Integrable systems and number theory in finite characteristic

The purpose of this paper is to give an overview of applications of the con cepts and techniques of the theory of integrable systems to number theory i n finite characteristic. The applications include explicit class field theo ry and Langlands conjectures for function fields, effect of the geometry of the theta divisor on factorization of analogs of Gauss sums, special value s of function field Gamma, zeta and L-functions, analogs of theorems of Wel l and Stickelberger, control of the intersection of the Jacobian torsion wi th the theta divisor. The techniques are the Krichever-Drinfeld dictionarie s and the theory of solitons, Akhiezer-Baker and tau functions developed in this context of arithmetic geometry by Anderson. (C) 2001 Elsevier Science B.V. All rights reserved.