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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,...
We describe an alternate approach to the solitons for F q [t] introduced by Anderson. The product of the monic polynomials in F q [t] of degreeiin a given congruence class can be expressed in a compact form by the solitons “for alliat once.” They also link the arithmetic of F q [t] gamma values at fractions to the analogues of Jacobians of Fermat curves in the setting of higher...
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