}}\) with the de Rham realization of a 1-motive, endowed with its Gauss-Manin connection one gets a homomorphism \[ {\mathcal M}_{{\mathbf M}, dR}:{\mathbf M}(S)\to \text{Ext}^1_S({\mathbf T}_{dR}({\mathbf M}^\vee,\nabla_{{\mathbf M}^\vee}),({\mathcal O}_S, d)). \] Let \(q: S\to T= \text{Spec}({\mathbf C})\) be a smooth morphism. Then there is a Hodge realization \({\mathbf T}_{{\mathbf Z}}({\mathbf M})\) which is a variation of MHS on \(S^{an}\) and is horizontal for \(\nabla_{{\mathbf M}}\). Composing \({\mathcal M}_{{\mathbf M}}\) with \({\mathbf T}_{{\mathbf Z}}\) instead of \({\mathbf T}_{dR}\) gives the definition of a Hodge-Marin map \[ {\mathcal M}_{{\mathbf M},{\mathbf Z}}:{\mathbf M}(S)\to \text{Ext}^1_{VMHAS/S_{an}}(T_{{\mathbf Z}}({\mathbf M}^\vee),{\mathbf Z}) \] which is compatible with \({\mathcal M}_{{\mathbf M},dR}\) and functorial in \({\mathbf M}\) and \(S\). Here the target is the category of estensions of variations of MHS of \({\mathbf T}_{{\mathbf Z}}({\mathbf M}^\vee)\) by \({\mathbf Z}\) on \(S^{an}\). The crystalline nature of the universal extension of \({\mathbf M}\) provides a functor \({\mathbf T}_{cris}\) from the category of 1-motives over \(S\) to the category of filtered crystals on the locally nilpotent crystalline site \(S^{crys}\). Composing \({\mathcal M}_{{\mathbf M}}\) with \({\mathbf T}_{crys}\) yields the crystalline Marvin map \[ {\mathcal M}_{{\mathbf M},crys}:{\mathbf M}(S)\to \text{Ext}^1_{S^{crys}}(T_{crys}({\mathbf M}^\vee,{\mathcal O}^{crys}_S). \]
scientific article; zbMATH DE number 5886350
scientific article