Traces of Eichler-Brandt matrices and type numbers of quaternion orders. (English) Zbl 0651.12011

Let A be a totally definite quaternion algebra over a totally real algebraic number field and M be the ring of algebraic integers of F. For any M-order G of A formulas are derived for the mass m(G) and the type number t(G) and for the trace of the Eichler-Brandt matrix B(G,J) (where J is an integral ideal) in terms of genus invariant of G and of invariants of F and J. Thus earlier results of Eichler and others are generalized. The proofs use integration on some quotient spaces of the idele group of A and an elementary version of Selberg’s trace formula.
Reviewer: M.Peters


11R52 Quaternion and other division algebras: arithmetic, zeta functions
11E41 Class numbers of quadratic and Hermitian forms
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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[1] Eichler M, Über die Klassenzahl total definiter Quaternionenalgebren,Math. Z. 43 (1937) 102–109 · JFM 63.0093.02
[2] Eichler M, Zur Zahlentheorie der Quaternionen algebren,J. reine angew. Math. 195 (1955) 127–151
[3] Eichler M, Quadratische Formen und orthogonale Gruppen 1974 2nd edn (Berlin: Springer Verlag)
[4] Fröhlich A, The Picard group of non-commutative rings, in particular of orders,Trans. Am. Math. Soc. 180 (1973) 1–45 · Zbl 0278.16016
[5] Hijikata H, Explicit formula of the traces of Hecke operators for Г0(N),J. Math. Soc. Jpn. 26 (1974) 56–82 · Zbl 0266.12009
[6] Kaplansky I, Submodules of quaternion algebras,Proc. London Math. Soc. (3)19 (1969) 219–232 · Zbl 0212.39102
[7] Körner O, Ordnungen von Quaternionenalgebren über lokalen Körpern,J. reine angew. Math. 333 (1982) 162–178
[8] Körner O, Über die zentrale Picard-Gruppe und die Einheiten lokaler Quaternionenordnungen,Manuscripta Math. 52 (1985) 203–225 · Zbl 0564.12021
[9] O’Meara O T, Introduction to quadratic forms 1963 (Berlin: Springer-Verlag)
[10] Peters M, Ternäre und quaternäre quadratische Formen und Quaternionen algebrenActa Arith. 15 (1969) 329–365
[11] Pizer A, Type numbers of Eichler orders,J. reine angew. Math. 264 (1973) 76–102
[12] Pizer A, On the arithmetic of quaternion algebras,Acta Arith. 31 (1976) 61–89 · Zbl 0348.12018
[13] Tamagawa T, On Selberg’s Trace Formula,J. Fac. Sci. Univ. Tokyo. Sect. I 8 (1960) 363–386 · Zbl 0118.11405
[14] Vignéras M F, Arithmétique des Algébres de Quaternions,Lecture Notes in Mathematics 800 (1980) (Berlin: Springer Verlag)
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