On sums of squares and on elliptic curves over function fields |
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| Affiliation: | Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, England;Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48104 USA;Mathematical Institute, University of Göttingen, Göttingen, Germany |
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| Abstract: | 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-Šafarevič group of this curve is nontrivial and infinitely divisible. |
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