Let SK(Γo(qν)) be the space of cusp forms of weight K with respect to the congruence subgroup Γ0(qν), and SK(Γ0(qν)) its subspace of all new forms in SK(Γ0(qν)), where q is a prime such that q ≥ 3. Now for f ∈ SK(Γ(qν)), put Then it is known that W induces an automorphism of SK(Γ0(qν)). On the other hand, for the character χ of (Z/qZ)× of order 2, let δx denote the “twisting operator” with respect to χ which was defined in [15] by Shimura, namely, for .

We are interested in determining, after a finite number of procedures, the number and the approximate positions of limit cycles for a given system.

Le résultat principal de ce travail est le suivant: les courbes elliptiques définies sur Q, et ayant bonne réduction en dehors de 3, vérifient la conjecture de Weil (cf. [2], Th. 2).

An important problem in the theory of arithmetic automorphic functions is to construct, for a reductive algebraic group over Q which defines a bounded symmetric domain, a system of canonical models [2], [6], [18]. For the similitude group of a hermitian form over a quaternion algebra whose center is a totally real field, this is solved by Shimura [17], and for the similitude group of a hermitian form with respect to an involution of the second kind of a central division algebra over a CM-field, by Miyake [8], In this paper, we show that this also can be done for the special Clifford group of a quadratic form Q over a totally real algebraic number field. (We have to impose certain conditions on the signature of Q in order that G defines a bounded symmetric domain, see 1.1.)

In [4] EGA IV (7.4.8) hat Grothendieck die folgende Frage gestellt: “A sei ein noetherscher Ring, I⊆ A ein Ideal, so daß A separiert und komplett in der I-adischen Topologie ist. A/I sei ein P-Ring. 1st dann A ebenfalls ein P-Ring?″ In dieser Arbeit beschäftigen wir uns mit dem Fall, daß A ein semilokaler noetherscher Ring ist und P die Eigenschaft “die formellen Fasern von A sind geometrisch regulär” bezeichnet. Wir wollen zeigen: “A sei ein semilokaler noetherscher I-adisch kompletter Ring, wobei I ein im Jacobsonradikal von A enthaltenes Ideal ist. Sind die formellen Fasern von A/I geometrisch regulär, so sind auch die formellen Fasern von A geometrisch regulär.”