Résume | Writing down a given modular forms as a linearcombination of theta series is a classical problem.The celebrated formula of Jacobi:"the number of ways of expressing an odd positive integer$n$ as sums of four squares is equal to $8$ times the sumof positive divisors of $n$"is an identity of the theta series(of the sum of four squares) and an Eisenstein series,which is the origin of the Siegel-Weil formula.Eichler solved this problem for non-Eisenstein seriesfor the norm forms of division quaternion algebrasover the rational numbers, and his work is vastly generalizedby Jacquet-Langlands to any quaternion algebra over any number field.I would like to present a simple argument (assuming the theoremof Jacquet-Langlands) how to express a given integral Hilbert modularform into an integral linear combination of such theta series. |