Special Functions and Gauss-Thakur Sums in Higher Rank and Dimension

Quentin Gazda, Andreas Maurischat

Published 2019-03-18Version 1

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 80s where they were at the heart of comparison isomorphisms, further important applications e.g. to transcendence theory have only been discovered recently. The Anderson-Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss-Thakur sums, as shown by Angl\`es and Pellarin. For Drinfeld-Hayes modules, generalizations of Anderson generating functions have been introduced by Green-Papanikolas and -- under the name of ``special functions'' -- by Angl\`es-Ngo Dac-Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. These links were verified for certain cases of Anderson $A$-modules in various papers by explicit computations. Our approach provides the theoretical reason why these computations work and is also explicit enough to recover the formulas obtained there. We also undertake the study of Gauss-Thakur sums for Anderson A-modules, and show that the result of Angl\`es-Pellarin relating values of the special functions to Gauss-Thakur sums holds in this generality.