Quadratic Diophantine equations and
class numbers
(Un exposé de Goro Shimura au séminaire de
théorie des nombres de Chevaleret le 10 mai 2004)
Résumé :
Gauss showed that the number of primitive representations of
an integer as sums of 3 squares is a simple factor times the class
number of binary forms of a fixed discriminant. We present a general
principle on a quadratic form of $n$ variables that connects
a primitive representation of an integer by the form to a
class of an orthogonal group in dimension $n-1$. The case of 3 squares
is an easiest example.