On sums of squares and on elliptic curves over function fields

It has long been known that every positive semidefinite function of R(x, y) is the sum of four squares. This paper gives the first example of such a function which is not expressible as the sum of three squares. The proof depends on the determination of the points on a certain elliptic curve defined over C(x). The 2-component of the Tate-Safarevic group of this curve is nontrivial and infinitely divisible.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33657/1/0000167.pd