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, k 2. 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.
1999 Academic Press
Given a finite field F q with q odd, we want to determine how many ways a polynomial in F q [T] can be written as a sum of k squares. For k 3 (or k=2, &1 not a square in F q ), the sum of k squares is an indefinite quadratic form, so there are infinitely many ways to write any polynomial over F q as a sum of k squares. Hence we refine our question; we seek a formula for the restricted representation numbers r k (:, m) where r k (:, m) denotes the number of ways a polynomial : of degree n can be written as a sum of squares of k polynomials whose degrees are strictly bounded by m>nΓ2.
In the 1940's Carlitz and Cohen studied this problem with n=2m&2 or m=2n&3. Using the circle method, Cohen obtained exact formulas ([C1], [C2]) but it is not clear whether these numbers are nonzero. More recently Serre showed these numbers are nonzero for k=3 (see [E-H]). In [M-W] elementary methods were used to give formulas in terms of