Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0871.11037
Harris, Michael
Period invariants of Hilbert modular forms. II. (English)
[J] Compos. Math. 94, No.2, 201-226 (1994). ISSN 0010-437X; ISSN 1570-5846/e

Let $E$ be a totally real field over $\Bbb Q$, and let $G= R_{E/\Bbb Q} GL(2,E)$. Let $\pi$ be an irreducible automorphic representation of $G$ attached to a holomorphic Hilbert modular cusp form with motivic central character. If $L$ is a subfield of $\Bbb C$ and $\lambda, \mu \in \Bbb C$, then the notation $\lambda \sim_L \mu$ means that either $\lambda \cdot \mu =0$ or $\lambda /\mu \in L^\times$. Let $I$ be the subset of the set $\Sigma$ of real embeddings of $E$ such that the weights of $\pi$ at places in $I$ are $\geq 2$, and let $D$ be a quaternion algebra over $E$ such that the Jacquet-Langlands transfer $\pi^D$ exists. In this paper, the author proves that $\nu^I (\pi) \sim_{\Bbb Q} q^D (\pi)$, where $\nu^I (\pi) \in \Bbb C^\times$ and $q^D (\pi) \in \Bbb C^\times/ \overline{\Bbb Q}^\times$ are the invariants described in Part I of the paper by the author [Lect. Notes Math. 1447, 155-202 (1990; Zbl 0716.11020)].\par Let $\omega$ be an algebraic Hecke character of a quadratic CM extension $\Cal K$ of $E$, and let $\pi (\omega, \Cal K)$ be the corresponding automorphic representation of $G$. Let $\rho$ be the nontrivial element of $\text{Gal} (\Cal K/E)$, and let $\widetilde{\omega} = \omega/ \omega^\rho$, where $\omega^\rho (x) = \omega (x^\rho)$. Then the author also proves that $i^{\eta (I,\pi)} \cdot \nu^I (\pi (\omega, \Cal K)) \sim_{\overline{\Bbb Q}} p_{\Cal K} (\widetilde{\omega}, I)$, where the invariants $\eta (I,\pi)$ and $p_{\Cal K} (\widetilde{\omega}, I)$ are described in another paper by the author [J. Am. Math. Soc. 6, 637-719 (1993; Zbl 0779.11023)].
[Min Ho Lee (Cedar Falls)]

MSC 2000:: *11F67 Special values of automorphic L-series, etc
11F41 Hilbert modular forms and surfaces
11F70 Representation-theoretic methods in automorphic theory

Keywords: Hilbert modular forms; automorphic representations; $L$-functions; theta functions

Citations: Zbl 0716.11020; Zbl 0779.11023

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]