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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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