The number of representations of an integer by a quadratic form

(Un exposé de Goro Shimura au séminaire de théorie des nombres de Chevaleret le 15 mai 2000)

Résumé :

This concerns the average of the representation numbers of a definite quadratic form over a totally real number field, as considered by Siegel. He gave a product formula for the average divided by the mass of the genus in question. He computed almost all factors, but some factors were given only as representation densities, without explicit formulas. We present an exact formula that gives the average as a finite product of certain explicitly defined quantities.