Lattice map for Anderson t-motives: First approach

Author(s)

Grishkov, A. ; Logachev, D.

• Document type: Journal article
• Source: JOURNAL OF NUMBER THEORY; v. 180, p. 373-402, NOV 2017.
• Web of Science Citations: 0

Abstract

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 bar right arrow L(M), where L(M) is a lattice in C-infinity(n). 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. (C) 2017 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 13/10596-8 - Algebraic loops and Drinfeld modules
• Grantee: Alexandre Grichkov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International