Frobenius Difference Equations and Difference Ga- lois Groups

This is a survey article on recent progress concerning transcendence problems over function fields in positive characteristic. We are interested in some special values that occur in the following two ways. One is the special values of certain special transcendental functions, e.g., Carlitz ζ-values at positive integers, which are specialization of Goss’ two-variable ζ-function, arithmetic (resp. geometric) Γ-functions at proper fractions (resp. proper rational functions), which are specialization of Goss’ two-variable Γ-function, and Drinfeld logarithms at algebraic points, etc. The other is from algebro-geometric objects that are defined over algebraic function fields. The suitable geometric objects here are Drinfeld modules and the special values are the entries of the period matrix of a Drinfeld module that is related to the comparison between the de Rham and Betti cohomologies of the given Drinfeld module. A natural question concerns the transcendence of these special values. In the 1980s and 1990s, Yu successfully developed methods of Gelfond-SchneiderLang type which can be applied to prove many important results on transcendence of the special values mentioned above. The breakthrough from transcendence of single values to linear independence of several special values is Yu’s sub-t-module theorem [37], which is an analogue of Wüstholz’ subgroup theorem [32]. Here t-modules are higher-dimension analogues of Drinfeld modules introduced by Anderson [1] and they play the analogous role of commutative algebraic groups in classical transcendence theory. The key ingredient when applying Yu’s sub-t-module theorem is to relate the special values in question to periods of certain t-modules. For more details we refer the readers to [38]. In 2004, Anderson-Brownawell-Papanikolas [3] developed a linear independence criterion over function fields, the so-called ABP criterion. It results from a system of Frobenius difference equations which are analogous to classical first-order linear differential equations. Passing from rigid analytically trivial abelian t-modules