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Gauss's theorem on sums of 3 squares, sheaves, and Gauss composition

Promotor: Bas Edixhoven, Promotor: Qing Liu

Albert Gunawan
08 maart 2016
Thesis in Leiden Repository

Gauss's theorem on sums of 3 squares relates the number of primitive integer points on the sphere of radius the square root of n with the class number of some quadratic imaginary order. In 2011, Edixhoven sketched a different proof of Gauss's theorem by using an approach from arithmetic geometry. He used the action of the special orthogonal group on the sphere and gave a bijection between the set of SO_3(Z)-orbits of such points, if non-empty, with the set of isomorphism classes of torsors under the stabilizer group. This last set is a group, isomorphic to the group of isomorphism classes of projective rank one modules over the ring Z[1/2,sqrt{-n}]. This gives an affine space structure on the set of SO_3(Z)-orbits on the sphere. In Chapter 3 we give a complete proof of Gauss's theorem following Edixhoven's work and a new proof of Legendre's theorem on the existence of a primitive integer solution of the equation x^2+y^2+z^2=n by sheaf theory. In Chapter 4 we make the action given by the sheaf method of the Picard group on the set of SO_3(Z)-orbits on the sphere explicit, in terms of SO_3(Q).

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