2011/09/04 by Aleksandr Grishkov, Grishkov, Aleksandr, Dmitry Logachev +1
Mathematics · #14K22 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Primary 11G09 #Secondary 11G15
paper · pdf · doi:10.48550/arxiv.1109.0679
openalex publication_date 2011/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1 -- 1. Namely, let M(0) be a t-motive of dimension n and rank r=2n --- the n-th power of the Carlitz module of rank 2, and let M be a t-motive which is in some sense "close" to M(0). We consider the lattice map M ↦ L(M), where L(M) is a lattice in Cn. We show that the lattice map is an isomorphism in a "neighborhood" of M(0). Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma.