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On a conjecture of Shimura concerning periods of Hilbert modular forms. (English) Zbl 0841.11024
Let \(F\) be a totally real number field of degree \(n\), and let \(F_{\mathbb A}\) be the adele ring of \(F\). Let \(\chi\) be a principal system of eigenvalues of Hecke operators on the space of holomorphic Hilbert modular cusp forms on \(\text{GL}(2, F_{\mathbb A})\) of weight \(k\), and let \({\mathbf f}\) be the primitive form that belongs to \(\chi\). If \(J_F\) denotes the set of all isomorphisms of \(F\) into \(\mathbb C\), then for each \(\varepsilon\in (\mathbb Z/ 2\mathbb Z)^{J_F}\), G. Shimura [Duke Math. J. 45, 637–679 (1978; Zbl 0394.10015)] introduced an invariant \(u(\varepsilon, {\mathbf f})\in \mathbb C^\times\) associated to the standard \(L\)-function \(D(m,{\mathbf f}, \varphi)\) attached to \({\mathbf f}\) twisted by a Hecke character \(\varphi\) of \(F^\times_{\mathbb A}\). When \(\chi\) occurs in the space of holomorphic automorphic forms on a quaternion algebra over \(F\) of signature \((\delta, J_F\setminus \delta)\), G. Shimura [Am. J. Math. 105, 253–285 (1983; Zbl 0518.10032)] introduced another invariant \(Q(\chi, \delta)\in \mathbb C^\times\) which appears in critical values of the Rankin-Selberg convolution of two Hilbert modular forms. In this paper, the author gives a description of \(u(\varepsilon, {\mathbf f})\) and \(Q(\chi, \delta)\), thereby proving an essential part of the conjecture of Shimura on \(P\)-invariants of Hilbert modular forms.

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F60 Hecke-Petersson operators, differential operators (several variables)
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