The Integral Basis Problem of Eichler
(Un exposé de Haruzo Hida au séminaire de
théorie des nombres de Chevaleret le 28 juin 2004)
Résumé :
Writing down a given modular forms as a linear
combination 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 sum
of 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 series
for the norm forms of division quaternion algebras
over the rational numbers, and his work is vastly generalized
by Jacquet-Langlands to any quaternion algebra over any number field.
I would like to present a simple argument (assuming the theorem
of Jacquet-Langlands) how to express a given integral Hilbert modular
form into an integral linear combination of such theta series.