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Anderson t-motives - a parallel world to abelian varieties, in finite characteristic.聽

Formally, Anderson t-motives (generalizations of Drinfeld modules) are some modules over a ring of non-commutative polynomials in two variables over a complete algebraically closed field of finite characteristic. Surprisingly, it turns out that their properties are very similar to the properties of abelian varieties (more exactly, of abelian varieties with multiplication by an imaginary quadratic field). For example, we can define Tate modules of Anderson t-motives, Galois action on them, lattices, modular curves, L-functions etc. Nevertheless, this analogy is far to be complete. There is no functional equation for their L-functions; notion of the algebraic rank is not known yet; 1 - 1 correspondence between Anderson t-motives and lattices also is known only for Drinfeld modules. A survey of the theory of Anderson t-motives and statements of some research problems will be given.