Le théorème de Gauss sur les sommes de 3 carrés, de faisceaux, et composition de Gauss

Abstract : 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 SO3(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, √- n]. This gives an affine space structure on the set of SO3(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 x2 + y2 + z2 = n by sheaf theory. In Chapter 4 we make the action given by the sheaf method of the Picard group on the set of SO3(Z)-orbits on the sphere explicit, in terms of SO3(Q).
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Albert Gunawan. Le théorème de Gauss sur les sommes de 3 carrés, de faisceaux, et composition de Gauss. Théorie des nombres [math.NT]. Université de Bordeaux, 2016. Français. ⟨NNT : 2016BORD0020⟩. ⟨tel-01310561⟩

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