On a conjecture of Shimura concerning periods of Hilbert modular forms.

Introduction. In this paper, we shall give an affirmative answer to an essential part of the conjecture of Shimura on P-invariants of Hilbert modular forms.

Let F be a totally real algebraic number field of degree n and [J.sub.F] be the set of all isomorphisms of F into C. Let [F.sub.A] (resp. [F.sup.x.sub.A]) be the adele ring (resp. the idele group) of F and [F.sup.x.[sub.infinity]] be the archimedean part of [F.sup.x.sub.A]. Let [chi] be a primitive system of eigenvalues of Hecke operators which occurs in the space of holomorphic Hilbert modular cusp forms on GL(2, [F.sub.A]) of weight k and f be the primitive form which belongs to [chi]. In [S1], Shimura introduced an invariant u([epsilon], f) [element of] [C.sup.x] for every [epsilon] [element of] [(Z/2Z).sup.J.sub.F] such that

[Mathematical Expression Omitted]

for certain critical values m [element of] Z whenever a Hecke character [Mathematical Expression Omitted] of [F.sup.x.sub.A] satisfies [Mathematical Expression Omitted] for [Mathematical Expression Omitted]. Here [Mathematical Expression Omitted] is the standard L-function attached to f twisted by [Mathematical Expression Omitted] and we write a ~ b for a, b [element of] C if b [not equal to] 0 and a/b [element of] Q. Put U([chi],[epsilon]) = u([epsilon], f).

In [S4], Shimura introduced another invariant Q([chi], [delta]) [element] [C.sup.x] for every subset [delta] of [J.sub.F] when [chi] occurs in the space of holomorphic automorphic forms on a quaternion algebra over F of signature ([delta], [J.sub.f]\[delta]) and showed that this invariant appears in critical values of the Rankin-Selberg convolution of two Hilbert modular forms. He conjectured further the following (Conjecture 5.12 of [S4], cf. also [S5], p. 293, (C1), (C2), (C3), (C4) and (C9))

Conjecture P. Assume k([tau]) [greater than or equal to] 2 for all [tau] [element] [J.sub.F] and k([tau]) mod 2 is independent of [tau]. Put [k.sub.0] = [max.sub.[tau][element].sup.J.sub.F].sup.(k([tau]). Then for every subset [delta] of [J.sub.F] and every [element] [element] [(Z/2Z).sup.[delta]], there exists a constant [Mathematical Expression Omitted] which satisfies the following properties.

[Mathematical Expression Omitted].

[Mathematical Expression Omitted].

[Mathematical Expression Omitted].

(P4) When [chi] is of CM-type, [Mathematical Expression Omitted] holds, where PK stands for the symbol of CM-periods introduced in [S2].

The principal result of this paper is:

Main Theorem. Assume k([tau]) [greater than or equal to] 3 for all [tau] [element] [J.sub.F] and k([tau]) mod 2 is independent of [tau]. Then, for every [tau] [element] [J.sup.F], there exist constants [c.sup.[+ or -].sub.[tau]] ([chi] [element] [C.sup.x]) determined uniquely mod [Q.sup.x] such that

[Mathematical Expression Omitted]

Here we understand that [Mathematical Expression Omitted] identifying Z/2Z with {0,1}. By this theorem, it is clear that P([chi], [delta], [epsilon]) satisfying (P1) ~ (P3) is given by

[Mathematical Expression Omitted]

We note that in [Y], [sections]6, we have defined Q([chi], [delta]) mod [Q.sup.x] assuming only k([tau]) [greater than or equal to] 3 for all [tau] [element] [delta].

Let us now outline our ideas of the proof and contents of each section. In [section]1, we shall review known properties of two basic period invariants Q([chi], [delta]) and U([chi], [epsilon]). In [section]2, Lemma 1, we shall show that a necessary and sufficient condition for the existence [c.sup.[+ or -].sub.[tau]] ([chi]) as in the Main Theorem is the following relations (R1) ~ (R3).

[Mathematical Expression Omitted].

[Mathematical Expression Omitted].

[Mathematical Expression Omitted].

We shall also prove (P4) in [section]2.

Now (R1) is already proved in [S1], Theorem 4.3. Harris [Ha2] proved (R2) under certain conditions, in particular when n, [[absolute value of][delta].sub.1]] and [[absolute value of][delta.sub.2]] are all even. In [section]3, using a base change lift of [chi] to a totally real quadratic extension of F, we shall remove this parity condition and obtain (R2) (Theorem 2).

In [seciton]4, we shall prove (R3). By (0), we see that (R3) follows if

[Mathematical Expression Omitted]

holds for one choice of a nonvanishing critical value m and of Hecke characters [Mathematical Expression Omitted] whose infinity types correspond to [[epsilon].sub.1], [[epsilon].sub.2], [[mu].sub.1]], [[mu].sub.2] respectively. Let K be a quadratic extension of F such that the Hecke character [eta] of [F.sup.x.sub.A] corresponding to K/F satisfies [[eta][infinity]] = [Mathematical Expression Omitted]. Again by (0), (4) reduces to

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the base change lift of f to K. By our choice of K, [Mathematical Expression Omitted] holds and we obtain (5) from a result of Hida [Hi] ([section]4, Theorem 3).

In [section]5, we shall prove the invariance of [c.sup.[+ or -].sub.[tau]] ([chi]) under the base change of [chi] to a totally real cyclic extension of F (Theorem 4). In [section]6, we shall discuss a possible generalization of the Main Theorem including the case where k([tau]) = 2 for some [tau].

Notation. Throughout the paper, we fix an algebraic closure [Mathematical Expression Omitted] of Q as the subfield of C. A finite extension of Q in [Mathematial Expression Omitted] will be called an algebraic number field. For an algebraic number field F, [F.sub.v], denotes the completion of F at a place v, [J.sub.F] the set of all isomorphisms of F into C and [I.sub.F] the free abelian group generated by [J.sub.F.] We denote by [a.sup.F.sub.r] (resp. [a.sup.F.sub.c]) the set of all real (resp. …