Introduction to Anderson t-motives: a survey

TL;DR

This survey explores Anderson t-motives, their structures, dualities, endomorphisms, and L-functions, providing explicit examples and new results in the context of function field analogs of abelian varieties.

Contribution

It offers a comprehensive overview of Anderson t-motives, including new results and explicit calculations, advancing the understanding of their algebraic and analytical properties.

Findings

Introduction of affine equations and T-divisible modules

Explicit calculations of Anderson t-motives

New results on duality and endomorphisms

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 $H^1$ and $H_1$, 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.