Introduction to Anderson t-motives: a survey

2020/08/24 by A. Grishkov, Grishkov, A., Dmitry Logachev +1
Computer Science · Mathematics · #11G09 #14M15 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2008.10657

openalex publication_date 2020/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a survey on Anderson t-motives -- high-dimensional generalizations of Drinfeld modules. They are the functional field analogs of abelian varieties with multiplication by an imaginary quadratic field. We describe their lattices, their groups H¹ and H₁, their tensor products, the duality functor and the duality theorem, endomorphisms of Drinfeld modules in finite characteristic, and their L-functions of a certain type. Further on, we introduce the notion of affine equations, T-divisible \Bbb F_q[[T]]-modules, holonomic sequences in the functional field case, analogs of Siegel matrices as elements of flag varieties, and some other notions (to be continued). Many examples of explicit calculations are given, some elementary research problems are stated. Some results (Sections 16; 19) are apparently new.

Citations

• Duality of Anderson T-motives
• Lattice of the dual of an Anderson t-motive in terms of a map to a flag variety
• Resultantal varieties related to zeroes of L-functions of Carlitz\n modules

Related

• A rapid introduction to Drinfeld modules, t-modules, and t-motives
• Determining t-motives and dual t-motives in Anderson's theory
• Pairing Anderson motives via formal residues in the Frobenius endomorphism
• Pure Anderson Motives and Abelian Sheaves over Finite Fields
• Isogenies of abelian Anderson A-modules and A-motives