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On the fundamental periods of automorphic forms of arithmetic type. (English) Zbl 0712.11028
Let B be a quaternion algebra over a totally real number field E and let $$\delta$$ resp. $$\delta '$$ be the set of archimedean primes of E unramified resp. ramified in B. Let B be the algebraic group consisting of invertible elements of B. Then $$G_{{\mathbb{R}}}=GL_ 2\quad ({\mathbb{R}})^ n\times ({\mathbb{H}}^*)^ m$$ where $$n=\#\delta$$, $$m=\#\delta '$$ and $${\mathbb{H}}$$ denotes the Hamilton quaternions. If h is an automorphic form on $$G_{{\mathbb{A}}}$$ which is $${\bar {\mathbb{Q}}}$$-rational and an eigenfunction of the Hecke algebra with eigencharacter $$\chi$$, then in his previous papers [Am. J. Math. 105, 253-285 (1983; Zbl 0518.10032), Invent. Math. 94, 245-305 (1988; Zbl 0656.10018)] the author investigated the plausibility of assigning certain non-zero complex numbers P($$\chi$$,$$\delta$$,$$\epsilon$$ ;B) determined modulo $${\bar {\mathbb{Q}}}^*to\chi$$,$$\delta$$ and each subset $$\epsilon\subset \delta$$. These numbers P should have several remarkable properties: in particular, the periods of h (suitably normalized) should be linear combinations of $$\pi^ nP(\chi,\delta,\epsilon;B)$$ for all $$\epsilon\subset \delta$$, and the value $$\pi^ nP(\chi,\delta,\epsilon;B)P({\bar \chi},\delta,\delta \setminus \epsilon;B)$$ up to an algebraic number should be equal to the natural scalar product $$<h,h>$$. In the last cited paper the author proves - among other things - the existence of such numbers P in the case $$n=1$$. In the present paper the invariants P are established for an arbitrary $$\delta\neq \emptyset$$ when B is a division algebra.
Reviewer: W.Kohnen

##### MSC:
 11F60 Hecke-Petersson operators, differential operators (several variables) 11R52 Quaternion and other division algebras: arithmetic, zeta functions
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##### References:
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