Automorphic Forms and Sums of Squares over Function Fields Original Research Article

We develop some of the theory of automorphic forms in the function field setting. As an application, we find formulas for the number of ways a polynomial over a finite field can be written as a sum of k squares, kgreater-or-equal, slanted2. As a consequence, we show every polynomial can be written as a sum of 4 squares. We also show, as in the classical case, that these representation numbers are asymptotic to the Fourier coefficients of the basic Eisenstein series.