Integrable systems and number theory in finite characteristic

The purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic. The applications include explicit class field theory and Langlands conjectures for function fields, effect of the geometry of the theta divisor on factorization of analogs of Gauss sums, special values of function field Gamma, zeta and L-functions, analogs of theorems of Weil and Stickelberger, control of the intersection of the Jacobian torsion with the theta divisor. The techniques are the Krichever–Drinfeld dictionaries and the theory of solitons, Akhiezer–Baker and tau functions developed in this context of arithmetic geometry by Anderson. © 2001 Elsevier Science B.V. All rights reserved.