Universal extension crystals of 1-motives and applications

Abstract

a b s t r a c tWe use the crystalline nature of the universal extension of a 1-motive M to define a canonical Gauss-Manin connection on the de Rham realization of M. As an application we provide a construction of the so-called Manin map from a motivic point of view.

Algebraization, Transcendence, and D-Group Schemes

“…This construction is alluded to in [Bry83] (2.2.2.1), appears in a "differential algebraic context" in [BP10] Lemma 3.4 (i-ii), and in a "geometric context" in [ABV05] (see also [AB11]). The construction of the D-structure on E(L × ) and of the extension (6.15) may be understood as follows in terms of moduli spaces of vector bundles with integrable connections.…”

On Deformations of 1-motives

“…(a) Let G be any smooth commutative R-group scheme. By [6, Lemma 1.1.2] the kernel of the reduction map 1) . In our hypothesis one can prove that it is killed by p s ′ −1 .…”

“…Indeed, by the theory of Greenberg functor the sections G(R/m i ), 1 ≤ i ≤ s ′ , can be identified with the group of k-rational sections of a smooth k-group scheme Gr i (G). Further there are so-called change of level morphisms 1) . We can improve also this estimate if p > 2.…”

“…Note that V(M) is killed by p s since the multiplication by p s morphism is the 0 map on G a,R . We recall that M ♮ is denoted by [1,2].…”