h(1) not equal h(1) for Anderson t-motives

Author(s)

Grishkov, A. ^{[1, 2]} ; Logachev, D. ^[3]

Affiliation

^[1] Omsk State Univ, Pr Mira 55-A, Omsk 644077 - Russia

^[2] Univ Sao Paulo, Dept Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo - Brazil

^[3] Univ Fed Amazonas, Dept Matemat, Manaus, Amazonas - Brazil

• Document type: Journal article
• Source: JOURNAL OF NUMBER THEORY; v. 225, p. 59-89, AUG 2021.
• Web of Science Citations: 0

Abstract

Let M be an Anderson t-motive of dimension n and rank r. Associated are two F-q{[}T]-modules H-1(M), H-1(M) of dimensions h(1)(M), h(1)(M) <= r - analogs of H-1(A, Z), H-1(A, Z) for an abelian variety A. There is a theorem (Anderson): h(1)(M) = r double left right arrow h(1)(M) = r; in this case M is called uniformizable. It is natural to expect that always h(1)(M) = h(1)(M). Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D. Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator N are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible. Finally, explicit formulas for calculation of h(1)(M), h(1)(M) obtained in the present paper will be used in future for systematic calculation of h(1), h(1) of all Anderson t-motives. Moreover, the first step of this calculation (for a class of t-motives) is already made in a forthcoming paper of the authors. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 17/19777-6 - Anderson t-motives, their L-functions and lattices
• Grantee: Alexandre Grichkov
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil